The Facts On File

DICTIONARYof

MATHEMATICS

The Facts On File

DICTIONARYof

MATHEMATICS

Edited byJohn Daintith

Richard Rennie

Fourth Edition

The Facts On File Dictionary of MathematicsFourth Edition

Copyright © 2005, 1999 by Market House Books Ltd

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PREFACE

This dictionary is one of a series designed for use in schools. It is intended for stu-dents of mathematics, but we hope that it will also be helpful to other science stu-dents and to anyone interested in science. Facts On File also publishes dictionaries ina variety of disciplines, including biology, chemistry, forensic science, marine sci-ence, physics, space and astronomy, and weather and climate.

The Facts On File Dictionary of Mathematics was first published in 1980 and thethird edition was published in 1999. This fourth edition of the dictionary has beenextensively revised and extended. The dictionary now contains over 2,000 head-words covering the terminology of modern mathematics. A totally new feature ofthis edition is the inclusion of over 800 pronunciations for terms that are not ineveryday use. A number of appendixes have been included at the end of the bookcontaining useful information, including symbols and notation, symbols for physicalquantities, areas and volumes, expansions, derivatives, integrals, trigonometric for-mulae, a table of powers and roots, and a Greek alphabet. There is also a list of Websites and a bibliography. A guide to using the dictionary has also been added to thislatest version of the book.

We would like to thank all the people who have cooperated in producing this book.A list of contributors is given on the acknowledgments page. We are also grateful tothe many people who have given additional help and advice.

v

ACKNOWLEDGMENTS

Contributors

Norman Cunliffe B.Sc.Eric Deeson M.Sc., F.C.P., F.R.A.S.Claire Farmer B.Sc.Jane Farrill Southern B.Sc., M.Sc.Carol Gibson B.Sc.Valerie Illingworth B.Sc., M.Phil.Alan Isaacs B.Sc., Ph.D.Sarah Mitchell B.A.Roger Adams B.Sc.Roger Picken B.Sc.Janet Triggs B.Sc.

Pronunciations

William Gould B.A.

vi

CONTENTS

Preface v

Acknowledgments vi

Guide to Using the Dictionary viii

Pronunciation Key x

Entries A to Z 1

Appendixes

I. Symbols and Notation 245

II. Symbols for Physical 248Quantities

III. Areas and Volumes 249

IV. Expansions 250

V. Derivatives 251

VI. Integrals 252

VII. Trigonometric Formulae 253

VIII. Conversion Factors 254

IX. Powers and Roots 257

X. The Greek Alphabet 260

XI. Web Sites 261

Bibliography 262

vii

GUIDE TO USING THE DICTIONARY

The main features of dictionary entries are as follows.

HeadwordsThe main term being defined is in bold type:

absolute Denoting a number or measure-ment that does not depend on a standardreference value.

PluralsIrregular plurals are given in brackets after the headword.

abscissa (pl. abscissas or abscissae) Thehorizontal or x-coordinate in a two-dimen-sional rectangular Cartesian coordinatesystem.

VariantsSometimes a word has a synonym or alternative spelling. This is placed in brackets afterthe headword, and is also in bold type:

angular frequency (pulsatance) Symbol:ω The number of complete rotations perunit time.

Here, ‘pulsatance’ is another word for angular frequency. Generally, the entry for the syn-onym consists of a simple cross-reference:

pulsatance See angular frequency.

AbbreviationsAbbreviations for terms are treated in the same way as variants:

cosecant (cosec; csc) A trigonometricfunction of an angle equal to the reciprocalof its sine ....

The entry for the synonym consists of a simple cross-reference:

cosec See cosecant.

Multiple definitionsSome terms have two or more distinct senses. These are numbered in bold type

base 1. In geometry, the lower side of atriangle, or other plane figure, or the lowerface of a pyramid or other solid.2. In a number system, the number of dif-ferent symbols used, including zero.

viii

Cross-referencesThese are references within an entry to other entries that may give additional useful in-formation. Cross-references are indicated in two ways. When the word appears in the de-finition, it is printed in small capitals:

Abelian group /ă-beel-ee-ăn / (commuta-tive group) A type of GROUP in which theelements can also be related to each otherin pairs by a commutative operation.

In this case the cross-reference is to the entry for ‘group’.

Alternatively, a cross-reference may be indicated by ‘See’, ‘See also’, or ‘Compare’, usu-ally at the end of an entry:

angle of depression The angle betweenthe horizontal and a line from an observerto an object situated below the eye level ofthe observer. See also angle.

Hidden entriesSometimes it is convenient to define one term within the entry for another term:

arc A part of a continuous curve. If the cir-cumference of a circle is divided into twounequal parts, the smaller is known as theminor arc and…

Here, ‘minor arc’ is a hidden entry under arc, and is indicated by italic type. The entry for‘minor arc’ consists of a simple cross-reference:

minor arc See arc.

PronunciationsWhere appropriate pronunciations are indicated immediately after the headword, en-closed in forward slashes:

abacus /ab-ă-kŭs/ A calculating deviceconsisting of rows of beads strung on wireand mounted in a frame.

Note that simple words in everyday language are not given pronunciations. Also head-words that are two-word phrases do not have pronunciations if the component words arepronounced elsewhere in the dictionary.

ix

/a/ as in back /bak/, active /ak-tiv//ă/ as in abduct /ăb-dukt/, gamma /gam-ă//ah/ as in palm /pahm/, father /fah-ther/,/air/ as in care /kair/, aerospace /air-ŏ-

spays//ar/ as in tar /tar/, starfish /star-fish/, heart

/hart//aw/ as in jaw /jaw/, gall /gawl/, taut /tawt//ay/ as in mania /may-niă/ ,grey /gray//b/ as in bed /bed//ch/ as in chin /chin//d/ as in day /day//e/ as in red /red//ĕ/ as in bowel /bow-ĕl//ee/ as in see /see/, haem /heem/, caffeine

/kaf-een/, baby /bay-bee//eer/ as in fear /feer/, serum /seer-ŭm//er/ as in dermal /der-măl/, labour /lay-ber//ew/ as in dew /dew/, nucleus /new-klee-ŭs//ewr/ as in epidural /ep-i-dewr-ăl//f/ as in fat /fat/, phobia /foh-biă/, rough

/ruf//g/ as in gag /gag//h/ as in hip /hip//i/ as in fit /fit/, reduction /ri-duk-shăn//j/ as in jaw /jaw/, gene /jeen/, ridge /rij//k/ as in kidney /kid-nee/, chlorine /klor-

een/, crisis /krÿ-sis//ks/ as in toxic /toks-ik//kw/ as in quadrate /kwod-rayt//l/ as in liver /liv-er/, seal /seel//m/ as in milk /milk//n/ as in nit /nit/

/ng/ as in sing /sing//nk/ as in rank /rank/, bronchus /bronk-ŭs//o/ as in pot /pot//ô/ as in dog /dôg//ŏ/ as in buttock /but-ŏk//oh/ as in home /hohm/, post /pohst//oi/ as in boil /boil//oo/ as in food /food/, croup /kroop/, fluke

/flook//oor/ as in pruritus /proor-ÿ-tis//or/ as in organ /or-găn/, wart /wort//ow/ as in powder /pow-der/, pouch

/powch//p/ as in pill /pil//r/ as in rib /rib//s/ as in skin /skin/, cell /sel//sh/ as in shock /shok/, action /ak-shŏn//t/ as in tone /tohn//th/ as in thin /thin/, stealth /stelth//th/ as in then /then/, bathe /bayth//u/ as in pulp /pulp/, blood /blud//ŭ/ as in typhus /tÿ-fŭs//û/ as in pull /pûl/, hook /hûk//v/ as in vein /vayn//w/ as in wind /wind//y/ as in yeast /yeest//ÿ/ as in bite /bÿt/, high /hÿ/, hyperfine /hÿ-

per-fÿn//yoo/ as in unit /yoo-nit/, formula /form-yoo-lă//yoor/ as in pure /pyoor/, ureter /yoor-ee-ter//ÿr/ as in fire /fÿr/

x

Bold type indicates a stressed syllable. In pronunciations, a consonant is sometimes dou-bled to prevent accidental mispronunciation of a syllable resembling a familiar word; forexample, /ass-id/ (acid), rather than /as-id/ and /ul-tră- sonn-iks/ (ultrasonics), rather than/ul-tră-son-iks/. An apostrophe is used: (a) between two consonants forming a syllable, asin /den-t’l/ (dental), and (b) between two letters when the syllable might otherwise be mis-pronounced through resembling a familiar word, as in /th’e-ră-pee/ (therapy) and /tal’k/(talc). The symbols used are:

Pronunciation Key

abacus /ab-ă-kŭs/ A calculating deviceconsisting of rows of beads strung on wireand mounted in a frame. An abacus withnine beads in each row can be used forcounting in ordinary arithmetic. The low-est wire counts the digits, 1, 2, … 9, thenext tens, 10, 20, … 90, the next hundreds,100, 200, … 900, and so on. The number342, for example, would be counted outby, starting with all the beads on the right,pushing two beads to the left hand side ofthe bottom row, four to the left of the sec-ond row, and three to the left of the thirdrow. Abaci, of various types, are still usedfor calculating in some countries; expertswith them can perform calculations veryrapidly.

Abelian group /ă-beel-ee-ăn / (commuta-tive group) A type of GROUP in which theelements can also be related to each otherin pairs by a commutative operation. Forexample, if the operation is multiplicationand the elements are rational numbers,then the set is an Abelian group because forany two elements a and b, a × b = b × a, andall three numbers, a, b, and a × b are el-ements in the set. All cyclic groups areAbelian groups. See also cyclic group.

abscissa /ab-siss-ă/ (pl. abscissas or ab-scissae) The horizontal or x-coordinate in atwo-dimensional rectangular Cartesian co-ordinate system. See Cartesian coordi-nates.

absolute Denoting a number or measure-ment that does not depend on a standardreference value. For example, absolutedensity is measured in kilograms per cubicmeter but relative density is the ratio ofdensity to that of a standard density (i.e.the density of a reference substance understandard conditions). Compare relative.

absolute convergence The convergenceof the sum of the absolute values of termsin a series of positive and negative terms.For example, the series:

1 – (1/2)2 + (1/3)3 – (1/4)4 + …is absolutely convergent because

1 + (1/2)2 + (1/3)3 + (1/4)4 + …is also convergent. A series that is conver-gent but has a divergent series of absolutevalues is conditionally convergent. For ex-ample,

1 – 1/2 + 1/3 – 1/4 + …is conditionally convergent because

1 + 1/2 + 1/3 + 1/4 + …

1

A

3

2

5

8

Abacus: the number 3258 is shown on the right-hand side.

absolute error

2

is divergent. See also convergent series.

absolute error The difference betweenthe measured value of a quantity and itstrue value. Compare relative error. See alsoerror.

absolute maximum See maximum point.

absolute minimum See minimum point.

absolute value The MODULUS of a realnumber or of a complex number. For ex-ample, the absolute value of –2.3, written|–2.3|, is 2.3. The absolute value of a com-plex number is also the modulus; for ex-ample the absolute value of 2 + 3i is √(22 +32).

abstract algebra See algebra; algebraicstructure.

abstract number A number regardedsimply as a number, without reference toany material objects or specific examples.For example, the number ‘three’ when itdoes not refer to three objects, quantities,etc., but simply to the abstract concept of‘three’.

acceleration Symbol: a The rate ofchange of speed or velocity with respect totime. The SI unit is the meter per secondper second (m s–2). A body moving in astraight line with increasing speed has apositive acceleration. A body moving in acurved path with uniform (constant) speedalso has an acceleration, since the velocity(a vector depending on direction) is chang-ing. In the case of motion in a circle the ac-celeration is v2/r directed toward the centerof the circle (radius r).

For constant acceleration:a = (v2 – v1)/t

v1 is the speed or velocity when timingstarts; v2 is the speed or velocity after timet. (This is one of the equations of motion.)This equation gives the mean accelerationover the time interval t. If the accelerationis not constant

a = dv/dt, or d2x/dt2

See also Newton’s laws of motion.

acceleration due to gravity See acceler-ation of free fall.

acceleration of free fall (acceleration dueto gravity) Symbol: g The constant accel-eration of a mass falling freely (withoutfriction) in the Earth’s gravitational field.The acceleration is toward the surface ofthe Earth. g is a measure of gravitationalfield strength – the force on unit mass. Theforce on a mass m is its weight W, where W= mg.

The value of g varies with distance fromthe Earth’s surface. Near the surface it isjust under 10 meters per second per second(9.806 65 m s–2 is the standard value). Itvaries with latitude, in part because theEarth is not perfectly spherical (it is flat-tened near the poles).

acceptance region When considering ahypothesis, the sample space is divided intotwo regions – the acceptance region andthe rejection region (or critical region). Theacceptance region is the one in which thesample must lie if the hypothesis is to be ac-cepted.

access time The time needed for the read-ing out of, or writing into, the memory ofa computer, i.e. the time it takes for thememory to transfer data from or to theCPU (see central processor).

accumulation point (cluster point) For agiven set S, a point that can be approachedarbitrarily closely by members of that set.Another way of saying this is that an accu-mulation point is the limit of a sequence ofpoints in the set. An accumulation point ofa set need not necessarily be a member ofthe set itself, although it can be. For exam-ple, any rational number is an accumula-tion point of the set of rationals. But 0 is anaccumulation point of the set 1,½,¼,⅛,… although it is not itself a member of theset.

accuracy The number of significant fig-ures in a number representing a measure-ment or value of a quantity. If a length iswritten as 2.314 meters, then it is normallyassumed that all of the four figures are

meaningful, and that the length has beenmeasured to the nearest millimeter. It is in-correct to write a number to a precision of,for example, four significant figures whenthe accuracy of the value is only three sig-nificant figures, unless the error in the esti-mate is indicated. For example, 2.310 ±0.005 meters is equivalent to 2.31 meters.

acre A unit of area equal to 4840 squareyards. It is equivalent to 0.404 68 hectare.

action An out-dated term for force. Seereaction.

action at a distance An effect in whichone body affects another through spacewith no apparent contact or transfer be-tween them. See field.

actuary An expert in statistics who calcu-lates insurance risks and relates them to thepremiums to be charged.

acute Denoting an angle that is less than aright angle; i.e. an angle less than 90° (orπ/2 radian). Compare obtuse; reflex.

addend /ad-end, ă-dend/ One of the num-bers added together in a sum. See also au-gend.

adder The circuitry in a computer thatadds digital signals (i.e. the ADDEND, AU-GEND and carry digit) to produce a sum anda carry digit.

addition Symbol: + The operation of find-ing the SUM of two or more quantities. Inarithmetic, the addition of numbers is com-mutative (4 + 5 = 5 + 4), associative (2 + (3+ 4) = (2 + 3) + 4), and the identity elementis zero (5 + 0 = 5). The inverse operation toaddition is subtraction. In vector addition,the direction of the two vectors affects thesum. Two vectors are added by placingthem head-to-tail to form two sides of a tri-angle. The length and direction of the thirdside is the VECTOR SUM. Matrix additioncan only be carried out between matriceswith the same number of rows andcolumns, and the sum has the same dimen-sions. The elements in corresponding posi-

tions in each MATRIX are added arithmeti-cally.

addition formulae Equations that ex-press trigonometric functions of the sum ordifference of two angles in terms of sepa-rate functions of the angles.

sin(x + y) = sinx cosy + cosx sinysin(x – y) = sinx cosy – cosx sinycos(x + y) = cosx cosy – sinx sinycos(x – y) = cosx cosy + sinx sinytan(x + y) = (tanx + tany)/(1 – tanx

tany)tan(x – y) = (tanx – tany)/(1 + tanx

tany)They are used to simplify trigonometric ex-pressions, for example, in solving an equa-tion. From the addition formulae thefollowing formulae can be derived:The double-angle formulae:

sin(2x) = 2 sinx cosxcos(2x) = cos2x – sin2xtan(2x) = 2tanx/(1 – tan2x)

The half-angle formulae:sin(x/2) = ±√[(1 – cosx)/2]cos(x/2) = ±√[(1 + cosx)/2]tan(x/2) = sinx/(1 + cosx) = (1 –

cosx)/sinxThe product formulae:

sinx cosy = ½[sin(x + y) + sin(x – y)]cosx siny = ½[sin(x + y) – sin(x – y)]cosx cosy = ½[cos(x + y) + cos(x – y)]sinx siny = ½[cos(x – y) – cos(x + y)]

addition of fractions See fractions.

addition of matrices See matrix.

address See store.

ad infinitum /ad in-fă-nÿ-tŭn/ To infinity;an infinite number of times. Often abbrevi-ated to ad inf.

adjacent 1. Denoting one of the sidesforming a given angle in a triangle. In aright-angled triangle it is the side betweenthe given angle and the right angle. Intrigonometry, the ratios of this adjacentside to the other side lengths are used to de-fine the cosine and tangent functions of theangle.

3

adjacent

2. Denoting two sides of a polygon thatshare a common vertex.3. Denoting two angles that share a com-mon vertex and a common side.4. Denoting two faces of a polyhedron thatshare a common edge.

adjoint (of a matrix) See cofactor.

admissible hypothesis Any hypothesisthat could possibly be true; in other words, an hypothesis that has not beenruled out.

aether /ee-th’er/ See ether.

affine geometry /ă-fÿn/ The study ofproperties left invariant under the group ofaffine transformations. See affine transfor-mation; geometry.

affine transformation A transformationof the form

x′ = a1x + b1y + c1,y′ = a2x + b2y + c2

where a1b2 – a2b1 ≠ 0. An affine transfor-mation maps parallel lines into parallellines, finite points into finite points, leavesthe line at infinity fixed, and preserves theratio of the distances separating threepoints that lie on a straight line. An affinetransformation can always be factored intothe product of the following importantspecial cases:1. translations: x′ = x + a, y′ = y + b2. rotations: x′ = xcosθ + ysinθ, y′ = –xsinθ

+ ycosθ3. stretchings or shrinkings: x′ = tx, y′ = ty4. reflections in the x-axis or y-axis: x′ = x,

y′ = –y or x′ = –x, y′ = y5. elongations or compressions: x′ = x, y′ =

ty or x′ = tx, y′ = y

aleph /ah-lef, ay-/ The first letter of theHebrew alphabet, used to denote transfi-nite cardinal numbers. '0, the smallesttransfinite cardinal number, is the numberof elements in the set of integers. '1, is thenumber of subsets of any set with '0, mem-bers. In general 'n+1 is defined in the sameway as the number of subsets of a set with'n members.

algebra The branch of mathematics inwhich symbols are used to represent num-bers or variables in arithmetical opera-tions. For example, the relationship:

3 × (4 + 2) = (3 × 4) + (3 × 2)belongs to arithmetic. It applies only to thisparticular set of numbers. On the otherhand the equation:

x(y + z) = xy + xzis an expression in algebra. It is true for anythree numbers denoted by x, y, and z. Theabove equation is a statement of the dis-tributive law of arithmetic; similar state-ments can be written for the associativeand commutative laws.

Much of elementary algebra consists ofmethods of manipulating equations to putthem in a more convenient form. For ex-ample, the equation:

x + 3y = 15can be changed by subtracting 3y fromboth sides of the equation, giving:

x + 3y – 3y = 15 – 3yx = 15 – 3y

The effect is that of moving a term (+3y)from one side of the equation to the otherand changing the sign. Similarly a multipli-cation on one side of the equation becomesa division when the term is moved to theother side; for example:

xy = 5becomes:

x = 5/y‘Ordinary’ algebra is a generalization of

arithmetic. Other forms of higher algebraalso exist, concerned with mathematicalentities other than numbers. For example,matrix algebra is concerned with the rela-tions between matrices; vector algebrawith vectors; Boolean algebra is applicableto logical propositions and to sets; etc. Analgebra consists of a number of mathemat-ical entities (e.g. matrices or sets) and oper-ations (e.g. addition or set inclusion) withformal rules for the relationships betweenthe mathematical entities. Such a system iscalled an algebraic structure.

algebra, Boolean See Boolean algebra.

algebraic structure A structure imposedon elements of a set by certain operationsthat act on or combine the elements. The

adjoint

4

combination of the set and the operationssatisfy particular axioms that define thestructure. Examples of algebraic structuresare FIELDS, GROUPS, and RINGS. The study ofalgebraic structure is sometimes called ab-stract algebra.

algorithm /al-gŏ-rith-’m/ A mechanicalprocedure for performing a given calcula-tion or solving a problem in a series ofsteps. One example is the common methodof long division in steps. Another is the Eu-clidean algorithm for finding the highestcommon factor of two positive integers.

allometry /ă-lom-ĕ-tree/ A relation be-tween two variables that can be expressedby the equation

y = axb

where x and y are the variables, a is a con-stant and b is a growth coefficient. it isused to describe the systematic growth ofan organism, in which case y is the mass ofa particular part of the organism and x isits total mass.

alphanumeric Describing any of thecharacters (or their codes) that stand forthe letters of the alphabet or numerals, es-pecially in computer science. Punctuationmarks and mathematical symbols are notregarded as alphanumeric characters.

alternate angles A pair of equal anglesformed by two parallel lines and a thirdline crossing both. For example, the twoacute angles in the letter Z are alternate an-gles. Compare corresponding angles.

alternate-segment theorem A result ingeometry stating that the angle between a

tangent to a CIRCLE and a chord drawnfrom the point of contact of the tangent isequal to any angle subtended by the chordin the alternate segment of the circle, wherethe alternative segment is on the side of thechord opposite (alternate to) the angle.

alternating series A series in which theterms are alternately positive and negative,for example:

Sn = –1 + 1/2 – 1/3 + 1/4 … + (–1)n/nSuch a series is convergent if the absolutevalue of each term is less than the preced-ing one. The example above is a convergentseries.

An alternating series can be constructedfrom the sum of two series, one with posi-tive terms and one with negative terms. Inthis case, if both are convergent separatelythen the alternating series is also conver-gent, even if the absolute value of eachterm is not always smaller than the one be-fore it. For instance, the series:

S1 = 1/2 + 1/4 + 1/8 + … + 1/2n

andS2 = –1/2 – 1/3 – 1/4 – 1/5 – …(–1)/(n + 1)are both convergent, and so their sum:

Sn = S1 + S2= 1/2 – 1/2 + 1/4 – 1/3 + 1/8 – 1/4 +…

is also convergent.

alternation See disjunction.

altitude The perpendicular distance fromthe base of a figure (e.g. a triangle, pyra-mid, or cone) to the vertex opposite.

ambiguous Having more than one possi-ble meaning, value, or solution. For exam-ple, an ambiguous case occurs in finding

5

ambiguous

x•

x •

Alternate angles: alternate angles formed by a line cutting two parallel lines.

the sides and angles of a triangle when twosides and an angle other then the includedangle are known. One solution is an acute-angled triangle and the other is an obtuse-angled triangle.

ampere /am-pair/ Symbol: A The SI baseunit of electric current, defined as the con-stant current that, maintained in twostraight parallel infinite conductors of neg-ligible circular cross-section placed onemeter apart in vacuum, would produce aforce between the conductors of 2 × 10–7

newton per meter. The ampere is namedfor the French physicist André Marie Am-père (1775–1836).

amplitude The maximum value of a vary-ing quantity from its mean or base value. Inthe case of a simple harmonic motion – awave or vibration – it is half the maximumpeak-to-peak value.

analog computer A type of COMPUTER inwhich numerical information (generallycalled data) is represented in the form of aquantity, usually a voltage, that can varycontinuously. This varying quantity is ananalog of the actual data, i.e. it varies in thesame manner as the data, but is easier tomanipulate in the mathematical operationsperformed by the analog computer. Thedata is obtained from some process, exper-iment, etc.; it could be the changing tem-perature or pressure in a system or thevarying speed of flow of a liquid. Theremay be several sets of data, each repre-sented by a varying voltage.

The data is converted into its voltageanalog or analogs and calculations andother sorts of mathematical operations, es-pecially the solution of differential equa-tions, can then be performed on thevoltage(s) (and hence on the data they rep-resent). This is done by the user selecting agroup of electronic devices in the computerto which the voltage(s) are to be applied.These devices rapidly add voltages, andmultiply them, integrate them, etc., as re-quired. The resulting voltage is propor-tional to the result of the operation. It canbe fed to a recording device to produce agraph or some other form of permanent

record. Alternatively it can be used to con-trol the process that produces the data en-tering the computer.

Analog computers operate in real timeand are used, for example, in the automaticcontrol of certain industrial processes andin a variety of scientific experiments. Theycan perform much more complicatedmathematics than digital computers butare less accurate and are less flexible in thekind of things they can do. See also hybridcomputer.

analog/digital converter A device thatconverts analog signals (the output froman ANALOG COMPUTER) into digital signals,so that they can be dealt with by a digitalcomputer. See computer.

analog electronics A branch of electron-ics in which inputs and outputs can have arange of voltages rather than fixed values.A frequently used circuit in analog elec-tronics is the operational amplifier, socalled because it can perform mathematicaloperations such as differentiation and inte-gration.

analogy A general similarity between twoproblems or methods. Analogy is used toindicate the results of one problem fromthe known results of the other.

analysis The branch of mathematics con-cerned with the limit process and the con-cept of convergence. It includes the theoryof differentiation, integration, infinite se-ries, and analytic functions. Traditionally,it includes the study of functions of realand complex variables arising from differ-ential and integral calculus.

analytic A function of a real or complexvariable is analytic (or holomorphic) at apoint if there is a neighborhood N of thispoint such that the function is differen-tiable at every point of N. An alternative(and equivalent) definition is that a func-tion is analytic at a point if it can be repre-sented in a neighborhood of this point byits Taylor series about it. A function is saidto be analytic in a region if it is analytic atevery point of that region.

ampere

6

analytical geometry (coordinate geome-try) The use of coordinate systems and al-gebraic methods in geometry. In a planeCartesian coordinate system a point is rep-resented by a set of numbers and a curve isan equation for a set of points. The geo-metric properties of curves and figures canthus be investigated by algebra. Analyticalgeometry also enables geometrical inter-pretations to be given to equations.

anchor ring See torus.

and See conjunction.

AND gate See logic gate.

angle (plane angle) The spatial relation-ship between two straight lines. If two linesare parallel, the angle between them iszero. Angles are measured in degrees or, al-ternatively, in radians. A complete revolu-tion is 360 degrees (360° or 2π radians). Astraight line forms an angle of 180° (π ra-dians) and a right angle is 90° (π/2 radi-ans).

The angle between a line and a plane isthe angle between the line and its orthogo-nal projection on the plane.

The angle between two planes is theangle between lines drawn perpendicularto the common edge from a point – one linein each plane. The angle between two in-tersecting curves is the angle between theirtangents at the point of intersection.

See also solid angle.

angle of depression The angle betweenthe horizontal and a line from an observerto an object situated below the eye level ofthe observer. See also angle.

angle of elevation The angle between thehorizontal and a line from an observer toan object situated above the eye level of theobserver. See also angle of depression.

ångstrom /ang-strŏm/ Symbol: Å A unitof length defined as 10–10 meter. Theångstrom is sometimes used for expressingwavelengths of light or ultraviolet radia-tion or for the sizes of molecules. The unitis named for the Swedish physicist and

astronomer Anders Jonas Ångstrom(1814–74).

angular acceleration Symbol: α The ro-tational acceleration of an object about anaxis; i.e. the rate of change of angular ve-locity with time:

α = dω/dtor

α = d2θ/dt2

where ω is angular velocity and θ is angu-lar displacement. Angular acceleration isanalogous to linear acceleration. See rota-tional motion.

angular displacement Symbol: θ The ro-tational displacement of an object about anaxis. If the object (or a point on it) movesfrom point P1 to point P2 in a plane per-pendicular to the axis, θ is the angle P1OP2,where O is the point at which the perpen-dicular plane meets the axis. See also rota-tional motion.

angular frequency (pulsatance) Symbol:ω The number of complete rotations perunit time. Angular frequency is often usedto describe vibrations. Thus, a simple har-monic motion of frequency f can be repre-sented by a point moving in a circular pathat constant speed. The foot of a perpendic-ular from the point to a diameter of the cir-cle moves with simple harmonic motion.The angular frequency of this motion isequal to 2πf, where f is the frequency of thesimple harmonic motion. The unit of angu-lar frequency, like frequency, is the hertz.

angular momentum Symbol: L Theproduct of the moment of inertia of a bodyand its angular velocity. Angular momen-tum is analogous to linear momentum, mo-ment of inertia being the equivalent ofmass for ROTATIONAL MOTION.

angular velocity Symbol: ω The rate ofchange of angular displacement with time:ω = dθ/dt. See also rotational motion.

anharmonic oscillator /an-har-mon-ik/A system whose vibration, while still peri-odic, cannot be described in terms of sim-ple harmonic motions (i.e. sinusoidal

7

anharmonic oscillator

annuity

8

straight angle

obtuse angle

reflex angle

acute angle

right angle

reflex angle

Angle: types of angle

motions). In such cases, the period of oscil-lation is not independent of the amplitude.

annuity A pension in which an insurancecompany pays the annuitant fixed regularsums of money in return for sums of moneypaid to it either in installments or as a lumpsum. An annuity certain is paid for a fixednumber of years as opposed to an annuitythat is payable only while the annuitant isalive.

annulus /an-yŭ-lŭs/ (pl. annuli or annu-luses) The region between two concentriccircles. The area of an annulus is π(R2 – r2),where R is the radius of the larger circleand r is the radius of the smaller.

antecedent In logic, the first part of aconditional statement; a proposition orstatement that is said to imply another. Forexample, in the statement ‘if it is rainingthen the streets are wet’, ‘it is raining’ is theantecedent. Compare consequent. See alsoimplication.

anticlockwise (counterclockwise) Rotat-ing in the opposite sense to the hands of aclock. See clockwise.

antiderivative /an-tee-dĕ-riv-ă-tiv/ Afunction g(x) that is related to a real func-tion f(x) by the fact that the derivative ofg(x) with respect to x, denoted by g′(x), is

equal to f(x) for all values of x in the do-main of f. The function g(x) is said to bethe antiderivative of f(x). The indefinite in-tegral ∫f(x)dx does not specify all the anti-derivatives of f(x) since an arbitraryconstant c can be added to any antideriva-tive. Thus, if both g1(x) and g2(x) are anti-derivatives of a continuous function f(x)then g1(x) and g2(x) can only differ by aconstant.

antilogarithm /an-tee-lôg-ă-rith-<m/ (an-tilog) The inverse function of a LOGA-RITHM. In common logarithms, theantilogarithm of x is 10x. In natural loga-rithms, the antilogarithm of x is ex.

antinode /an-tee-nohd/ A point of maxi-mum vibration in a stationary wave pat-tern. Compare node. See also stationarywave.

antinomy /an-tin-ŏ-mee/ See paradox.

antiparallel /an-tee-pa-ră-lel/ Parallel butacting in opposite directions, said of vec-tors.

antisymmetric matrix /an-tee-să-met-rik/(skew-symmetric matrix) A square matrixA that satisfies the relation AT = –A, whereAT is the TRANSPOSE of A. The definition ofan antisymmetric matrix means that all en-tries aij of the matrix have to satisfy aij =–aji for all the i and j in the matrix. This, inturn, means that all diagonal entries in thematrix must have a value of zero, i.e. aii =0 for all i in the matrix.

apex The point at the top of a solid, suchas a pyramid, or of a plane figure, such asa triangle.

Apollonius’ circle /ap-ŏ-loh-nee-ŭs/ Acircle defined as the locus of all points Pthat satisfy the relation AP/BP = c, where Aand B are points in a plane and c is a con-stant. In the case of c = 1 a straight line isobtained. This case can be considered to bea particular case of a circle or can be ex-plicitly left out as a special case.

9

Apollonius’ circle

R

r

Annulus (shown shaded)

The circle is named for the Greek mathe-matician Apollonius of Perga (c. 261 BC–c.190 BC).

Apollonius’ theorem /ap-ŏ-loh-nee-ŭs/The equation that relates the length of amedian in a triangle to the lengths of itssides. If a is the length of one side and b isthe length of another, and the third side isdivided into two equal lengths c by a me-dian of length m, then:

a2 + b2 = 2m2 + 2c2

apothem /ap-ŏ-th’em/ (short radius) Aline segment from the center of a regularpolygon perpendicular to the center of aside.

applications program A computer pro-gram designed to be used for a specific pur-pose (such as stock control or wordprocessing).

applied mathematics The study of themathematical techniques that are used tosolve problems. Strictly speaking it is theapplication of mathematics to any ‘real’system. For instance, pure geometry is thestudy of entities – lines, points, angles, etc.– based on certain axioms. The use of Eu-

clidean geometry in surveying, architec-ture, navigation, or science is appliedgeometry. The term ‘applied mathematics’is used especially for mechanics – the studyof forces and motion. Compare pure math-ematics.

approximate Describing a value of somequantity that is not exact but close enoughto the correct value for some specific pur-pose, as within certain boundaries of error.It is also used as a verb meaning ‘to find thevalue of a quantity within certain boundsof accuracy, but not exactly’. For example,one can approximate an irrational number,such as π, by finding its decimal expansionto a certain number of places.

approximate integration Any of varioustechniques for finding an approximatevalue for a definite integral. There aremany integrals that cannot be evaluatedexactly and approximation techniques areused to estimate values for such integrals.Both analytical and numerical approxi-mate integration techniques exist. In someanalytical approximate integration tech-niques the value of the integral is expressedas an ASYMPTOTIC SERIES. Two examples ofnumerical approximate integration areSIMPSON’S RULE and the TRAPEZIUM RULE.See also numerical integration.

approximately equal to Symbol ≅ Asymbol used to relate two quantities inwhich one of the quantities is a good ap-proximation to the other quantity but isnot exactly equal to it. An example of theuse of this symbol is π ≅ 22/7.

approximation /ă-proks-ă-may-shŏn/ Acalculation of a quantity that gives values

Apollonius’ theorem

10

am

c c

b

Apollonius’ theorem: a2 + b2 = 2m2 + 2c2

P

A B

Apollonius’ circle

that are not, in general, exact but are closeto the exact values. There are many inte-grals and differential equations in mathe-matics and its physical applications thatcannot be solved exactly and require theuse of approximation techniques. SeeNewton’s method; numerical integration;Simpson’s rule; trapezium rule.

apse /aps/ Any point on the orbit at whichthe motion of the orbiting body is at rightangles to the central radius vector of theorbit. The distance from an apse to the cen-tre of motion (the apsidal distance) equalsthe maximum or minimum value of the RA-DIUS VECTOR.

arc A part of a continuous curve. If the cir-cumference of a circle is divided into twounequal parts, the smaller is known as theminor arc and the larger is known as themajor arc.

arc cosecant (arc cosec; arc csc) An in-verse cosecant. See inverse trigonometricfunctions.

arc cosech An inverse hyperbolic cose-cant. See inverse hyperbolic functions.

arc cosh An inverse hyperbolic cosine. Seeinverse hyperbolic functions.

arc cosine (arc cos) An inverse cosine. Seeinverse trigonometric functions.

arc cotangent (arc cot) An inverse cotan-gent. See inverse trigonometric functions.

arc coth An inverse hyperbolic cotangent.See inverse hyperbolic functions.

Archimedean solid /ar-kă-mee-dee-ăn, -m…brevel-dee-ăn/ See semi-regular poly-hedron. Archimedean solids are named forthe Greek mathematician Archimedes (287BC–212 BC).

Archimedean spiral A particular type ofSPIRAL that is described in POLAR COORDI-NATES by the equation r = aθ, where a is apositive constant. It can be considered torepresent the locus of a point moving alonga radius vector with a uniform velocitywhile the radius vector itself is movingabout a pole with a constant angular ve-locity. A spiral of this type asymptoticallyapproaches a circle of radius a.

Archimedes’ principle /ar-kă-mee-deez/The upward force of an object totally orpartly submerged in a fluid is equal to theweight of fluid displaced by the object. Theupward force, often called the upthrust, re-sults from the fact that the pressure in afluid (liquid or gas) increases with depth. Ifthe object displaces a volume V of fluid ofdensity ρ, then:

upthrust u = Vρgwhere g is the acceleration of free fall. If theupthrust on the object equals the object’sweight, the object will float.

arc secant (arc sec) An inverse secant. Seeinverse trigonometric functions.

arc sech An inverse hyperbolic secant. Seeinverse hyperbolic functions.

arc sine (arc sin) An inverse sine. See in-verse trigonometric functions.

arc sinh An inverse hyperbolic sine. Seeinverse hyperbolic functions.

arc tangent (arc tan) An inverse tangent.See inverse trigonometric functions.

11

arc tangent

majorarc

minorarc

Arc: major and minor arcs of a circle.

arc tanh An inverse hyperbolic tangent.See inverse hyperbolic functions.

are /air/ A metric unit of area equal to 100square meters. It is equivalent to 119.60 sqyd. See also hectare.

area Symbol: A The extent of a plane fig-ure or a surface, measured in units oflength squared. The SI unit of area is thesquare meter (m2). The area of a rectangleis the product of its length and breadth.The area of a triangle is the product of thealtitude and half the base. Closed figuresbounded by straight lines have areas thatcan be determined by subdividing theminto triangles. Areas for other figures canbe found by using integral calculus.

Argand diagram /ar-gănd/ See complexnumber.

argument (amplitude) 1. In a complexnumber written in the form r(cosθ + i sinθ),the angle θ is the argument. It is thereforethe angle that the vector representing thecomplex number makes with the horizon-tal axis in an Argand diagram. See alsocomplex number; modulus.2. In LOGIC, a sequence of propositions orstatements, starting with a set of premisses(initial assumptions) and ending with aconclusion.

arithmetic The study of the skills neces-sary to manipulate numbers in order to

solve problems containing numerical infor-mation. It also involves an understandingof the structure of the number system andthe facility to change numbers from oneform to another; for example, the changingof fractions to decimals, and vice versa.

arithmetic and logic unit (ALU) Seecentral processor.

arithmetic mean See mean.

arithmetic sequence (arithmetic progres-sion) A SEQUENCE in which the differencebetween each term and the one after it isconstant, for example, 9, 11, 13, 15, ….The difference between successive terms iscalled the common difference. The generalformula for the nth term of an arithmeticsequence is:

nn = a + (n – 1)dwhere a is the first term of the sequenceand d is the common difference. Comparegeometric sequence. See also arithmetic se-ries.

arithmetic series A SERIES in which thedifference between each term and the oneafter it is constant, for example, 3 + 7 + 11+ 15 + …. The general formula for an arith-metic series is:

Sn = a + (a + d) + (a + 2d) + …+ [a + (n – 1)d]= ∑[a + (n – 1)d]

In the example, the first term, a, is 3, thecommon difference, d, is 4, and so the

arc tanh

12

A curved area can be found by dividing it into rectangles and adding the areas of the rectangles.The more rectangles, the better the approximation.

nth term, a + (n – 1)d, is 3 + (n – 1)4. Thesum to n terms of an arithmetic series isn[2a + (n – 1)d]/2 or n(a + l)/2 where l is the last (nth) term. Compare geometric series.

arm One of the lines forming a givenangle.

array An ordered arrangement of num-bers or other items of information, such asthose in a list or table. In computing, anarray has its own name, or identifier, andeach member of the array is identified by asubscript used with the identifier. An arraycan be examined by a program and a par-ticular item of information extracted byusing this identifier and subscript.

artificial intelligence The branch ofcomputer science that is concerned withprograms for carrying out tasks which re-quire intelligence when they are done byhumans. Many of these tasks involve a lotmore computation than is immediately ap-parent because much of the computation isunconscious in humans, making it hard tosimulate. Programs now exist that playchess and other games at a high level, takedecisions based on available evidence,prove theorems in certain branches ofmathematics, recognize connected speechusing a limited vocabulary, and use televi-sion cameras to recognize objects. Al-though these examples sound impressive,the programs have limited ability, no cre-ativity, and each can only carry out a lim-ited range of tasks. There is still a lot moreresearch to be done before the ultimategoal of artificial intelligence is achieved,which is to understand intelligence wellenough to make computers more intelli-gent than people. In fact there is consider-able controversy about the whole subject,with many people postulating that thehuman thought process is different in kindto the computational operation of com-puter processes.

assembler See program.

assembly language See program.

associative Denoting an operation that isindependent of grouping. An operation • isassociative if

a•(b•c) = (a•b)•cfor all values of a, b, and c. In ordinaryarithmetic, addition and multiplication areassociative operations. This is sometimesreferred to as the associative law of addi-tion and the associative law of multiplica-tion. Subtraction and division are notassociative. See also commutative; distrib-utive.

astroid /ass-troid/ A star-shaped curveddefined in terms of the parameter θ by: x =a cos3θ, y = a sin3θ, where a is a constant.

astronomical unit (au, AU) The meandistance between the Sun and the Earth,used as a unit of distance in astronomy formeasurements within the solar system. It isdefined as 149 597 870 km.

asymmetrical /ay-să-met-ră-kăl/ Denot-ing any figure that cannot be divided intotwo parts that are mirror images of eachother. The letter R, for example, is asym-metrical, as is any solid object that has aleft-handed or right-handed characteristic.Compare symmetrical.

asymptote /ass-im-toht/ A straight linetowards which a curve approaches butnever meets. A hyperbola, for example, hastwo asymptotes. In two-dimensionalCartesian coordinates, the curve with theequation y = 1/x has the lines x = 0 and y =0 as asymptotes, since y becomes infinitelysmall, but never reaches zero, as x in-creases, and vice versa.

asymptotic series /ass-im-tot-ik/ A seriesof the form a0 + a1/x + a2/x2 + ... an/xn rep-resenting a function f(x) is an asymptoticseries if |f(x) – Sn(x)| tends to zero as |x|→∞for a fixed n, where Sn(x) is the sum of thefirst n terms of the series. Asymptotic seriescan be defined for either real or complexvariables. They are usually divergent al-though some are convergent. Asymptoticseries are used extensively in mathematicalanalysis and its physical applications. For example, many integrals that cannot

13

asymptotic series

be calculated precisely can be calculatedapproximately in terms of asymptoticseries.

atmosphere A unit of pressure equal to 760 mmHg. It is equivalent to 101 325 newtons per square meters(101 325 N m–2).

atmospheric pressure See pressure of theatmosphere.

atto- Symbol: a A prefix denoting 10–18.For example, 1 attometer (am) = 10–18

meter (m).

attractor The point or set of points inphase space to which a system moves withtime. The attractor of a system may be asingle point (in which case the systemreaches a fixed state that is independent oftime), or it may be a closed curve known asa limit cycle. This is the type of behavior

atmosphere

14

y

4 —

3 —

2 —

1 —

0-4 -3 -2 -1

1 2 3 4 5 6 7 8

— -1

— -2

— -3

— -4

— — — — — — — —— — — — — x

Asymptote: the x-axis and the y-axis are asymptotes to this curve.

Attractor: an example of a strange attractor. The variable shown on the left displays chaoticbehavior. The plot in phase space on the right shows a non-intersecting curve.

found in oscillating systems. In some sys-tems, the attractor is a curve that is notclosed and does not repeat itself. This,known as a strange attractor, is character-istic of chaotic systems. See chaos theory.

AU (au) See astronomical unit.

augend /aw-jend, aw-jend/ A number towhich another, the ADDEND, is added in asum.

automorphism /aw-tŏ-mor-fiz-ăm/ Invery general terms, a transformation on el-ements belonging to a set that has somestructure, in which the structural relationsbetween the elements are unchanged by thetransformations. The concept of automor-phism applies to structure in algebra andgeometry. An example of an automor-phism is a transformation among thepoints of space that takes every figure intoa similar figure. The set of automorphismson a set of elements is a GROUP called theautomorphism group. In physics the set oftransformations between frames of refer-ence forms the physical automorphismgroup.

auxiliary equation An equation that isused in the solution of an inhomogeneoussecond-order linear DIFFERENTIAL EQUATION

of the form ad2y/dx2 + bdy/dx + cy = f(x)where a, b, and c are constants and f(x) isa function of x. The solution is found interms of the solution to the simpler homo-geneous equation ad2y/dx2 + bdy/dx + cy =0. A result in the theory of differentialequations shows that the general solutionof the first equation can be written in theform y = g(x) + y1(x), where g(x) is a func-tion, known as the complementary func-tion, which is the general solution of thesecond equation, and y1(x) is a particularsolution of the first equation, known as theparticular integral. The complementaryfunction can be found by taking y = exp(mx) to be a solution of the second equa-tion. Taking this solution gives rise to theauxiliary equation am2 + bm + c = 0. Theform the complementary function takes de-pends on the nature of the roots of the aux-iliary equation. If the equation has two

different real roots, m1 and m2, then thecomplementary function is given by y = Aexp (m1x) + B exp (m2x), where A and Bare constants. If the auxiliary equation hasone root m that occurs twice then the com-plementary function is given by y = (A +Bx) exp (mx).

average See mean.

axial plane /aks-ee-ăl/ A fixed referenceplane in a three-dimensional coordinatesystem. For example, in rectangular Carte-sian coordinates, the planes defined by x =0, y = 0, and z = 0 are axial planes. The x-coordinate of a point is its perpendiculardistance from the plane x = 0, and the y-and z-coordinates are the perpendiculardistances from the y = 0 and z = 0 planes re-spectively. See also coordinates.

axial vector (pseudovector) A quantitythat acts like a vector but changes sign ifthe coordinate system is changed from aright-handed system to a left-handed sys-tem. An example of an axial vector is a vec-tor formed by the (cross) vector product oftwo vectors. See also polar vector.

axiom /aks-ee-ŏm/ (postulate) In a math-ematical or logical system, an initial propo-sition or statement that is accepted as truewithout proof and from which furtherstatements, or theorems, can be derived. Ina mathematical proof, the axioms are oftenwell-known formulae.

axiom of choice See choice; axiom of.

axis (pl. axes) 1. A line about which a fig-ure is symmetrical.2. One of the fixed reference lines used in agraph or a coordinate system. See coordi-nates.3. A line about which a curve or body ro-tates or revolves.4. The line of intersection of two or morecoaxial planes.

azimuth /az-ă-mŭth/ The angle θ meas-ured in a horizontal plane from the x-axisin spherical polar coordinates. It is thesame as the longitude of a point.

15

azimuth

azimuth

16

z

z = 0

y

0x

z

y

x0

x = 0

z

y = 0

y

x0

In three-dimensional rectangularCartesion coordinates, the x- and y-axes lie in the axial plane z = 0,the y- and z-axes in the axial planex = 0, and the x- and z-axes in theaxial plane y = 0.

Axial planes

backing store See disk; magnetic tape;store.

ballistic pendulum A device for measur-ing the momentum (or velocity) of a pro-jectile (e.g. a bullet). It consists of a heavypendulum, which is struck by the projec-tile. The momentum can be calculated bymeasuring the displacement of the pendu-lum and using the law of constant momen-tum. If the mass of the projectile is knownits velocity can be found.

ballistics The study of the motion of ob-jects that are propelled by an external force(i.e. the motion of projectiles).

bar A unit of pressure defined as 105 pas-cals. The millibar (mb) is more common; itis used for measuring atmospheric pressurein meteorology.

bar chart A GRAPH consisting of barswhose lengths are proportional to quanti-ties in a set of data. It can be used when oneaxis cannot have a numerical scale; e.g. toshow how many pink, red, yellow, and

white flowers grow from a packet of mixedseeds. See also histogram.

barn Symbol: b A unit of area defined as10–28 square meter. The barn is sometimesused to express the effective cross-sectionsof atoms or nuclei in the scattering or ab-sorption of particles.

barrel A US unit of capacity used to meas-ure solids or liquids. It is equal to 7056cubic inches (0.115 6 m3). In the petroleumindustry, 1 barrel = 42 US gallons, 35 im-perial gallons, or 159.11 liters.

barrel printer See line printer.

barycenter /ba-ră-sen-ter/ See center ofmass. Used in particular for the center ofmass of a system of separate objects con-sidered as a whole.

barycentric coordinates /ba-ră-sen-trik/Coordinates that relate the center of massof separate objects to the center of mass ofthe system of several objects as a whole.Consider three objects with masses m1, m2,

17

B

15 –

10 –

5 –

0 1 2 3 4number of books read in a week

frequency

Bar chart

This bar chart shows the results when a groupof 40 school students were asked how manybooks they each had read in the previousweek. 13 had read none, 13 had read one, 8had read two, 5 had read three, 1 had readfour, and no-one had read five or more.

and m3 with m1 + m2 + m3 = 1, and theircenters of mass at the points p1 = (x1,y1,z1),p2 = (x2,y2,z2), p3 = (x3,y3,z3). Then the cen-ter of mass of the three objects together isthe point

p = m1p1 + m2p2 + m3p3 =(m1x1+m2x2+m3x3, m1y1+ m2y2+m3y3,

m1z1+m2z2+m3z3)(m1,m2,m3) are said to be the barycentriccoordinates of the point p with respect tothe points p1, p2, and p3.

base 1. In geometry, the lower side of atriangle, or other plane figure, or the lowerface of a pyramid or other solid. The alti-tude is measured from the base and at rightangles to it.2. In a number system, the number of dif-ferent symbols used, including zero. Forexample, in the decimal number system,the base is ten. Ten units, ten tens, etc., aregrouped together and represented by thefigure 1 in the next position. In binarynumbers, the base is two and the symbolsused are 0 and 1.3. In logarithms, the number that is raisedto the power equal to the value of the log-arithm. In common logarithms the base is10; for example, the logarithm to the base10 of 100 is 2:

log10100 = 2100 = 102

base unit A unit that is defined in terms ofreproducible physical phenomena or pro-totypes, rather than of other units. The sec-ond, for example, is a base unit (of time) inthe SI, being defined in terms of the fre-quency of radiation associated with a par-ticular atomic transition. Conventionally,seven units are chosen as base units in theSI. See also SI units.

BASIC See program.

basis vectors In two dimensions, twononparallel VECTORS, scalar multiples ofwhich are added to form any other vectorin the same plane. For example, the unitvectors i and j in the directions of the x-and y-axes of a Cartesian coordinate sys-tem are basis vectors. The position vectorOP of the point P(2,3) is equal to 2i + 3j.

Similarly, in three dimensions a vector canbe written as the sum of multiples of threebasis vectors.

batch processing A method of operation,used especially in large computer systems,in which a number of programs are col-lected together and input to the computeras a single unit. The programs forming abatch can either be submitted at a centralsite or at a remote job entry site; there canbe any number of remote job entry sites,which can be situated at considerable dis-tance from the computer. The programsare then executed as time becomes avail-able in the system. Compare time sharing.

Bayes’ theorem /bayz/ A formula ex-pressing the probability of an intersectionof two or more sets as a product of the in-dividual probabilities for each. It is used tocalculate the probability that a particularevent Bi has occurred when it is knownthat at least one of the set B1, B2, … Bnhas occurred and that another event A hasalso occurred. This conditional probabilityis written as P(Bi|A). B1, … Bn form a par-tition of the sample space s such thatB1∪B2∪…∪Bn = s and Bi∩Bj = 0, for all iand j. If the probabilities of B1, B2, … Bnand all of the conditional probabilitiesP(A|Bj) are known, then P(Bi|A) is given by

P(Bi)P(A|Bj)The theorem is named for the English

mathematician Thomas Bayes (1702–61).

beam compass See compasses.

bearing The horizontal angle between aline and a fixed reference direction, usuallymeasured clockwise from the direction ofnorth. Bearings are expressed in degrees(e.g. 125°); bearings of less than 100° arewritten with an initial 0 (e.g. 030°). Theyhave important applications in radar,sonar, and surveying.

bel See decibel.

Bernoulli trial An experiment in whichthere are two possible independent out-comes, for example, tossing a coin. Theexperiment is named for the Swiss mathe-

base

18

matician Jakob (or Jacques) Bernoulli(1654–1705), who made an importantcontribution to the discipline of probabil-ity theory.

Bessel functions /bess-ăl/ A set of func-tions, denoted by the letter J, that are solu-tions to equations involving waves, whichare expressed in cylindrical polar coordi-nates. The solutions form an infinite seriesand are listed in tables. The set of functionsis named for the German astronomerFriedrich Wilhelm Bessel (1784–1846),who used them in his work on planetarymovements. See also partial differentialequation.

beva- Symbol: B A prefix used in the USAto denote 109. It is equivalent to the SI pre-fix giga-.

Bezier curve /bez-ee-ay/ A type of curveused in computer graphics. A simple Beziercurve is defined by four points. Two ofthese are the end points of the curve. Theother two are control points and lie off thecurve. A way of producing the curve is tofirst join the four points by three straightlines. If the midpoints of these three linesare taken, together with the original twoend points, a better approximation is todraw five straight lines joining these

points. The midpoints of these lines canthen be used to get an even better approxi-mation, and so on. The Bezier curve is thelimit of this recursive process. Bezier curvescan be defined mathematically by cubicpolynomials. A similar type of curve ob-tained by using control points is known asa B-spline. The curve is named for theFrench engineer and mathematician PierreBezier (1910–99).

bias A property of a statistical sample thatmakes it unrepresentative of the wholepopulation. For example, if medical data isbased on a survey of patients in a hospital,then the sample is a biased estimate of thegeneral population, since healthy peoplewill be excluded.

biconditional /bÿ-kŏn-dish-ŏ-năl/ Sym-bol: ↔ or ≡ In logic, the relationship if andonly if (often abbreviated to iff) that holdsbetween a pair of propositions or state-ments P and Q only when they are both

19

biconditional

E

E

C

C

E

C

E

CE

C

E

C

E

C

E

C

Bezier curve: construction of a curve by successive iterations.

T TT FF TF F

P ≡ QTFFT

P Q

Biconditional

true or both false. It is also the relationshipof logical equivalence; the truth of P is botha necessary and a sufficient CONDITION forthe truth of Q (and vice versa). The truth-table definition for the biconditional isshown. See also truth table.

bilateral symmetry A type of symmetryin which a shape is SYMMETRICAL about aplane, i.e. each half of the shape is a mirrorimage of the other.

billion A number equal to 109 (i.e. onethousand million). This has always beenthe definition in the USA. In the UK a bil-lion was formerly 1012 (i.e. one million mil-lion). In the 1960s, the British Treasurystarted using the term billion in the Ameri-can sense of 109, and this usage has nowsupplanted the original ‘million million’.

bimodal /bÿ-moh-d’l/ Describing distribu-tions of numerical data that have twopeaks (modes) in frequency.

binary /bÿ-nă-ree, -nair-ee/ Denoting orbased on two. A binary number is made upusing only two different digits, 0 and 1, in-stead of ten in the decimal system. Eachdigit represents a unit, twos, fours, eights,sixteens, etc., instead of units, tens, hun-dreds, etc. For example, 2 is written as 10,3 is 11, 16 (= 24) is 10 000. Computers cal-culate using binary numbers. The digits 1and 0 correspond to on/off conditions in anelectronic switching circuit or to presenceor absence of magnetization on a disk ortape. Compare decimal; duodecimal; hexa-decimal; octal.

binary logarithm A LOGARITHM to thebase two. The binary logarithm of 2 (writ-ten log22) is 1.

binary operation A mathematical pro-cedure that combines two numbers, quan-tities, etc., to give a third. For example,multiplication of two numbers in arith-metic is a binary operation.

binary relation A relation between twoelements a and b of a set S. For example,the symbol > for ‘greater than’ indicates a

binary relation between two real numbers.In general, the existence of a relation be-tween a and b can be indicated by aRb.

binomial /bÿ-moh-mee-ăl/ An algebraicexpression with two variables in it. For ex-ample, 2x + y and 4a + b = 0 are binomials.Compare trinomial.

binomial coefficient The factor multi-plying the variables in a term of a BINOMIAL

EXPANSION. For example, in (x + y)2 = x2 +2xy + y2 the binomial coefficients are 1, 2,and 1 respectively.

binomial distribution The distributionof the number of successes in an experi-ment in which there are two possible out-comes, i.e. success and failure. Theprobability of k successes is

b(k,n,p) = n!/k!(n – k)! × pn × qn–k

where p is the probability of success and q(= 1 – p) the probability of failure on eachtrial. These probabilities are given by theterms in the binomial theorem expansionof (p + q)n. The distribution has a mean npand variance npq. If n is large and p smallit can be approximated by a Poisson distri-bution with mean np. If n is large and p isnot near 0 or 1, it can be approximated bya normal distribution with mean np andvariance npq

binomial expansion A rule for the ex-pansion of an expression of the form (x +y)n. x and y can be any real numbers and nis an integer. The general formula, some-times called the binomial theorem, is

(x + y)n = xn + nxn–1y +[n(n – 1)/2!]xn–2y2 + … + yn

This can be written in the form:xn + nC1xn–1y + nC2xn–2y2 + …

+ nCrxn–ryr + …The coefficients nC1, nC2, etc., are called bi-nomial coefficients. In general, the rth co-efficient, nCr, is given by

n!/(n – r)!r! See also combination.

binomial theorem See binomial expan-sion.

birectangular /bÿ-rek-tang-gyŭ-ler/ Hav-ing two right angles. See spherical triangle.

bilateral symmetry

20

bisector /bÿ-sek-ter, bÿ-sek-/ A straightline or a plane that divides a line, a plane,or an angle into two equal parts.

bistable circuit /bÿ-stay-băl/ An elec-tronic circuit that has two stable states.The circuit will remain in one state untilthe application of a suitable pulse, whichwill cause it to assume the other state.

Bistable circuits are used extensively incomputer equipment for storing data andfor counting. They usually have two inputterminals to which pulses can be applied. Apulse on one input causes the circuit tochange state; it will remain in that stateuntil a pulse on the other input causes it toassume the alternative state. These circuitsare often called flip-flops.

bit Abbreviation of binary digit, i.e. eitherof the digits 0 or 1 used in binary notation.Bits are the basic units of information incomputing as they can represent the statesof a two-valued system. For example, thepassage of an electric pulse along a wirecould be represented by a 1 while a 0would mean that no pulse had passed.Again, the two states of magnetization ofspots on a magnetic tape or disk can berepresented by either a 1 or 0. See also bi-nary; byte; word.

bitmap /bit-map/ See computer graphics.

bivariate /bÿ-vair-ee-ayt/ Containing twovariable quantities. A plane vector, for ex-ample, is bivariate because it has both mag-nitude and direction.

A bivariate random variable (X,Y) hasthe joint probability P(x,y); i.e. the proba-bility that X and Y have the values x and yrespectively, is equal to P(x) × P(y), when Xand Y are independent.

Board of Trade unit (BTU) A unit of en-ergy equivalent to the kilowatt-hour (3.6 ×106 joules). It was formerly used in the UKfor the sale of electricity.

Bolzano–Weierstrass theorem /bohl-tsah-noh vÿ-er-shtrahss/ The theorem thatany bounded infinite set has an accumula-tion point. The accumulation point need

not be in the set; e.g. the set 1,½,¼,… isbounded and infinite and it does have anaccumulation point, namely 0, but thatpoint is not in the set. The theorem isnamed for the Czech philosopher, mathe-matician and theologian Bernard Bolzano(1781–1848) and the German mathemati-cian Karl Weierstrass (1815–97).

Boolean algebra /boo-lee-ăn/ A system ofmathematical logic that uses symbols andset theory to represent logic operations in amathematical form. It was the first systemof logic using algebraic methods for com-bining symbols in proofs and deductions.Various systems have been developed andare used in probability theory and comput-ing. Boolean algebra is named for theBritish mathematician George Boole(1815–64).

bound 1. In a set of numbers, a value be-yond which there are no members of theset. A lower bound is less than or equal toevery number in the set. An upper bound isgreater than or equal to every number inthe set. The least upper bound (or supre-mum) is the lowest of its upper bounds andthe greatest lower bound (or infimum) isthe largest of its lower bounds. For exam-ple, the set 0.6, 0.66, 0.666, … has a leastupper bound of 2/3.2. A bound of a function is a bound of theset of values that the function can take forthe range of values of the variable. For ex-ample, if the variable x can be any realnumber, then the function f(x) = x2 has alower bound of 0.3. In formal logic a variable is said to bebound if it is within the scope of a quanti-fier. For example, in the sentence Fy →(∃x)Fx, x is a bound variable, whereas y isnot. A variable which is not bound is saidto be free.

boundary condition In a DIFFERENTIAL

EQUATION, the value of the variables at acertain point, or information about theirrelationship at a point, that enables the ar-bitrary constants in the solution to be de-termined. For example, the equation

d2y/dx2 – 4dy/dx + 3y = 0has a general solution

21

boundary condition

y = Ae–x + Be–3x

where A and B are arbitrary constants. Ifthe boundary conditions are y = 1 at x = 0and dy/dx = 3 at x = 0, the first can be sub-stituted to obtain B = 1 – A. Differentiatingthe general solution for y gives

dy/dx = –Ae–x – 3Be–3x

and substituting the second boundary con-dition then gives

3 = –A – 3B = 2A – 3That is, A = 3 and B = –2.

boundary line A line on a graph that in-dicates the boundary at which an inequal-ity in two dimensions holds. A commonconvention for drawing the boundary lineis that if the inequality is of the form‘greater than or equal to’ (or ‘less than orequal to’), the line is solid, to indicate thatpoints on the line are included. If the in-equality is of the type ‘greater than’ (or‘less than’) the line is dotted to show thatpoints on the line are not included.

bounded function A real function f (x)defined on a domain D for which there is anumber M such that f(x) ≤ M for all x in thedomain D. It is an important result inmathematical analysis that if f(x) is a con-tinuous function on a closed interval be-tween a and b then it is also a boundedfunction on that interval.

bounded set A set that is bounded bothabove and below; i.e. the set has both anupper BOUND and a lower bound.

brackets In mathematical expressions,brackets are used to indicate the order inwhich operations are to be carried out. Forexample:

9 + (3 × 4) = 9 + 12= 21

(9 + 3) × 4 = 12 × 4= 48

Brackets are also used in FACTORIZATION.For example:

4x3y2 – 10x2y2 = 2x2y(2xy – 5y)

branch 1. A section of a curve separatedfrom the remainder of the curve by discon-tinuities or special points such as vertices,

maximum points, minimum points, orcusps.2. A departure from the normal sequentialexecution of instructions in a computerprogram. Control is thus transferred to an-other part of the program rather than pass-ing in strict sequence from one instructionto the next. The branch will be either un-conditional, i.e. it will always occur, or itwill be conditional, i.e. the transfer of con-trol will depend on the result of some arith-metical or logical test. See also loop.

breadth A horizontal distance, usuallytaken at right angles to a length.

Briggsian logarithm /brig-zee-ăn/ Seelogarithm.

British thermal unit (Btu) A unit of en-ergy equal to 1.055 06 × 103 joules. It wasformerly defined by the heat needed toraise the temperature of one pound of air-free water by one degree Fahrenheit atstandard presure. Slightly different ver-sions of the unit were in use depending onthe temperatures between which the degreerise was measured.

B-spline See Bezier curve.

BTU See Board of Trade unit.

Btu See British thermal unit.

buffer store (buffer) A small area of themain STORE of a computer in which infor-mation can be stored temporarily before,during, and after processing. A buffer canbe used, for instance, between a peripheraldevice and the CENTRAL PROCESSOR, whichoperate at very different speeds.

bug An error or fault in a computer pro-gram. See debug.

buoyancy // The tendency of an object tofloat. The term is sometimes also used forthe upward force (UPTHRUST) on a body.See center of buoyancy. See alsoArchimedes’ principle.

boundary line

22

bushel A unit of capacity usually used forsolid substances. In the USA it is equal to64 US dry pints or 2150.42 cubic inches. Inthe UK it is equal to 8 UK gallons.

butterfly effect Any effect in which asmall change to a system results in a dis-proportionately large disturbance. Theterm comes from the idea that the Earth’satmosphere is so sensitive to initial condi-tions that a butterfly flapping its wings in

one part of the world may be the cause ofa tornado in another part of the world. Seechaos theory.

byte /bÿt/ A subdivision of a WORD incomputing, often being the number of BITS

used to represent a single letter, number, orother CHARACTER. In most computers a byteconsists of a fixed number of bits, usuallyeight. In some computers bytes can havetheir own individual addresses in the store.

23

byte

calculus /kal-kyŭ-lŭs/ (infinitesimal calcu-lus) (pl. calculuses) The branch of mathe-matics that deals with the DIFFERENTIATION

and INTEGRATION of functions. By treatingcontinuous changes as if they consisted ofinfinitely small step changes, differentialcalculus can, for example, be used to findthe rate at which the velocity of a body ischanging with time (acceleration) at a par-ticular instant.

Integral calculus is the reverse process,that is finding the end result of known con-tinuous change. For example, if a car’s ac-celeration a varies with time in a knownway between times t1 and t2, then the totalchange in velocity is calculated by the inte-gration of a over the time interval t1 to t2.Integral calculus is also used to find thearea under a curve. See differentiation; in-tegration.

calculus of variations The branch ofmathematics that uses calculus to find min-imum or maximum values for a system. Ithas many applications in physical scienceand engineering. For example, FERMAT’SPRINCIPLE in optics can be regarded as anapplication of the calculus of variations.See also variational principle.

calibration /kal-ă-bray-shŏn/ The mark-ing of a scale on a measuring instrument.For example, a thermometer can be cali-brated in degrees Celsius by marking thefreezing point of water (0°C) and the boil-ing point of water (100°C).

calorie /kal-ŏ-ree/ Symbol: cal A unit ofenergy approximately equal to 4.2 joules.It was formerly defined as the energyneeded to raise the temperature of onegram of water by one degree Celsius. Be-cause the specific thermal capacity of waterchanges with temperature, this definition isnot precise. The mean or thermochemicalcalorie (calTH) is defined as 4.184 joules.The international table calorie (calIT) is de-fined as 4.186 8 joules. Formerly the meancalorie was defined as one hundredth of theheat needed to raise one gram of waterfrom 0°C to 100°C, and the 15°C calorieas the heat needed to raise it from 14.5°Cto 15.5°C.

cancellation Removing a common factorin a numerator and denominator, or re-moving the same quantity from both sidesof an algebraic equation. For example,xy/yz can be simplified, by the cancellation

24

C

−3

3

0

0

2

0

0

0

2

multiplyrow 1by −3

add row 1to row 2

0

0

2

0

2

0

−3

0

0

−3

3

0

0

2

0

0

0

2

1

3

0

0

2

0

0

0

2

Reduction of a matrix to canonical form.

of y, to x/z. The equation z + x = 2 + x issimplified to z = 2 by canceling (subtract-ing) x from both sides.

candela Symbol: cd The SI base unit of lu-minous intensity, defined as the intensity(in the perpendicular direction) of theblack-body radiation from a surface of1/600 000 square meter at the temperatureof freezing platinum and at a pressure of101 325 pascals.

canonical form (normal form) In matrixalgebra, the DIAGONAL MATRIX derived by aseries of transformations on anotherSQUARE MATRIX of the same order.

Cantor’s diagonal argument /kan-terz/An argument to show that the real num-bers, unlike the rationals, are not count-able, and hence that there are more realnumbers than rational numbers. The argu-ment proceeds by assuming that the realsare countable and showing that this leadsto a contradiction.

Every real number can be expressed asan infinite decimal expansion. We supposethat all reals are expressed in this way andsince they are countable they can bearranged in order in a list as shown.We now define a real number b1 . b2 b3 b4… by saying that b1 must be any numberdifferent from a1. Hence our new numberwill not be equal to the first real number inour list. b2 must be any number different

from a21, and hence the new number willnot be equal to the second number on thelist. Continuing in this way we have amethod of producing an infinite decimalthat must define a real number, but is notequal to any of the real numbers in our list.But we assumed that all real numbers oc-curred somewhere in the list; hence there isa contradiction, and so it cannot be truethat the real numbers are countable.

Cantor’s diagonal argument is namedfor the Russian mathematician Georg Can-tor (1845–1918). See also Cantor set.

Cantor set /kan-ter/ The set obtained bytaking a closed interval in the real line, e.g.[0,1], and removing the middle third, i.e.the open interval (1/3, 2/3), and then doingthe same to the two remaining closed inter-vals [0,1/3] and [2/3,1], and so on ad in-finitum. The set generated in this way hasthe remarkable property of containing un-countably many points – i.e. the same num-ber of points as the whole real line – yetbeing nowhere dense – i.e. for any point inthe set one can always find a point not inthe set arbitrarily close to it. This set is alsosometimes known as the Cantor discontin-uum

capital 1. The total sum of all the assets ofa person or company, including cash, in-vestments, household goods, land, build-ings, machinery, and finished or unfinishedgoods.2. A sum of money borrowed or lent onwhich interest is payable or received. Seecompound interest; simple interest.3. The total amount of money contributedby the shareholders when a company isformed, or the amount contributed to apartnership by the partners.

carat /ka-răt/ (metric) A unit of mass usedfor precious stones. It is equal to 200 mil-ligrams.

cardinality /kar-dă-nal-ă-tee/ Symboln(A). The number of elements in a finite setA. The symbols |A| and #A are sometimesused to denote the cardinality of A. Thecardinality is sometimes called the cardinalnumber of the set. If A, B, and C are all fi-

25

cardinality

a1 a11 a12 a13 a14

a2 a21 a22 a23 a24

a3 a31 a32 a33 a34

a4 a41 a42 a43 a44

Cantor’s diagonal argument

nite subsets of a set S then the following re-lations between cardinal numbers hold:

n(A∪B) = n(A) + n(B) – n(A∩B)and

n(A∪B∪C) = n(A) + n(B) + n(C) –n(A∩B) – n(A∩C) – n(B∩C) +

n(A∩B∩C).

cardinal numbers Whole numbers thatare used for counting or for specifying atotal number of items, but not the order inwhich they are arranged. For example,when one says ‘three books’, the three is acardinal number.

Two sets are said to have the same car-dinal number if their elements can be putinto one-to-one correspondence with eachother. Compare ordinal numbers. See alsoaleph.

cardioid /kar-dee-oid/ An EPICYCLOID thathas only one loop, formed by the path of apoint on a circle rolling round the circum-ference of another that has the same radius.

Cartesian coordinates /kar-tee-zhăn/ Amethod of defining the position of a pointby its distance from a fixed point (origin) inthe direction of two or more straight lines.On a flat surface, two straight lines, calledthe x-axis and the y-axis, form the basis ofa two-dimensional Cartesian coordinatesystem. The point at which they cross is theorigin (O). An imaginary grid is formed bylines parallel to the axes and one unitlength apart. The point (2,3), for example,is the point at which the line parallel to they-axis two units in the direction of the x-axis, crosses the line parallel to the x-axisthree units in the direction of the y-axis.Usually the x-axis is horizontal and the y-axis is perpendicular to it. These areknown as rectangular coordinates. If theaxs are not at right angles, they are obliquecoordinates

In three dimensions, a third axis, the z-axis, is added to define the height or depthof a point. The COORDINATES of a point arethen three numbers (x,y,z). A right-handedsystem is one for which if the thumb of theright hand points along the x-axis, the fin-gers of the hand fold in the direction inwhich the y-axis would have to rotate to

point in the same direction as the z-axis. Aleft-handed system is the mirror image ofthis. In a rectangular system, all three axsare mutually at right-angles. The Cartesiancoordinate system is named for the Frenchmathematician, philosopher, and scientistRené du Perron Descartes (1596–1650).See also cylindrical polar coordinates;polar coordinates.

Cartesian product The Cartesian prod-uct of two sets A and B, which is written A× B, is the set of ordered pairs ⟨x,y⟩ wherex belongs to A and y belongs to B.

A × B = ⟨x,y⟩ | x ∈A and y ∈B

catastrophe theory The study of theways in which discontinuities appear, bothin mathematics and its physical applica-tions. Catastrophe theory deals with prob-lems that cannot be analyzed usingcalculus since calculus is largely concernedwith continuity. Catastrophe theory can beregarded as a branch of TOPOLOGY. It hasbeen applied to several physical problems,including problems in the theory of waves,especially optics. It has also been applied tothe evolution of complex systems such asbiological systems and (far more dubi-ously) to social and economic changes.

category A category consists of twoclasses: a class of objects and a class ofmorphisms – i.e. mappings that are in somesense structure-preserving. Associated witheach pair of objects are a set of the mor-phisms and a law of composition for thesemorphisms. Category theory is the study ofsuch entities. It provides a model for manysituations where sets with certain struc-tures are studied along with a class of map-pings that preserve these structures.Examples of categories are sets with func-tions and groups with homomorphisms.

catenary /kat-ĕ-nair-ee/ The plane curveof a flexible uniform line suspended fromtwo points. For example, an empty wash-ing line attached to two poles and hangingfreely between them follows a catenary.The catenary is symmetrical about an axisperpendicular to the line joining the twopoints of suspension. In Cartesian coordi-

cardinal numbers

26

27

CD-ROM

x

y

zz

x

y

left-handed right-handed

In three-dimensional Cartesian coordinates, a right-handedsystem is the mirror image of a left-handed system.

P(a,b)

x

y

0

absicca of P = a units

ordinate of Pordinate of Pordinate of P= b units= b units= b units

Two-dimensional rectangular Cartesian coordinates showing a point P(a,b).

nates, the equation of a catenary that hasits axis of symmetry lying along the y-axisat y = a, is

y = (a/2)(ex/a + e–x/a)

catenoid /kat-ĕ-noid/ The curved surfaceformed by rotating a catenary about itsaxis of symmetry.

Cauchy’s inequality /koh-sheez, koh-sheez/ For sums, the inequality has theform that if a1, a2, ..., an and b1, b2, ..., bnare two sets of real numbers then the in-

equality (Σaibi)2 ≤ (Σai2)(Σbi

2). This in-equality only becomes an equality in thecase when there are constants k and l thatsatisfy kai = lbi for all i, i.e. if the ai and thebi are all proportional. It is also called theCauchy–Schwarz inequality.

CD-ROM A type of compact disc with aread-only memory, which provides read-only access to up to 640 megabytes ofmemory for use on a computer. The diskcan carry data as text, images (video) andsound (audio).

Celsius degree /sel-see-ŭs/ Symbol: °C Aunit of temperature difference equal to onehundredth of the difference between thetemperatures of freezing and boiling waterat one atmosphere pressure. It was for-merly known as the degree centigrade andis equivalent to 1 K. On the Celsius scalewater freezes at 0°C and boils at 100°C. Itis named for the Swedish astronomer An-ders Celsius (1701–44).

center A point about which a geometricfigure is symmetrical.

center of buoyancy For an object in afluid, the center of mass of the displacedvolume of fluid. In order for a floating ob-ject to be stable the center of mass of theobject must lie below the center of buoy-ancy; when the object is in equilibrium, the two lie on a vertical line. See alsoArchimedes’ principle.

center of curvature See curvature.

center of gravity See center of mass.

center of mass (barycentre) A point in abody (or system) at which the whole massof the body may be considered to act.Often the term center of gravity is used.This is, strictly, not the same unless thebody is in a constant gravitational field.The center of gravity is the point at whichthe weight may be considered to act.

The center of mass coincides with thecenter of symmetry if the symmetrical bodyhas a uniform density throughout. In othercases the principle of moments may be usedto locate the point. For instance, twomasses m1 and m2 a distance d apart havea center of mass on the line between them.If this is a distance d1 from m1 and d2 fromm2 then m1d1 = m2d2 or:

m1d1 = m2(d – d1)d1 = m2d/(m1 + m2)

A more general relationship can be appliedto a number of masses m1, m2,…, mi thatare respectively distances r1, r2,…, ri froman origin. The distance r from the origin tothe center of mass is given by:

r = ∑rimi/∑mi

In the case of a body having a uniform den-sity an integration must be used to obtainthe position of the center of mass, whichcoincides with the CENTROID.

center of pressure For a body or surfaceof a fluid, the point at which the resultantof pressure forces acts. If a surface lies hor-izontally in a fluid, the pressure at allpoints will be the same. The resultant forcewill then act through the centroid of thesurface. If the surface is not horizontal, thepressure on it will vary with depth. The re-sultant force will now act through a differ-ent point and the center of pressure is notat the centroid.

center of projection The point at whichall the lines forming a central projectionmeet. See central projection.

centi- Symbol: c A prefix denoting 10–2.For example, 1 centimeter (cm) = 10–2

meter (m).

central conic A conic with a center ofsymmetry; e.g. an ellipse or hyperbola.

central enlargement A central projec-tion. See also scale factor.

central force A force that acts on any af-fected object along a line to an origin. Forinstance, the motion of electric forces be-tween charged particles are central; fric-tional forces are not.

central limit theorem A result of proba-bility theory that if X1, X2, ... Xn is a ran-dom sample of size n from a distribution,which is not necessarily a normal distribu-tion, in which the finite mean is µ and thefinite variance is σ2, then as the sample sizeincreases the distribution is approximatelya normal distribution. This theorem canalso be stated in the forms that X

_= (ΣXi)/n

is approximately equal to N(µ,σ2/n) andΣ(X

_– µ)/(σ/√n) is approximately equal to

N(0,1). The size that n needs to be for thetheorem to be useful depends on how closethe distribution is to a normal distribution.If the distribution is close to a normal dis-tribution then for n = 10 it is a reasonably

Celsius degree

28

good approximation. If it is a long wayfrom being a normal distribution it is not agood approximation until well after n =100.

central processor (central processingunit; CPU) A highly complex electronic de-vice that is the nerve center of a COMPUTER.It consists of the control unit and the arith-metic and logic unit (ALU). Also some-times considered part of the centralprocessor is the main store, or memory,where a program or a section of a programis stored in binary form.

The control unit supervises all activitieswithin the computer, interpreting the in-structions that make up the program. Eachinstruction is automatically brought, inturn, from the main store and kept tem-porarily in a small store called a register.Electronic circuits analyze the instructionand determine the operation to be carriedout and the exact location or locations instore of the data on which the operation isto be performed. The operation is actuallyperformed by the ALU, again using elec-tronic circuitry and a set of registers. It maybe an arithmetical calculation, such as theaddition of two numbers, or a logical oper-ation, such as selecting or comparing data.This process of fetching, analyzing, and ex-ecuting instructions is repeated in the re-quired order until an instruction to stop isexecuted.

The size of central processors has di-minished considerably with advances intechnology. It is now possible to form acentral processor on a single silicon chip afew millimeters square in area. This tinydevice is known as a microprocessor.

central projection (conical projection) Ageometrical transformation in which astraight line from a point (called the centerof projection) to each point in the figure iscontinued to the point at which it passesthrough a second (image) plane. Thesepoints form the image of the original fig-ure. When a photographic image is createdfrom a film using an enlarger, this is thekind of PROJECTION that takes place. Thelight source is at the center of projection,the light rays are the straight lines, the film

is the first plane, and the screen or print isthe second. In this case the two planes areusually parallel, but this is not always so incentral projection.

central quadric A quadric surface that isnot a DEGENERATE QUADRIC and has a cen-ter of symmetry. This definition means thata central quadric has to be either an ellip-soid or a hyperboloid. The hyperboloidcan have either one sheet or two. See coni-coid.

centrifugal force A force supposed to actradially outward on a body moving in acurve. In fact there is no real force acting;centrifugal force is said to be a ‘fictitious’force, and the use is best avoided. The ideaarises from the effect of inertia on an objectmoving in a curve. If a car is movingaround a bend, for instance, it is forced ina curved path by friction between thewheels and the road. Without this friction(which is directed toward the center of thecurve) the car would continue in a straightline. The driver also moves in the curve,constrained by friction with the seat, re-straint from a seat belt, or a ‘push’ from thedoor. To the driver it appears that there isa force radially outward pushing his or herbody out – the centrifugal force. In fact thisis not the case; if the driver fell out of thecar he or she would move straight forwardat a tangent to the curve. It is sometimessaid that the centrifugal force is a ‘reaction’to the CENTRIPETAL FORCE – this is not true.(The ‘reaction’ to the centripetal force is anoutward push on the road surface by thetires of the car.)

centripetal force A force that causes anobject to move in a curved path rather thancontinuing in a straight line. The force isprovided by, for instance:– the tension of the string, for an objectwhirled on the end of a string;– gravity, for an object in orbit round aplanet or a star;– electric force, for an electron in the shellof an atom.

The centripetal force for an object ofmass m with constant speed v and path ra-dius r is mv2/r, or mω2r, where ω is angular

29

centripetal force

speed. A body moving in a curved path hasan acceleration because the direction of thevelocity changes, even though the magni-tude of the velocity may remain constant.This acceleration, which is directed towardthe center of the curve, is the centripetal ac-celeration. It is given by v2/r or ω2r.

centroid /sen-troid/ (mean center) Thepoint in a figure or solid at which the CEN-TER OF MASS would be if the figure or bodywere of uniform-density material. The cen-troid of a symmetrical figure is at the cen-ter of symmetry; thus, the centroid of acircle is at its center. The centroid of a tri-angle is the point at which the mediansmeet.

For non-symmetrical figures or bodiesintegration is used to find the centroid. Thecentroid of a line, figure, or solid is thepoint that has coordinates that are themean values of the coordinates of all thepoints in the line, figure, etc. For a surface,the coordinates of the centroid are givenby:

x_

= [∫∫xdxdy]/A, etc.the integration being over the surface, andA being the area. For a volume, a triple in-tegral is used to obtain the coordinates ofthe centroid:

x_

= [∫∫∫xdxdydz]/V, etc.

c.g.s. system A system of units that usesthe centimeter, the gram, and the second asthe base mechanical units. Much early sci-entific work used this system, but it hasnow almost been abandoned.

chain A former unit of length equal to 22yards. It is equivalent to 20.116 8 m.

chain rule A rule for expressing the de-rivative of a function z = f(x) in terms ofanother function of the same variable,u(x), where z is also a function of u. Thatis:

dz/dx = (dz/du)(du/dx)This is often called the ‘function of a func-tion’ rule.

For a function z = f(x1,x2,x3,…) of sev-eral variables, in which each of the vari-ables x1, x2, x3,… is itself a function of asingle variable, t, the derivative dz/dt,

called the total derivative, is given by thechain rule for partial differentiation, whichis:

dz/dt = (∂z/∂x1)(dx1/dt) +(∂z/∂x2)(dx2/dt) + …

changing the subject of a formula Re-arranging a formula so that the single termon the left-hand side (the subject) is re-placed by another term from the formula.For example, the formula for the area of asphere is A = 4πr2 (where A is the area andr is the radius). Changing the subject of theformula to r gives r = ½√(A/π).

channel A path along which informationcan travel in a computer system or com-munications system.

chaos theory The theory of systems thatexhibit apparently random unpredictablebehavior. The theory originated in studiesof the Earth’s atmosphere and the weather.In such a system there are a number of vari-ables involved and the equations describ-ing them are nonlinear. As a result, thestate of the system as it changes with timeis extremely sensitive to the original condi-tions. A small difference in starting condi-tions may be magnified and produce alarge variation in possible future states ofthe system. As a result, the system appearsto behave in an unpredictable way and mayexhibit seemingly random fluctuations(chaotic behavior). The study of such non-linear systems has been applied in a num-ber of fields, including studies of fluiddynamics and turbulence, random electri-cal oscillations, and certain types of chem-ical reaction. See also attractor.

character One of a set of symbols that canbe represented in a computer. It can be aletter, number, punctuation mark, or a spe-cial symbol. A character is stored or ma-nipulated in the computer as a group ofBITS (i.e. binary digits). See also byte; word;store.

characteristic 1. See logarithm.2. See eliminant.

centroid

30

chi-square distribution /kÿ-skwair/ (χ2

distribution) The distribution of the sumof the squares of random variables withstandard normal distributions. For exam-ple, if x1, x2,…xi,… are independent vari-ables with standard normal distribution,then

χ2 = ∑xi2

has a chi-square distribution with n de-grees of freedom, written χn

2. The meanand variance are n and 2n respectively. Thevalues χn

2(α) for which P(χ2 ≤ χn2(α)) = α

are tabulated for various values of n.

chi-square test A measure of how well atheoretical probability distribution fits aset of data. For i = 1, 2, … m the value xioccurs oi times in the data and the theorypredicts that it will occur ei times. Providedei ≥ 5 for all values of i (otherwise valuesmust be combined), then

χ2 = ∑ (oi – ei)2/eihas a chi-square distribution with n de-grees of freedom. See also chi-square distri-bution.

choice, axiom of The axiom states that,given any collection of sets, one can form anew set by choosing one element fromeach. This axiom may seem intuitively ob-vious and it was presupposed in many clas-sical mathematical works. However, it hasbeen a point of debate and controversysince many of its consequences appeared tobe paradoxical. An example is the Banach-Tarski theorem, which proves that it is pos-sible to cut a solid sphere into a finitenumber of pieces and to reassemble thesepieces to form two solid spheres the samesize as the original sphere. Despite theseapparent paradoxes the axiom is widelyaccepted. It has many equivalents includ-ing the well-ordering principle and ZORN’SLEMMA.

chord A straight line joining two pointson a curve, for example, the line segmentjoining two points on the circumference ofa circle.

chord of contact The line that connectsthe points of contact between two tangentsto a curve from a point P (p,q). For a conic

the chord of contact is also known as thepolar of P with respect to the conic and thepoint P is known as the pole. For a generalconic section described by the equation

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0,the equation of the chord of contact for P(p,q) is

apx + h(py + qx) + bqy + g(p + x) +f(q + y) + c = 0.

circle The plane figure formed by a closedcurve consisting of all the points that are ata fixed distance (the radius, r) from a par-ticular point in the plane; the point is thecenter of the circle. The diameter of a circleis twice its radius; the circumference is 2πr;and its area is πr2. In Cartesian coordi-nates, the equation of a circle centred at theorigin is

x2 + y2 = r2

The circle is the curve that encloses thelargest possible area within any givenperimeter length. It is a special case of anellipse with eccentricity 0.

circular argument An argument that,tacitly or explicitly, assumes what it is try-ing to prove, and is consequently invalid.

circular cone A CONE that has a circularbase.

circular cylinder A CYLINDER in whichthe base is circular.

circular functions See trigonometry.

31

circular functions

γ x rγ x rγ x r

γ

rrr

Circular measure: in a circle of radius r and cir-cumference 2πr, the angle γ radians subtendsan arc length γ × r.

circular measure The measurement of anangle in radians.

circular mil See mil.

circular motion A form of periodic (orcyclic) motion; that of an object moving ina circular path. For this to be possible, apositive central force must act. If the objecthas a uniform speed v and the radius of thecircle is r, the angular velocity (ω) is v/r.There is an acceleration toward the centerof the circle (the centripetal acceleration)equal to v2/r or ω2r. See also centripetalforce; rotational motion.

circumcenter /ser-kŭm-sen-ter/ See cir-cumcircle.

circumcircle /ser-kŭm-ser-kăl/ (circum-scribed circle) The circle that passesthrough all three vertices of a triangle orthrough the vertices of any other cyclicpolygon. The figure inside the circle is saidto be inscribed. The point in the figure thatis the center of the circle is called the cir-cumcenter. For a triangle with side lengthsa, b, and c the radius r of the circumcircleis given by:

r = abc/4√[s(s – a)(s – b)(s – c)]where s is (a + b + c)/2.

circular measure

32

•

•

secant

tangentdiameter

radius

chord

segmentsegmentsegment

sectorsectorsector

•

E

D

CA

B

xx

x

Angles in thesame segmentof a circle areequal.

Properties of circles

33

circumcircle

An angle that an arc subtends at the center of a circle is twice the angle that it subtends at thecircumference: ACB = 2ADBAn angle in a semicircle is a right angle: XPY (=½XCY) = 90o

∧ ∧∧ ∧

P

Y

B

A

X

D

C

A

DC

B

P

Two tangents from an external point:(1) are equal, PA = PB(2) subtend equal angles at the center, PCA = PCB(3) the line from the point to the center bisects the line AB, AD = DB

∧∧

A tangent and a secant from an external point: PC.PB = PA2

Two intersecting chords: FX.GX = DX.XE

P

B

C

G

D

X

E

A

Properties of circles

circumference The boundary, or lengthof the boundary, of a closed curve, usuallya circle. The circumference of a circle isequal to 2πr, where r is the radius of thecircle.

circumscribed Describing a geometricfigure that is drawn around and enclosinganother geometrical figure. For example,in a square, a circle can be drawn throughthe vertices. This is called the circum-scribed circle, and the square is called theinscribed square of the circle. Similarly, aregular polyhedron might have a circum-scribed sphere, and a rectangular pyramida circumscribed cone. Compare inscribed.

cissoid /sis-oid/ A curve that is defined bythe equation y2 = x3/2(a–x). It has the linex = 2a as an ASYMPTOTE and a CUSP at theorigin.

class 1. A grouping of data that is taken asone item in a FREQUENCY TABLE or HIS-TOGRAM.2. Often used simply as a synonym for set.However, in set theory it is sometimes de-sirable, to avoid paradoxes, to allow theexistence of collections that are not sets.Such collections are known as classes orproper classes. For example, the collectionof all sets is a proper class not a set.

classical mechanics A system of me-chanics that is based on Newton’s laws ofmotion. Relativity effects and quantumtheory are not taken into account in classi-cal mechanics.

class mark See frequency table.

clock pulse One of a series of regularpulses produced by an electronic devicecalled a clock and are used to synchronizeoperations in a computer. Every instruc-tion in a computer program causes a num-ber of operations to be done by theCENTRAL PROCESSOR of the computer. Eachof these operations, performed by the con-trol unit or arithmetic and logic unit, aretriggered by one clock pulse and must becompleted before the next clock pulse. Theinterval at which the pulses occur is usually

a few microseconds (millionths of a sec-ond).

clockwise Rotating in the same sense asthe hands of a clock. For example, the headof an ordinary screw is turned clockwise(looking at the head of the screw) to driveit in. Looking at the other end the rotationappears to be anticlockwise (counterclock-wise).

closed Describing a set for which a givenoperation gives results in the same set. Forexample, the set of positive integers isclosed with respect to addition and multi-plication. Adding or multiplying any twomembers gives another positive integer.The set is not closed with respect to divi-sion since dividing certain integers does notgive a positive integer (e.g. 4/5). The set ofpositive integers is also not closed with re-spect to subtraction (e.g. 5–7 = –2). Seealso closed interval; closed set.

closed curve (closed contour) A curve,such as a circle or an ellipse, that forms acomplete loop. It has no end points. A sim-ple closed curve is a closed curve that doesnot cross itself. Compare open curve.

closed interval A set consisting of thenumbers between two given numbers (endpoints), including the end points. For ex-ample, all the real numbers greater than orequal to 2 and less than or equal to 5 con-stitute a closed interval. The closed intervalbetween two real numbers a and b is writ-ten [a,b]. On a number line the end pointsare marked by a blacked-in circle. Com-pare open interval. See also interval.

closed set A set in which the limits thatdefine the set are included. The set of ra-tional numbers greater than or equal to 0and less than or equal to ten, written x:0≤x≤10; x ∈R, and the set of points onand within a circle are examples of closedsets. Compare open sets.

closed surface A surface that has noboundary lines or curves, for example asphere or an ellipsoid.

circumference

34

closed system (isolated system) A set ofone or more objects that may interact witheach other, but do not interact with theworld outside the system. This means thatthere is no net force from outside or energytransfer. Because of this the system’s angu-lar momentum, energy, mass, and linearmomentum remain constant.

closure See group.

cluster point See accumulation point.

coaxial /koh-aks-ee-ăl/ 1. Coaxial circlesare circles such that all pairs of the circleshave the same RADICAL AXIS.2. Coaxial planes are planes that passthrough the same straight line (the axis).

COBOL /koh-bôl/ See program.

coding The writing of instructions in acomputer programming language. The per-son doing the coding starts with a writtendescription or a diagram representing thetask to be carried out by the computer.This is then converted into a precise andordered sequence of instructions in the lan-guage selected. See also flowchart; pro-gram.

coefficient /koh-i-fish-ĕnt/ A multiplyingfactor. For example, in the equation 2x2 +3x = 0, where x is a variable, the coefficientof x2 is 2 and the coefficient of x is 3. Some-times the value of the coefficients is notknown, although they are known to stayconstant as x changes, for example, ax2 +bx = 0. In this case a and b are constant co-efficients. See also constant.

coefficient, binomial See binomial coef-ficient.

coefficient of friction See friction.

coefficient of restitution See restitution,coefficient of.

cofactor /koh-fak-ter/ The DETERMINANT

of the matrix obtained by removing therow and column containing the element.The matrix formed by all the cofactors of

the elements in a matrix is called the ad-joint of the matrix.

coherent units A system or subset ofunits (e.g. SI units) in which the derivedunits are obtained by multiplying or divid-ing together base units, with no numericalfactor involved.

colatitude /koh-lat-ă-tewd/ See sphericalpolar coordinates.

collinear /kŏ-lin-ee-er/ Lying on the samestraight line. Any two points, for example,could be said to be collinear because thereis a straight line that passes through both.Similarly, two vectors are collinear if theyare parallel and both act through the samepoint.

35

collinear

A =a b cd e fg h i

The cofactors a’, b’, and c’ of the elementsa, b, and c in a 3 x 3 matrix A.

The adjoint of A.

a’ b’ c’d’ e’ f’g’ h’ i’

a’ =e fh i

= ei – hf

b’ =d fg i

= di – gf

c’ =d eg h

= dh – ge

Cofactor

cologarithm /koh-lôg-ă-rith-<m/ TheLOGARITHM of the reciprocal of a givennumber; i.e. the negative of the logarithm.It is sometimes used in logarithmic compu-tation to avoid the use of negative mantis-sas or of subtraction of logarithms.

column matrix See column vector.

column vector (column matrix) A num-ber (m) of quantities arranged in a singlecolumn; i.e. an m × 1 matrix. For example,the vector that defines the displacement ofthe point (x,y,z) from the origin of a Carte-sian coordinate system is usually written asa column vector.

combination Any subset of a given set ofobjects regardless of the order in whichthey are selected. If r objects are selectedfrom n, and each object can only be chosenonce, the number of different combina-tions is

n!/[r!(n–r)!]written as nCr or C(n,r). For instance, ifthere are 15 students in a class and only 5books, then each book has to be shared by3 students. The number of ways in whichthis can happen – i.e. the number of com-binations of 3 from 15 – is 15!/3!12!, or455. If each object can be selected morethan once the number of different combi-nations is n+r–1Cr. See also factorial, permu-tation.

combinatorics /kŏm-bÿ-nă-tor-iks, -toh-riks/ (combinatorial analysis) The branchof mathematics that studies the number ofpossible configurations or arrangements ofa certain type. It forms the basis for the

theory of probability since we have toknow how to calculate the total number ofdifferent ways an event can happen beforewe can hope to predict how it is likely tohappen. There are many unsolved prob-lems in combinatorics that at first appearsimple. For example, a rectangular grid ofsome fixed dimension, m × n, is made up ofunit squares each of one of two colors.How many different color patterns arethere if the number of boundary edges be-tween the two colors is a certain fixednumber? This problem was completelysolved in two dimensions in the 1960s, butthe solution for the three-dimensionalproblem is still unknown.

commensurable /kŏ-men-shŭ-ră-băl/Able to be measured in the same way andin terms of the same units. For example, a30-centimeter rule is commensurable witha 1-meter length of rope, because both canbe measured in centimeters. Neither iscommensurable with an area.

common denominator A whole numberthat is a common multiple of the denomi-nators of two or more fractions. For exam-ple, 6 and 12 are both commondenominators of 1/2 and 1/3. The lowest(or least) common denominator (LCD) isthe smallest number that is a common mul-tiple of the denominators of two or morefractions. For example, the LCD of 1/2,1/3, and 1/4 is 12. Fractions are put interms of the LCDs when they are to beadded or subtracted:

1/2 + 1/3 + 1/4 = 6/12 + 4/12 + 3/12 = 13/12

common difference The difference be-tween successive terms in an arithmetic se-quence or arithmetic series.

common factor 1. A whole number thatdivides exactly into two or more givennumbers. For example, 7 is a common fac-tor of 14, 49, and 84. Since 7 is the largestnumber that divides into all three exactly,it is the highest common factor (HCF). Seealso factor.2. A number or variable by which severalparts of an expression are multiplied. For

cologarithm

36

The column vector that defines thedisplacement of a point (x, y, z) from theorigin of a Cartesian coordinate system.

xyz

Column vector

example, in 4x2 + 4y2, 4 is a common fac-tor of x2 and y2, and from the distributivelaw for multiplication and addition,

4x2 + 4y2 = 4(x2 + y2)

common fraction See fraction.

common logarithm See logarithm.

common multiple A whole number thatis a multiple of each of a group of numbers.For example, 100 is a common multiple of5, 25, and 50. The lowest (or least) com-mon multiple (LCM) is the smallest num-ber that is a common multiple; in this caseit is 50.

common ratio The ratio of successiveterms in a geometric sequence or geometricseries.

common tangent A single line that formsa tangent to two or more separate curves.The term is also used for the length of theline joining the two tangential points.

commutative /kŏ-myoo-tă-tiv, kom-yŭ-tay-tiv/ Denoting an operation that is in-dependent of the order of combination. Abinary operation • is commutative if a•b =b•a for all values of a and b. In ordinaryarithmetic, multiplication and addition arecommutative operations. This is sometimesreferred to as the commutative law of mul-tiplication and the commutative law of ad-dition. Subtraction and division are notcommutative operations. See also associa-tive; distributive.

commutative group See Abelian group.

compact A set S of real numbers is com-pact if, given any collection of open setswhose union contains S, we can find a fi-nite subcollection of those open sets whoseunion also contains S. The concept can begeneralized to any topological space, andalso to mathematical logic. In logic a for-mal system is said to be compact if it issuch that, when a given sentence is a logi-cal consequence of a given set of sentences,it is a consequence of some finite subset ofthem.

compasses An instrument used for draw-ing circles. It consists of two rigid armsjoined by a hinge. At one end is a sharppoint, which is placed at the center of thecircle. At the other end is a pencil or othermarker, which traces out the circumferencewhen the compasses are pivoted aroundthe point. In a beam compass, used fordrawing large circles, the sharp point andthe marker are attached to opposite ends ofa horizontal beam.

compiler See program.

complement The set of all the elementsthat are not in a particular set. If the set A= 1, 2, 3, and the universal set, E, is takenas containing all the natural numbers, thenthe complement of A, written A′ or , is 4,5, 6, …. See Venn diagram.

complementary angles A pair of anglesthat add together to make a right angle(90° or π/2 radians). Compare conjugateangles; supplementary angles.

37

complementary angles

EEE

A

A’A’A’

The shaded area in the Venn diagram is thecomplement A′ of the set A.

αβ

Complementary angles: α + β = 90°

complementary function See auxiliaryequation.

complete In mathematical logic a formalsystem is said to be complete if every truesentence in the system is also provablewithin the system. Not all logical systemshave this property. See Gödel’s incomplete-ness theorem.

completing the square A way of solvinga QUADRATIC EQUATION, by dividing bothsides by the coefficient of the square termand adding a constant, in order to expressthe equation as a single squared term. Forexample, to solve 3x2 + 6x + 2= 0:

x2 + 2x + ⅔ = 0(x + 1)2 – 1 + ⅔ = 0x + 1 = +√⅓ or –√⅓

x = –1 + √⅓ or –1 – √⅓

complex analysis The branch of analysisthat is specifically concerned with COMPLEX

FUNCTIONS. Some of the results and tech-niques of real analysis can be extended tocomplex analysis but many results arisespecifically because of the mathematicalstructure imposed by dealing with complexfunctions. Complex analysis is used in thetheory of ASYMPTOTIC SERIES and has manyapplications in the physical sciences and inengineering.

complex fraction See fraction.

complex function A FUNCTION in whichthe variable quantities are complex num-bers. See complex analysis.

complexity theory The mathematicaltheory of systems that can exhibit complexbehavior. It includes such topics as chaostheory, ideas about how organization andorder can emerge in systems, and howemergent laws of phenomena arise at dif-ferent levels of description. Complexitytheory is a subject of great interest to math-ematicians, physicists, chemists, biologists,and economists, amongst others. Comput-ers are used extensively in complexitytheory since the systems being analyzed donot, in general, have exact solutions. At thetime of writing, there is not yet a consensus

as to whether there are general ‘laws ofcomplexity’, which are applicable to allcomplex systems.

complex number A number that hasboth a real part and an imaginary part. Theimaginary part is a multiple of the squareroot of minus one (i). Some algebraic equa-tions cannot be solved with real numbers.For example, x2 + 4x + 6 = 0 has the solu-tions x = –2 + √(–2) and x = –2 – √(–2). Ifthe number system is extended to include i= √–1, all algebraic equations can besolved. In this case the solutions are x = –2+ i√2 and x = –2 – i√2. The real part is –2and the imaginary part is +i√2 or –i√2.

Complex numbers are sometimes repre-sented on an Argand diagram, which issimilar to a graph in Cartesian coordinates,but with the horizontal axis representingthe real part of the number and the verticalaxis the imaginary part. The diagram isnamed for the Swiss mathematician JeanArgand (1768–1822).

Any complex number can also be writ-ten as a function of an angle θ, just asCartesian coordinates can be convertedinto polar coordinates. Thus r(cosθ +i sinθ) is equivalent to x + iy, where x =rcosθ and y = rsinθ. Here, r is the modulusof the complex number and θ is the argu-ment (or amplitude). This can also be writ-ten in the exponential form r = eiθ.

component /kŏm-poh-nĕnt/ The resolvedpart of a VECTOR in a particular direction,

complementary function

38

| | | | |1 2 3 4 5

x

•r

4 –

3 –

2 –

1 –

0#

P

iy

The point P(4,3) on an Argand diagram repre-sents the complex number z = 4 + 3i. In thepolar form z = r (cos θ + i sin θ).

often one of two components at right an-gles.

component forces See component vec-tors.

component vectors The components ofa given VECTOR (such as a force or velocity)are two or more vectors with the same ef-fect as the given vector. In other words thegiven vector is the resultant of the compo-nents. Any vector has an infinite number ofsets of components. Some sets are more usethan others in a given case, especially pairsat 90°. The component of a given vector(V) in a given direction is the projection ofthe vector on to that direction; i.e. Vcosθ,where θ is the angle between the vector andthe direction.

component velocities See componentvectors.

composite function A function formedby combining two functions f and g. Thereare several notations for such a compositefunction: for example, g[f(x)], gof, g.f, gf.For a composite function to exist it is nec-essary that the domain of g contains therange of f. The order of the functions f andg in a composite function has to be speci-fied since, in general, g[f(x)] ≠ f[g(x)]. Thismeans that to find the composite functiong[f(x)], f(x) is found first and then g[f(x)] isfound. The non-commuting nature of com-posite functions can be seen by consideringthe functions f(x) = sinx and g(x) = x2 forwhich g[f(x)] = (sinx)2 and f[g(x)] = sin(x2).

composite number An integer that hasmore than one prime factor. For example,4 (= 2 × 2), 6 (= 2× 3), 10 (= 2 × 5) are com-posite numbers. The prime numbers and±1 are not composite.

composition The process of combiningtwo or more functions to obtain a new one.For example the composition of f(x) andg(x), which is written f•g, is obtained byapplying g(x) and then applying f(x) to theresult. If f(x) = x – 2 and g(x) = x3 + 1 thenf•g(x) = f(x3+1) = x3 – 1, whereas g•f(x) =g(x–2) = (x – 2)3 + 1. As can be seen, these

two resulting functions are not the same. Ingeneral, composing functions in a differentorder will produce different results.

compound growth The growth ofmoney that is associated with COMPOUND

INTEREST. Compound growth can be tabu-lated in tables called compound growth ta-bles, which give the multiplying factors fordifferent periods of time (usually years)and different percentage rates. These tablescan be drawn in the form of a matrix, withthe number of years increasing from left toright and the percentage rates increasingfrom top to bottom. Compound growth ta-bles can also be used to record the growthin population with time for different popu-lation growth rates.

compound interest The interest earnedon capital, when the interest in each periodis added to the original capital as it isearned. Thus the capital, and therefore theinterest on it, increases year by year. If P isthe principal (the original amount ofmoney invested), R percent the interest rateper annum, and n the number of interestperiods, then the compound interest is

P(1 + R/100)n

This formula is a geometric progressionwhose first term is P (when n = 0) andwhose common ratio is (1 + R/100). Com-pare simple interest.

compound proposition See proposition.

compound transformation A transfor-mation on a figure that can be regarded asone transformation followed by anothertransformation. For example, a compoundtransformation could consist of a rotationfollowed by a translation through space.Another example is a translation followedby a reflection. If the first and second trans-formations on the object O are denoted T1and T2 respectively, then the compoundtransformation is denoted by T2T1(0). Ingeneral, performing the two transforma-tions in different order leads to two differ-ent compound transformations. This isconsistent with the fact that transforma-tions can be described by matrices.

39

compound transformation

computability /kŏm-pyoo-tă-bil-ă-tee/Intuitively, a problem or function is com-putable if it is capable of being solved by anideal machine (computer) in a finite time.In the 1930s it was discovered that someproblems had no algorithmic solution andcould not be solved by computers. This ledmany mathematicians to try to formulate aprecise definition of the intuitive conceptof computability, and Turing, Gödel, andChurch independently came up with threevery different abstract definitions, whichall turned out to define exactly the same setof functions. The definition given by Tur-ing is that a function f is computable if foreach element x of its domain, when somerepresentation of x is placed on the tape ofa TURING MACHINE, the machine stops in afinite time with a representation of f(x) onthe tape.

computer Any automatic device or ma-chine that can perform calculations andother operations on data. The data must bereceived in an acceptable form and isprocessed according to instructions. Themost versatile and most widely used com-puter is the digital computer, which is usu-ally referred to simply as a computer. Seealso analog computer; hybrid computer.

A digital computer is an automaticallycontrolled calculating machine in which in-formation, generally known as data, is rep-resented by combinations of discreteelectrical pulses denoted by the binary dig-its 0 and 1. Various operations, both arith-metical and logical, are performed on thedata according to a set of instructions (aprogram). Instructions and data are fedinto the main store or memory of the com-puter, where they are held until required.The instructions, coded like the data in bi-nary form, are analyzed and carried out bythe central processor of the computer. Theresult of this processing is then delivered tothe user.

The technology used in digital comput-ers is so highly advanced that they can op-erate at extremely high speeds and canstore a huge amount of information. Thetube valves used in early computers werereplaced by transistors; transistors, resis-tors, etc., were subsequently packed into

integrated circuits, which have becomemore and more complicated. As the elec-tronic circuits used in the various devices ina computer system have decreased in sizeand increased in complexity, so the com-puters themselves have grown smaller,faster, and more powerful. The microcom-puter has been developed as a somewhatsimpler version of the full-size mainframecomputer. Computers now have an im-mense range of uses in science, technology,industry, commerce, education, and manyother fields.

computer graphics The creation and re-production of pictures, photographs, anddiagrams using a computer. There aremany different formats for storing imagesbut they fall into two main classes. In rastergraphics the picture is stored as a series ofdots (or pixels). The information in thecomputer file is a stream of data indicatingthe presence or absence of a dot and thecolor if present. Images of this type aresometimes known as bitmaps. This formatis used for high-quality artwork and forphotographs. Diagrams are more conve-niently stored using vector graphics, inwhich the information is stored as mathe-matical instructions. For example, it is pos-sible to specify a circle by its center, itsradius, and the thickness of the line form-ing the circumference. More complicatedcurves are usually drawn using BEZIER

CURVES. Vector images are easier to changeand take up less storage space than rasterimages.

computer modeling The development ofa description or mathematical representa-tion (i.e. a model) of a complicated processor system, using a computer. This modelcan then be used to study the behavior orcontrol of the process or system by varyingthe conditions in it, again with the aid of acomputer.

concave /kong-kayv, kong-kayv/ Curvedinwards. For example, the inner surface ofa hollow sphere is concave. Similarly intwo dimensions, the inside edge of the cir-cumference of a circle is concave. A con-cave polygon is a polygon that has one (or

computability

40

more) interior angles greater than 180°.Compare convex.

concentric Denoting circles or spheresthat have the same center. For example, ahollowed out sphere consists of two con-centric spherical surfaces. Compare eccen-tric.

conclusion The proposition that is as-serted at the end of an argument; i.e. whatthe argument sets out to prove.

condition In logic, a proposition or state-ment, P, that is required to be true in orderthat another proposition Q be true. If P isa necessary condition then Q could not betrue without P. If P is a sufficient condi-tion, then whenever P is true Q is also true,but not vice versa. For example, for aquadrilateral to be a rectangle it must sat-isfy the necessary condition that two of itssides be parallel, but this is not a sufficientcondition. A sufficient condition for aquadrilateral to be a rhombus is that all itssides have a length of 5 centimeters, butthis is not a necessary condition. For a rec-tangle to be a square it is both a necessaryand a sufficient condition that all its sidesare of equal length.

In formal terms, if P is a necessary con-dition for Q, then Q → P. If P is a sufficientcondition, then P → Q. If P is a necessaryand sufficient condition for Q then P ≡ Q.See also biconditional, symbolic logic.

conditional (conditional statement; con-ditional proposition) An if… then… state-ment.

conditional convergence See absoluteconvergence.

conditional equation See equation.

conditional probability See probability.

cone A solid defined by a closed planecurve (forming the base) and a point out-side the plane (the vertex). A line segmentfrom the vertex to a point on the planecurve generates a curved lateral surface asthe point moves around the plane curve.The line is the generator of the cone andthe plane curve is its directrix. Any line seg-ment from the vertex to the directrix is anelement of the cone.

If the directrix is a circle the cone is acircular cone. If the base has a center, a linefrom the vertex to this is an axis of thecone. If the axis is at right angles to thebase the cone is a right cone; otherwise it isan oblique cone. The volume of a cone isone third of the base area multiplied by thealtitude (the perpendicular distance fromthe vertex to the base). For a right circularcone

V = πr2h/3where r is the radius of the base and h thealtitude. The area of the curved (lateral)surface of a right circular cone is πrs, wheres is the length of an element (the slantheight).

41

cone

convex concave

Concave and convex curvatures

hyperbolaellipse

parabola

The three sections of a cone – the ellipse, theparabola, and the hyperbola.

If an extended line is used to generatethe curved surface (i.e. extending beyondthe directrix and beyond the vertex), an ex-tended surface is produced with two parts(nappes) each side of the vertex. Morestrictly, this is called a conical surface

confidence interval An interval that isthought, with a preselected degree of con-fidence, to contain the value of a parame-ter being estimated. For example, in abinomial experiment the α% confidenceinterval for the probability of success P liesbetween P – a and P + a, where

a = z√[P(1 – P)/N]N is the sample size, P the proportion ofsuccesses in the sample, and z is given by atable of area under the standard normalcurve. P will lie in this interval α times outof every 100.

confocal conics /kon-foh-kăl/ Two ormore conics that have the same focus.

conformable matrices See matrix.

conformal mapping A geometricaltransformation that does not change theangles of intersection between two lines orcurves. For example, Mercator’s projectionis a conformal mapping in which any anglebetween a line on the spherical surface anda line of latitude or longitude will be thesame on the map.

congruence /kong-groo-ĕns/ The propertyof being congruent.

congruent /kong-groo-ĕnt/ 1. Denotingtwo or more figures that are identical insize and shape. Two congruent plane fig-ures will fit into the area occupied by eachother; i.e. one could be brought into coin-cidence with the other by moving it with-out change of size. Two circles arecongruent if they have the same radius. Theconditions for two triangles to be congru-ent are: 1. Two sides and the includedangle of one are equal to two sides and theincluded angle of the other.2. Two angles and the included side of oneare equal to two angles and the includedside of the other.

3. Three sides of one are equal to threesides of the other.

In solid geometry, two figures are con-gruent if they can be brought into coinci-dence in space.

Sometimes the term directly congruentis used to describe identical figures; indi-rectly congruent figures are ones that aremirror images of each other. Compare sim-ilar.2. Two elements a and b of a ring are con-gruent modulo d if there exist elements inthe ring, p, q, and r, such that a = dp + r, b= dq + r. Intuitively, this means that theyboth leave the same remainder when di-vided by d.3. Two square matrices A and B are con-gruent if A can be transformed into B by acongruent transformation; i.e. there existsa nonsingular matrix C such that B =CTAC, where CT is the transpose of C.

conic /kon-ik/ A type of plane curve de-fined so that for all points on the curve thedistance from a fixed point (the focus) hasa constant ratio to the perpendicular dis-tance from a fixed straight line (the direc-trix). The ratio is the eccentricity of theconic, e; i.e. the eccentricity is the distancefrom curve to focus divided by distancefrom curve to directrix.

The type of conic depends on the valueof e: when e is less than 1 it is an ELLIPSE;when e equals 1 it is a PARABOLA; when e isgreater than 1 it is a HYPERBOLA. A circle isa special case of an ellipse with eccentricity0.

The original definition of conics was asplane sections of a conical surface – hencethe name conic section. In a conical surfacehaving an apex angle of 2θ, the cross-sec-tion on a plane that makes an angle θ withthe axis of the cone, (i.e. a plane parallel tothe slanting edge of the cone) is a parabola.A cross-section in a plane that makes anangle greater than θ with the axis is an el-lipse. A cross-section in a plane making anangle less than θ with the axis is a hyper-bola and because this plane cuts bothhalves (nappes) of the cone, the hyperbolahas two arms.

confidence interval

42

There are various ways of writing theequation of a conic. In Cartesian coordi-nates:

(1 – e2)x2 + 2e2qx + y2 = e2qwhere the focus is at the origin and the di-rectrix is the line x = q (a line parallel to they-axis a distance q from the origin). Thegeneral equation of a conic (i.e. the generalconic) is:

ax2 + bxy + cy2 + dx + ey + f = 0where a, b, c, d, e, and f are constants (heree is not the eccentricity). This includes de-generate cases (degenerate conics), such asa point, a straight line, and a pair of inter-secting straight lines. A point, for example,is a section through the vertex of the coni-cal surface. A pair of intersecting straightlines is a section down the axis of the sur-face. The tangent to the general conic at thepoint (x1,y1) is:

ax1x + b(xy1 + x1y) + cy1y +d(x + x1) + e(y + y1) + f = 0

conical helix See helix.

conical projection See central projec-tion.

conical surface See cone.

conicoid A type of surface in which sec-tions of the surface are conics. Conicoidscan have equations in three-dimensionalCartesian coordinates as follows:elliptic paraboloid

x2/a2 + y2/b2 = 2z/chyperbolic paraboloid

x2/a2 – y2/b2 = 2z/chyperboloid

x2/a2 + y2/b2 – z2/c2 = 1ellipsoid

x2/a2 + y2/b2 + z2/c2 = 1Conicoids are also known as quadric

surfaces or quadrics. The simplest types areconicoids of revolution, formed by rotatinga conic about an axis. A sphere is a specialcase of an ELLIPSOID in which all three axesare equal (just as a circle is a special case ofan ellipse). See also paraboloid. See illus-tration overleaf.

conic sections See conic.

conjugate angles A pair of angles thatadd together to make a complete revolu-tion (360° or 2π radians). Compare com-plementary angles; supplementary angles.

conjugate axis See hyperbola.

conjugate complex numbers Two com-plex numbers of the form x + iy and x – iy,which when multiplied together have a realproduct x2 + y2. If z = x + iy, the complexconjugate of z is z

_= x – iy.

conjugate diameter Two diameters ABand CD of an ellipse are said to be conju-gate diameters if CD is the diameter thatcontains the mid-points of the set of chordsparallel to AB, where a diameter of an el-lipse is defined to be a locus of the mid-points of a set of parallel chords. For anequation for an ellipse of the form x2/a2 +y2/b2 = 1 the product of the gradients oftwo conjugate diameters is –b2/a2. Thismeans that, in general, conjugate diametersare not perpendicular (except in the case ofa = b; i.e. a circle).

conjugate hyperbola See hyperbola.

conjunction /kŏn-junk-shŏn/ Symbol: ∧In logic, the relationship and between two

43

conjunction

αβ

Conjugate angles: α + β = 360°

P Q P ∧ Q

T T TT F FF T FF F F

Conjunction

connected

44

hyperboloid of two sheets

hyperboloid of one sheet

ellipsoid

hyperbolic paraboloid

elliptic paraboloid

Conicoid surfaces

or more propositions or statements. Theconjunction of P and Q is true when P istrue and Q is true, and false otherwise. Thetruth table definition for conjunction isshown in the illustration. Compare dis-junction. See also truth table.

connected Intuitively, a connected set is aset with only one piece. More rigorously, aset in a topological space is said to be con-nected if it is not the union of two non-empty disjoint closed sets. For example,the set of rational numbers is not con-nected since the set of all rational numbersless than √3 and the set of all rational num-bers greater than √3 are both closed in theset of all rational numbers. However, theset of all real numbers is connected since nosuch decomposition is possible. A subset Sof a topological space is path-connected ifany two points in S can be joined by a pathin S, where a path from a to b in S is a con-tinuous map f:[0,1]→S such that f(0)=aand f(1)=b.

A simply connected set is a path-con-nected set such that any closed curvewithin it can be deformed continuously toa point of the set without leaving the set. Apath-connected set that is not simply con-nected is multiply connected and the con-nectivity of the set is one plus themaximum number of points that can bedeleted if the set is a curve, or one plus themaximum number of closed cuts that canbe made if the set is a surface, without sep-arating the set so that it is no longer path-connected. For example, the regionbetween two concentric circles has connec-tivity two, or is doubly connected, sinceone closed cut can be made which stillleaves a connected region.

connected particles Particles that areconnected by a light inextensible string. Ifparticles are connected then their motionsare not independent. An example is themotion of two particles joined by a light in-extensible string which passes over a fixedpulley. If the two particles have masses m1and m2 then the heavier particle moves up.If m2 > m1, the acceleration down of m2(and the acceleration up of m1) = a, the ten-sion in the string is denoted T and the ac-

celeration due to gravity is denoted g then:a = [(m2 – m1)g]/(m1 + m2) and T =2m1m2g/(m1 + m2).

connectivity The number of cuts neededto break a shape in two parts. For example,a rectangle, a circle, and a sphere, all havea connectivity of one. A flat disk with ahole in it or a torus has a connectivity oftwo. See also topology.

consequent In logic, the second part of aconditional statement; a proposition orstatement that is said to follow from or beimplied by another. For example, in thestatement ‘if Jill is happy, then Jack ishappy’, ‘Jack is happy’ is the consequent.Compare antecedent. See also implication.

conservation law A law stating that thetotal value of some physical quantity isconserved (i.e. remains constant) through-out any changes in a closed system. Theconservation laws applying in mechanicsare the laws of constant mass, constant en-ergy, constant linear momentum, and con-stant angular momentum.

conservation of angular momentum,law of See constant angular momentum;law of.

conservation of energy, law of See con-stant energy; law of.

conservation of linear momentum,law of See constant linear momentum;law of.

conservation of mass, law of See con-stant mass; law of.

conservation of mass and energy Thelaw that the total energy (rest mass energy+ kinetic energy + potential energy) of aclosed system is constant. In most chemicaland physical interactions the mass changeis undetectably small, so that the measur-able rest-mass energy does not change (it isregarded as ‘passive’). The law then be-comes the classical law of conservation ofenergy. In practice, the inclusion of mass inthe calculation is necessary only in the case

45

conservation of mass and energy

of nuclear changes or systems involvingvery high speeds. See also mass-energyequation; rest mass.

conservation of momentum, law ofSee constant linear momentum; law of.

conservative field A field such that thework done in moving an object betweentwo points in the field is independent of thepath taken. See conservative force.

conservative force A force such that, if itmoves an object between two points, theenergy transfer (work done) does not de-pend on the path between the points. Itmust then be true that if a conservativeforce moves an object in a closed path(back to the starting point), the energytransfer is zero. Gravitation is an exampleof a conservative force; friction is a non-conservative force.

consistent Describing a theory, system, orset of propositions giving rise to no contra-dictions. Arithmetic, for example, isthought to be a consistent logical systembecause none of its axioms nor any of thetheorems that are derived from these by therules, are believed to be contradictory. Seecontradiction.

consistent equations A set of equationsthat can be satisfied by at least one set ofvalues for the variables. For example, theequations x + y = 2 and x + 4y = 6 are sat-isfied by x = 2/3 and y = 4/3 and they aretherefore consistent. The equations x + y =4 and x + y = 9 are inconsistent.

constant A quantity that does not changeits value in a general relationship betweenvariables. For example, in the equation y =2x + 3, where x and y are variables, thenumbers 2 and 3 are constants. In this casethey are absolute constants because theirvalues never change. Sometimes a constantcan take any one of a number of values indifferent applications of a general formula.In the general quadratic equation

ax2 + bx + c = 0a, b, and c are arbitrary constants becauseno values are specified for them. An INDEF-

INITE INTEGRAL includes an arbitrary con-stant (the constant of integration), whichdepends on the limits chosen.

constant angular momentum, law of(law of conservation of angular momen-tum) The principle that the total angularmomentum of a system cannot change un-less a net outside torque acts on the system.See also constant linear momentum; lawof.

constant energy, law of (law of conser-vation of energy) The principle that thetotal energy of a system cannot change un-less energy is taken from or given to theoutside. See also mass-energy equation.

constant linear momentum, law of(law of conservation of linear momentum)The principle that the total linear momen-tum of a system cannot change unless a netoutside force acts.

constant mass, law of (law of conserva-tion of mass) The principle that the totalmass of a system cannot change unlessmass is taken from or given to the outside.See also mass-energy equation.

constant momentum, law of See con-stant linear momentum; law of.

construct In geometry, to draw a figure,line, point, etc., meeting certain conditions;e.g. a line that bisects a given line. Usuallycertain specific restrictions are imposed onthe method used; e.g. using only a straightedge and compasses. There is an importantclass of problems concerning questions ofwhether certain things can be constructedusing given methods. Examples are twocelebrated problems of whether it is possi-ble to construct two lines that trisect agiven angle, and to construct a squareequal in area to a given circle – in bothcases using only a straight edge and com-passes. Both these constructions have beenshown to be impossible.

constructive proof A proof that not onlyshows that a certain mathematical entity,such as a root of an equation or a fixed-

conservation of momentum, law of

46

point of a transformation, exists, but alsoexplicitly produces it. Constructive proofsare usually considerably longer and morecomplicated and harder to find than non-constructive proofs of the same results.Many results that have been proved non-constructively have yet to be given con-structive proofs.

constructivist mathematics An ap-proach to mathematics that insists thatonly constructive proofs are acceptableand rejects as meaningless nonconstructiveproofs. Constructivist mathematics is con-siderably more restricted than classicalmathematics and rejects many of its theo-rems. Different varieties of constructivismdiffer over what exactly counts as an ac-ceptably constructive proof. One of thebest known examples of mathematicalconstructivism is the intuitionism ofBrouwer.

contact force A force between bodies inwhich the bodies are in contact. By con-trast, in a non-contact force such as thegravitational force between the Earth andthe Sun, the two bodies are separated by‘empty’ space. Examples of contact forcesinclude tension in which a particle is hang-ing in equilibrium at the end of a string andthrust in which a particle is suspended by aspring underneath it. In the first case thetension in the string acts upwards on theparticle, as does the thrust in the spring inthe second case.

continued fraction An expression thatcan be written in the form:

where all the a’s and b’s are numbers(which are usually positive integers). If thisfraction terminates after a finite number ofterms it is said to be terminating or finite.If it does not terminate it is said to be non-terminating or infinite. The values of a con-tinued fraction that are obtained bytruncating the fraction at a finite point

such as b0, b1, b2, etc., are called conver-gents.

continued product Symbol: Π The prod-uct of a number of related terms. For ex-ample, 2 × 4 × 6 × 8… is a continuedproduct, written:

Πk

an

This means the product of k terms, withthe nth term, an = 2n.

Π∞

1an

has an infinite number of terms.

continuous function A function that hasno sudden changes in values as the variableincreases or decreases smoothly. More pre-cisely, a function f(x) is continuous at apoint x = a if the limit of f(x) as x ap-proaches a is f(a). When a function doesnot satisfy this condition at a point, it issaid to be discontinuous, or to have a dis-continuity, at that point. For example, tanθhas discontinuities at θ = π/2, 3π/2, 5π/2,…A function is continuous in an interval of xif there are no points of discontinuity inthat interval.

continuous stationery A length of fan-folded paper with sprocket holes alongeach side for transporting it through theprinter of a computer. It may be perforatedfor tearing it into separate sheets afterprinting; there may also be perforationsalong the sides to tear off the sprocketholes.

continuum /kŏn-tin-yoo-ŭm/ (pl. con-tinua) A compact connected set with atleast two points. The conditions that theset has at least two points and is connectedimply that the set has an infinite number ofpoints. Any closed interval of the real num-bers is a continuum and the set of all realnumbers is called the real continuum.

The continuum hypothesis is the con-jecture that every infinite subset of the realcontinuum has the CARDINAL NUMBER ei-ther of the positive integers or of the entireset of real numbers. This is equivalent to

47

continuum

b0 a1

b1 a2

b2 a3

b3 . . .

the statement that 2'0 is the lowest cardi-nal number greater than '. See aleph.

contour integral See line integral.

contour line A line on a map joiningpoints of equal height. Contour lines areusually drawn for equal intervals of height,so that the steeper a slope, the closer to-gether the contour lines. See illustrationoverleaf.

contradiction In LOGIC, a proposition,statement, or sentence that both assertssomething and denies it. It is a form ofwords or symbols that cannot possibly betrue; for example, ‘if I can read the bookthen I cannot read the book’ and ‘he iscoming and he is not coming’. Comparetautology.

contradiction, law of See laws ofthought.

contrapositive /kon-tră-poz-ă-tiv/ Inlogic, a statement in which the antecedentand consequent of a conditional are re-versed and negated. The contrapositive ofA → B is ∼B → ∼A (not B implies not A),and the two statements are logically equiv-alent. See implication. See also bicondi-tional.

control theory A branch of appliedmathematics that is concerned with trying

to obtain a specific type of desired dynam-ical behavior from a physical system. Somedevice is added to the system to affect itsbehavior, and the concept of feedback isapplied very widely in control systems.There are many important applications ofcontrol theory, including industrial ma-chinery, robots, chemical processes, andthe stability of cars, trains, and aircraft. Seealso cybernetics.

control unit See central processor.

convergent sequence A sequence inwhich the difference between each termand the one following it becomes smallerthroughout the SEQUENCE; i.e. the differ-ence between the nth term and the (n + 1)thterm decreases as n increases. For example,1, ½, ¼, ⅛, … is a convergent sequence,but 1, 2, 4, 8, … is not. A convergent se-quence has a limit; i.e. a value towardswhich the nth term tends as n becomes in-finitely large. In the first example here thelimit is 0. Compare divergent sequence. Seealso convergent series; geometric sequence.

convergent series An infinite SERIES a1 +a2 + … is convergent if the partial sums a1+ a2 + … + an tend to a limit value as ntends to infinity. For example, the series S= 1 + ½ + ¼ + ⅛ + … is a convergent serieswith sum 2, since 2 is the limit approachedby the sum of the first n terms, namely1–(1/2n) as n tends to infinity. The series 1

contour integral

48

50

4030

2010

5040302010

height in meters

contour lines

map

cross-section along the dotted line

A hill shown as contour lines on a map and as a cross-section.

+ (–1) + 1 + (–1) + 1 + … is not convergent.Compare divergent series. See also conver-gent sequence; geometric series.

converse A logical IMPLICATION taken inthe reverse order. For example, the con-verse ofif I am under 16, then I go to schoolisif I go to school, then I am under 16.

The converse of an implication is not al-ways true if the implication itself is true.There are a number of theorems in mathe-matics for which both the statement andthe converse are true. For example, the the-orem:if two chords of a circle are equidistantfrom the center, then they are equalhas a true converse:if two chords of a circle are equal, thenthey are equidistant from the center of thecircle.

conversion factor The ratio of a meas-urement in one set of units to the equiva-lent numerical value in other units. Forexample, the conversion factor from inchesto centimeters is 2.54 because 1 inch = 2.54centimeters (to two decimal places).

conversion graph A graph showing a re-lationship between two variable quantities.If one quantity is known, the correspond-ing value of the other can be read directly

from the graph. For example, air pressuredepends on height above sea level. A stan-dard curve of altitude against air pressuremay be plotted on a graph. An air-pressuremeasurement can then be converted to anindication of height by reading the appro-priate value from the graph.

convex /kon-veks, kon-veks/ Curved out-wards. For example, the outer surface of asphere is convex. Similarly, in two dimen-sions, the outside of a circle is its convexside. A convex polygon is one in which nointerior angle is greater than 180°. Com-pare concave.

coordinate geometry See analyticalgeometry.

coordinates Numbers that define the po-sition of a point, or set of points. A fixedpoint, called the origin, and fixed lines,called axs, are used as a reference. For ex-ample, a horizontal line and a vertical linedrawn on a page might be defined as the x-axis and the y-axis respectively, and thepoint at which they cross as the origin (O).Any point on the page can then be giventwo numbers – its distance from O alongthe x-axis and its distance from O alongthe y-axis. Distances to the right of the ori-gin for x and above the origin for y are pos-itive; distances to the left of the origin for xand below the origin for y are negative.These two numbers would be the x and ycoordinates of the point. This type of coor-dinate system is known as a rectangularCartesian coordinate system. It can havetwo axs, as on a flat surface, such as a map,or three axs, when depth or height alsohave to be specified. Another type of coor-dinate system (POLAR COORDINATES) ex-presses the position of a point as radialdistance from the origin (the pole), with itsdirection expressed as an angle or angles(positive when anticlockwise) between theradius and a fixed axis (the polar axis). Seealso Cartesian coordinates.

coplanar /koh-play-ner/ Lying in the sameplane. Any set of three points, for example,could be said to be coplanar because thereis a plane in which they all lie. Two vectors

49

coplanar

4000 –

3000 –

2000 –

1000 –

0 – – – – –

0.06 0.07 0.08 0.09 0.10air pressure in megapascals (MPa)

height in metersabove sea level

A conversion graph for finding altitude fromair pressure measurements. (Standard airpressure at sea level is 0.101325 megapascals.)

are coplanar if there is a plane that con-tains both.

coplanar forces Forces in a single plane.If only two forces act through a point, theymust be coplanar. So too are two parallelforces. However, nonparallel forces thatdo not act through a point cannot be copla-nar. Three or more nonparallel forces act-ing through a point may not be coplanar. Ifa set of coplanar forces act on a body, theiralgebraic sum must be zero for the body tobe in equilibrium (i.e. the resultant in onedirection must equal the resultant in theopposite direction). In addition there mustbe no couple on the body (the moment ofthe forces about a point must be zero).

Coriolis force /kor-ee-oh-lis, ko-ree-/ A‘fictitious’ force used to describe the mo-tion of an object in a rotating system. Forinstance, air moving from north to southover the surface of the Earth would, to anobserver outside the Earth, be moving in astraight line. To an observer on the Earththe path would appear to be curved, as theEarth rotates. Such systems can be de-scribed by introducing a tangential Corio-lis ‘force’. The idea is used in meteorologyto explain wind directions. It is named forthe French physicist Gustave-GaspardCoriolis (1792–1843).

corollary /kŏ-rol-ă-ree/ A result that fol-lows easily from a given theorem, so that itis not necessary to prove it as a separatetheorem.

correction A quantity added to a previ-ously obtained approximation to yield abetter approximation. When using loga-rithmic or trigonometric tables the correc-tion is the number added to a logarithm orto a trigonometric function in the table togive the logarithm or trigonometric func-tion of a number or angle that is not in thetable.

correlation In statistics, the correlationcoefficient of two random variables X andY is defined by

r(X,Y) = cov(X,Y)/√(var(X)var(Y)

where cov and var denote covariance andvariance respectively. It satisfies –1 ≤ r ≤ 1and is a measurement of the interdepen-dence between random variables, or theirtendency to vary together. If r ≠ 0 then Xand Y are said to be correlated: they arecorrelated positively if 0 < r ≤ 1 and nega-tively if –1 ≤ r < 0. If r = 0 then X and Y aresaid to be uncorrelated.

For two sets of numbers (x1,…xn) and(y1,…yn) the correlation coefficient is

Σn

i=1(xi – x-) (yi – y-)

r = –––––––––––––––––––––––– [Σ

n

i=1(xi – x-)2 Σ

n

i=1(yi – y-)2]

where x- and y- are the correspondingmeans. It measures how near the points(x1,y1)…(xn,yn) are to lying on a straightline. If r = 1 the points lie on a line and thetwo sets of data are said to be in perfectcorrelation.

correspondence See function.

corresponding angles A pair of angleson the same side of a line (the transversal)that intersects two other lines; they are be-tween the transversal and the other lines. Ifthe intersected lines are parallel, the corre-sponding angles are equal. Compare alter-nate angles.

cos /koz/ See cosine.

cosec /koh-sek/ See cosecant.

cosecant /koh-see-kănt, -kant/ (cosec; csc)A trigonometric function of an angle equalto the reciprocal of its sine; i.e. cosecα =1/sinα. See also trigonometry.

cosech /koh-sech, -sek/ A hyperbolic cose-cant. See hyperbolic functions.

cosh /kosh, kos-aych/ A hyperbolic cosine.See hyperbolic functions.

cosine /koh-sÿn/ (cos) A trigonometricfunction of an angle. The cosine of an angleα (cosα) in a right-angled triangle is theratio of the side adjacent to it, to the hy-

coplanar forces

50

potenuse. This definition applies only toangles between 0° and 90° (0 and π/2 radi-ans). More generally, in rectangular Carte-sian coordinates, the x-coordinate of anypoint on the circumference of a circle of ra-dius r centred on the origin is rcosα, whereα is the angle between the x-axis and theradius to that point. In other words, the co-sine function is the horizontal componentof a point on a circle. Cosα varies periodi-cally in the same way as sinα, but 90°ahead. That is: cosα is 1 when α is 0°, fallsto zero when α = 90° (π/2) and then to –1when α = 180° (π), returning to zero at α =270° (3π/2) and then to +1 again at α =360° (2π). This cycle is repeated every com-plete revolution. The cosine function hasthe following properties:

cosα = cos(α + 360°) = sin(α + 90°)cosα = cos(–α)

cos(90° + α) = –cosαThe cosine function can also be defined

as an infinite series. In the range from +1 to–1:

cosx = 1 – x2/2! + x4/4! – x6/6! + …

cosine rule In any triangle, if a, b, and care the side lengths and γ is the angle op-posite the side of length c, then

c2 = a2 + b2 – 2abcosγ

cot /kot/ See cotangent.

cotangent /koh-tan-jĕnt/ (cot) A trigono-metric function of an angle equal to thereciprocal of its tangent; i.e. cotα = 1/tanα.See also trigonometry.

cotangent rule A rule for triangles thatstates that if the side AB of a triangle is di-vided into the ratio m:n by a point D then(m + n) cot θ = m cot α – n cot β, where θ,α, and β are defined in the diagram. The

cotangent rule can also be expressed in theform: (m + n) cot θ = n cot A–m cot B,where A is the angle at A and B the angleat B.An important special case of the cotangentrule is that in which D is the mid-point ofAB. In this case the rule becomes 2 cot θ =cot α – cot β.The cotangent rule can be used to analyzesome problems in mechanics involvingequilibrium, particularly in the case whenD is the mid-point of AB.

coth /koth/ A hyperbolic cotangent. Seehyperbolic functions.

coulomb /koo-lom/ Symbol: C The SI unitof electric charge, equal to the chargetransported by an electric current of oneampere flowing for one second. 1 C = 1 As. The unit is named for the French physi-cist Charles Augustin de Coulomb (1736–1806).

countable (denumerable) A set is count-able if it can be put in one-one correspon-dence with the integers. The set of rationalnumbers, for example, is countablewhereas the set of real numbers is not (seeCantor’s diagonal argument). To showthat the rationals are countable we need toshow how they can be arranged in a seriessuch that every rational number will be in-cluded somewhere.

If we consider the array in the diagramit is clear that every rational number willoccur in it somewhere. But by starting with1/1 and following the path indicated wecan enumerate every number in the array.If we reduce each fraction to its lowestterms and then remove any that has al-

51

countable

–1

0

+1

2π 3π 4ππ

y

x

Cosine: The graph of y = cos x.

C

BA m nD#

α β

Cotangent rule

ready occurred in the list it is clear that thiswill give a list in which each rational num-ber occurs once and only once.

counterclockwise See anticlockwise.

couple A pair of equal parallel forces inopposite directions and not acting througha single point. Their linear resultant is zero,but there is a net turning effect (moment).The net turning effect T (the torque) isgiven by:

T = Fd1 + Fd2F being the magnitude of each force and d1and d2 the distances from any point to thelines of action of each force. This is equiv-alent to:

T = Fdwhere d is the distance between the forces.

covariance /koh-vair-ee-ăns/ A statisticthat measures the association between twovariables. If for x and y there are n pairs ofvalues (xnyi), then the covariance is definedas

[1/(n – 1)]∑(xi – x′)(yi – y′)where x′ and y′ are the mean values.

CPU /see-pee-yoo/ See central processor.

Cramer’s rule /kray-merz/ A rule thatgives the solution of a set of linear equa-tions in terms of a matrix. If a set of n lin-ear equations for n unknown variables x1,x2, ... xn can be written in the form of a ma-trix equation Ax = b, where A is an invert-ible matrix the solution for the equations

can be written uniquely as x = A–1 = b.Cramer’s rule states that the xi can be writ-ten as:

xi = (b1A1i + b2A2i + ...bnAni)/(det A),for all i between 1 and n, where the bj arethe entries in the b column and the Aji arethe cofactors of A. This is the case becauseA–1 = adjA/detA, where adjA is the adjointof the matrix A. In turn, this means that x= (adj A) b/detA. In the case of two linearequations with two unknowns x1, and x2of the form

ax1 + bx2 = l,cx1 + dx2 = m,

with ad – bc ≠ 0, Cramer’s rule leads to theresult:

x1 = (ld–bm)/(ad–bc),x2 = (am–lc)/(ad–bc).

critical damping See damping.

critical path The sequence of operationsthat should be followed in order to com-plete a complicated process, task, etc., inthe minimum time. It is usually determinedby using a computer.

critical region See acceptance region.

cross multiplication A way of simplify-ing an equation in which one or both termsare fractions. The product of the numera-tor on the left-hand side of the equationand the denominator on the right-handside equals the product of the denominatoron the left-hand side and the numerator onthe right-hand side. For example, crossmultiplying the equation

4x—3

= 3y—2

gives4x × 2 = 3y × 3

or8x – 9y

cross product See vector product.

cross-section (section) A plane cuttingthrough a solid figure or the plane figureproduced by such a cut. For example, thecross-section through the middle of asphere is a circle. A vertical cross-section

counterclockwise

52

1 2 3 4 5 ...

1/1 1/2 1/3 1/4 1/5 ...

2/1 2/2 2/3 2/4 2/5 ...

3/1 3/2 3/3 3/4 3/5 ...

4/1 4/2 4/3 4/4 4/5 ...

5/1 5/2 5/3 5/4 5/5 ...

1

2

3

4

5 . . . . .. . . . .

Countable

through an upright cone and off the axis isa hyperbola.

crystallographic group /kris-tă-lŏ-graf-ik/ A POINT GROUP that is compatible withthe symmetry of a crystal. This restrictionmeans that there are 32 possible pointgroups. Crystallographic groups can onlyhave twofold, threefold, fourfold, or six-fold rotational symmetry since any otherrotational symmetry would be incompati-ble with the translational symmetry exist-ing in a crystal. However, fivefold andicosahedral symmetries are possible inQUASICRYSTALLINE SYMMETRY. The SPACE

GROUP of a crystal is characterized by thecombination of the crystallographic pointgroup and the translational symmetry ofthe crystal lattice.

crystallographic symmetry The symme-try that is associated with the regular three-dimensional structure of a crystal. The setof crystallographic symmetry operations ofa crystal makes up the SPACE GROUP of thecrystal. This consists of the symmetry op-erations of the CRYSTALLOGRAPHIC GROUP

and the symmetry operations associatedwith the translational symmetry of thecrystal.

csc See cosecant.

cube 1. The third power of a number orvariable. The cube of x is x × x × x = x3 (xcubed).2. In geometry, a solid figure that has sixsquare faces. The volume of a cube is l3,where l is the length of a side.

cube root An expression that has a thirdpower equal to a given number. The cuberoot of 27 is 3, since 33 = 27.

cube root of unity A complex number zfor which z3 = 1. There are three cube rootsof unity, which can be denoted as 1, ω, andω2, where ω and ω2 are defined by

ω = exp(2πi/3) = cos(2π/3) + i sin (2π/3)= + 1/2 –√3i/2,

ω2 = exp(4πi/3) = cos(4π/3) + i sin(4π/3) = –1/2 – √3i/2.

The cube roots of unity satisfy the follow-ing relations:

ω_

= ω2

ω2 + ω + 1 = 0

cubic equation A polynomial equation inwhich the highest power of the unknownvariable is three. The general form of acubic equation in a variable x is

ax3 + bx2 + cx + d = 0where a, b, c, and d are constants. It is alsosometimes written in the reduced form

x3 + bx2/a + cx/a + d/a = 0In general, there are three values of x

that satisfy a cubic equation. For example,2x3 – 3x2 – 5x + 6 = 0

can be factorized to(2x + 3)(x – 1)(x – 2) = 0

and its solutions (or roots) are –3/2, 1, and2. On a Cartesian coordinate graph, thecurve

y = 2x3 – 3x2 – 5x + 6crosses the x-axis at x = –3/2, x = +1, and x= +2.

cubic graph A graph of the equationy = ax3 + bx2 + cx + d,

where a, b, c, and d are constants. As x be-comes very large y→ax3. This means that ifa > 0 then y→∞ as x→∞ and y→–∞ asx→–∞ and that if a < 0 then y→–∞ as x→∞and y→∞ as x→–∞. Since dy/dx = 3ax2 +2bx + c the nature of the stationary pointsof the cubic graph are determined by thenature of the solutions of the quadraticequation 3ax2 +2bx + c = 0. If the qua-dratic equation has two real roots that aredistinct then the curve has two turningpoints that are distinct, one of which is amaximum and one of which is a minimum.If the quadratic equation has two real rootsthat are equal the curve has a point of in-flexion. If the quadratic equation does nothave any real roots the curve does not haveany real stationary points. The curve iscontinuous since y does not tend to infinityfor any finite value of x. A cubic curve caneither: (1) cross the x-axis three times; (2)cross and touch the x-axis; (3) cross the x-axis once; or (4) touch the x-axis at thepoint of inflexion. This means that therehas to be at least one point of intersectionwith the x-axis. This result means, in turn,

53

cubic graph

that a CUBIC EQUATION ax3 + bx2 + cx + d =0 must have at least one real root.

cuboid A box-shaped solid figurebounded by six rectangular faces. The op-posite faces are congruent and parallel. Ateach of the eight vertices, three faces meetat right angles to each other. The volume ofa cuboid is its length, l, times its breadth, b,times its height, h. The surface area is thesum of the areas of the faces, that is

2(l × b) + 2(b × h) + 2(l × h)In the special case in which l = b = h, all

the faces are square and the cuboid is acube of volume l3 and surface area 6l2.

cumulative distribution See distributionfunction.

cumulative frequency The total fre-quency of all values up to and including theupper boundary of the class interval underconsideration. See also frequency table.

curl (Symbol ∇∇) A vector operator on avector function that, for a three-dimen-sional function, is equal to the sum of thevector product (cross product) of the unitvectors and partial derivatives in each ofthe component directions. That is:

curl F = ∇F = i×∂F/∂x +j×∂F/∂∂y + k×∂F/∂z

where i, j, and k are the unit vectors in thex, y, and z directions respectively. Inphysics, the curl of a vector arises in the re-lationship between electric current andmagnetic flux, and in the relationship be-tween the velocity and angular momentumof a moving fluid. See also div; grad.

curvature The rate of change of the slopeof the tangent to a curve, with respect todistance along the curve. For each point ona smooth curve there is a circle that has thesame tangent and the same curvature atthat point. The radius of this circle, calledthe radius of curvature, is the reciprocal ofthe curvature, and its center is known asthe center of curvature. If the graph of afunction y = f(x) is a continuous curve, theslope of the tangent at any point is given bythe derivative dy/dx and the curvature isgiven by:

(d2y/dx2)/[1 + (dy/dx)2]3/2

curve A set of points forming a continu-ous line. For example, in a graph plotted inCartesian coordinates, the curve of theequation y = x2 is a parabola. A curved sur-face may similarly represent a function oftwo variables in three-dimensional coordi-nates.

curve sketching Sketching the graph of afunction y = f(x) in such a way as to indi-cate the main features of interest of thatcurve. This usually involves determiningthe general shape of the curve and investi-gating how it behaves at points of specialinterest. To be more specific, symmetry ofthe curve about the x-axis and the y-axisare investigated, as is symmetry about theorigin. The behavior as x and y becomevery large is investigated in both the posi-tive and negative directions. The points atwhich the curve cross the x-axis and the y-axis are established. The problems ofwhether there are any values of x for whichy is infinite and any values of y for which xis infinite are investigated. The nature ofany stationary points is also investigated.There are also a number of other featuresthat can be looked at, including: establish-ing the intervals for which the function isalways decreasing or always increasing; theconcave or convex nature of the curve; andall the asymptotes of the curve. Usuallycurve sketching is done using Cartesian co-ordinates. It is a way of visualizing howfunctions behave without calculating theexact values.

curvilinear integral /ker-vă-lin-ee-er/ Seeline integral.

cusp A sharp point formed by a disconti-nuity in a curve. For example, two semicir-cles placed side by side and touching forma cusp at which they touch.

cut See Dedekind cut.

cybernetics /sÿ-ber-net-iks/ The branch ofscience concerned with control systems, es-pecially with regard to the comparisons be-tween those of machines and those of

cuboid

54

human beings and other animals. In a se-ries of operations, information gained atone stage can be used to modify later per-formances of that operation. This is knownas feedback and enables a control system tocheck and possibly adjust its actions whenrequired.

cycle A series of events that is regularly re-peated (e.g. a single orbit, rotation, vibra-tion, oscillation, or wave). A cycle is acomplete single set of changes, startingfrom one point and returning to the samepoint in the same way.

cyclic function See periodic function.

cyclic group A group in which each el-ement can be expressed as a power of anyother element. For example, the set of allnumbers that are powers of 3 could bewritten as …31/3, 31/2, 3, 32, 33, … or …91/6, 91/4, 91/2, 9, 93/2, …, etc. See alsoAbelian group.

cyclic polygon A polygon for which thereis a circle on which all the vertices lie. Alltriangles are cyclic. All regular polygonsare cyclic. All squares and rectangles arecyclic quadrilaterals. However, not allquadrilaterals are cyclic. Convex quadri-laterals are cyclic if the opposite angles aresupplementary. For a cyclic quadrilateralwith sides of length a, b, c, and d (in order)the expression (ac + bd) is equal to theproduct of the diagonals. This is known asPtolemy’s theorem, named for the Egypt-ian astronomer Ptolemy (or ClaudiusPtolemaeus) (fl. 2nd century AD).

cyclic quadrilateral A four-sides figurewhose corners (vertices) lie on a circum-scribed circle. The opposite angles are sup-plementary, i.e. they add to 180°. Seecircumcircle; supplementary angles.

cycloid /sÿ-kloid/ The curve traced out bya point on a circle rolling along a straightline, for example, a point on the rim of awheel rolling along the ground. For a circle

55

cycloid

y

P

O

2r2r2r

2πrx

r#

Cycloid

y

xO

In this graph a cusp occurs at the origin O.

of radius r along a horizontal axis, the cy-cloid produced is a series of continuousarcs that rises from the axis to a height 2πand fall to touch the axis again at a cusppoint, where the next arc begins. The hori-zontal distance between successive cusps is2πr, the circumference of the circle. Thelength of the cycloid between adjacentcusps is 8r. If θ is the angle formed by theradius to a point P (x,y) on the cycloid andthe radius to the point of contact with thex-axis, the parametric equations of the cy-cloid are:

x = r(θ – sinθ)y = r(1 – cosθ)

cylinder A solid defined by a closed planecurve (forming a base) with an identicalcurve parallel to it. Any line segment froma point on one curve to a correspondingpoint on the other curve is an element ofthe cylinder. If one of these elements movesparallel to itself round the base it sweepsout a curved lateral surface. The line is agenerator of the cylinder and the planeclosed curve forming the base is called thedirectrix

If the bases are circles the cylinder is acircular cylinder. If the bases have centersthe line joining them is an axis of the cylin-der. A right cylinder is one with its axis atright angles to the base; otherwise it is anoblique cylinder. The volume of a cylinderis Ah, where A is the base area and h the al-titude (the perpendicular distance betweenthe bases). For a right circular cylinder, thecurved lateral surface area is 2πrh, where ris the radius.

If the generator is an indefinitely ex-tended line it sweeps out an extended sur-face – a cylindrical surface

cylindrical helix See helix.

cylindrical polar coordinates A methodof defining the position of a point in spaceby its horizontal radius r from a fixed ver-tical axis, the angular direction θ of the ra-dius from an axis, and the height z above afixed horizontal reference plane. Startingat the origin O of the coordinate system,the point P(r,θ,z) is reached by moving outalong a fixed horizontal axis to a distancer, following the circumference of the hori-zontal circle radius r centred at O throughan angle θ, and then moving vertically up-ward by a distance z. For a point P(r,θ,z),the corresponding rectangular CARTESIAN

COORDINATES (x,y,z) are:x = rcosθy = rsinθ

z = zCompare spherical polar coordinates.

See also coordinates; polar coordinates.

cylindrical surface See cylinder.

cylinder

56

P(r, #, z)

rO

z

#

A point P(r,θ,z) in cylindrical polar coordinates.

d’Alembertian /dal-ahm-bair-ti-ăn/ Sym-bol 2. An operator that acts on the func-tion u(x,y,z,t) for the displacement of awave. It is related to the LAPLACIAN opera-tor ∇2 by:

2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2 – (1/c2)∂2/∂t2

= ∇2 – (1/c2) ∂2/∂t2,where c is the speed of the wave. Using thisoperator the WAVE EQUATION can be writ-ten very concisely as 2u = 0. Sometimesthe d’Alembertian is taken to refer specifi-cally to the wave equation for electromag-netic waves, with c being the speed of lightin that case. It is named for the Frenchmathematician, encyclopedist, and phil-osopher Jean Le Rond d’Alembert (1717–83).

d’Alembert’s principle /dal-ahm-bairz/A principle that combines Newton’s sec-ond and third laws of motion into theequation F–ma = 0. Using this principle, allproblems involving forces can be treated asequilibrium problems, with the –ma beingregarded as an inertial reaction force. Forbodies in motion this type of equilibrium iscalled dynamic equilibrium or kinetic equi-librium to distinguish it from static equilib-rium. The application of d’Alembert’sprinciple can be used to simplify manyproblems in Newtonian mechanics.

d’Alembert’s ratio test /dal-ahm-bairz/(generalized ratio test) A method of show-ing whether a series is convergent or diver-gent. The absolute value of the ratio ofeach term to the one before it is taken:

|un+1/un|If the LIMIT of this is l as n tends to infinityand l is less than 1, then the series is con-vergent. If l is greater than 1, the series isdivergent. If l is equal to 1, the test fails andsome other method has to be used.

damped oscillation An oscillation withan amplitude that progressively decreaseswith time. See damping.

damping The reduction in amplitude of avibration with time by some form of resis-tance. A swinging pendulum will at lastcome to rest; a plucked string will not vi-brate for long – in both cases internaland/or external resistive forces progres-sively reduce the amplitude and bring thesystem to equilibrium.

In many cases the damping force(s) willbe proportional to the object’s speed. Inany event, energy must be transferred fromthe vibrating system to overcome the resis-tance. Where damping is an asset (as inbringing the pointer of a measuring instru-ment to rest), the optimum situation occurswhen the motion comes to zero in theshortest time possible, without vibration.This is critical damping. If the resistiveforce is such that the time taken is longerthan this, overdamping occurs. Con-versely, underdamping involves a longertime with vibrations of decreasing ampli-tude.

data /day-tă/ (now often used as a singularnoun) (sing. datum) The facts that refer toor describe an object, idea, condition, situ-ation, etc. In computing, data can be re-garded as the facts on which a PROGRAM

operates as opposed to the instructions inthe program. It can only be accepted andprocessed by the computer in binary form.Data is sometimes considered to be numer-ical information only.

data bank A large collection of organizedcomputer data, from which particularpieces of information can be readily ex-tracted. See also database.

57

D

database A large collection of organizeddata providing a common pool of informa-tion for users, say, in the various sectionsof a large organization. Information can beadded, deleted, and updated as required.The management of a database is verycomplicated and costly so that computerprograms have been developed for thispurpose. These programs allow the infor-mation to be extracted in many differentways. For example, a request could be putin for an alphabetical list of people over acertain age and living in a specified area, inwhich their employment and incomeshould be given. Alternatively the requestcould be for an alphabetical list of peopleover a certain age and income level inwhich their address and form of employ-ment should be given.

data processing The sequence of opera-tions performed on data in order to extractinformation or to achieve some form oforder. The term usually means the process-ing of data by computers but can also in-clude its observation and collection.

data transmission Any method of trans-ferring data from one computer to anotheror from an outstation (such as a cash-pointmachine) to a central computer.

debug To detect, locate, and correct er-rors or faults (bugs) that occur in computerprograms or in pieces of computer equip-ment. Since PROGRAMS and equipment areoften highly complicated, debugging canbe a tedious and lengthy job. Programmingerrors may result from the incorrect codingof an instruction (known as a syntax error)or from using instructions that will not givethe required solution to a problem (a logicerror). Syntax errors can usually be de-tected and located by the compiler; logicerrors can be more difficult to find.

deca- Symbol: da A prefix denoting 10.For example, 1 decameter (dam) = 10 me-ters (m).

decagon /dek-ă-gon/ A plane figure withten straight sides. A regular decagon hasten equal sides and ten equal angles of 36°.

decahedron /dek-ă-hee-drŏn/ A polyhe-dron that has ten faces. See polyhedron.

deci- Symbol: d A prefix denoting 10–1.For example, 1 decimeter (dm) = 10–1

meter (m).

decibel /des-ă-bel/ Sybol: dB A unit ofpower level, usually of a sound wave orelectrical signal, measured on a logarithmicscale. The threshold of hearing is taken as0 dB in sound measurement. Ten times thispower level is 10 dB. The fundamental unitis the bel, but the decibel is almost exclu-sively used (1 dB = 0.1 bel).

A power P has a power level in decibelsgiven by:

10 log10(P/P0)where P0 is the reference power.

decimal Denoting or based on the numberten. The numbers in common use forcounting form a decimal number system. Adecimal fraction is a rational number writ-ten as units, tenths, hundredths, thou-sandths, and so on. For example, ¼ is 0.25in decimal notation. This type of decimalfraction (or decimal) is a finite decimal be-cause the third and subsequent digits afterthe decimal point are 0. Some rationalnumbers, such as 5/27 (= 0.185 185 185…)cannot be written as an exact decimal, butresult in a number of digits that repeat in-definitely. These are called repeating deci-mals. All rational numbers can be writtenas either finite decimals or repeating deci-mals. A decimal that is not finite and doesnot repeat is an irrational number and canbe quoted to any number of decimal places,but never exactly. For example, π to an ac-curacy of six decimal places is 3.141 593and to seven decimal places is 3.141 592 7.

A decimal measure is any measuringsystem in which larger and smaller unitsare derived by multiplying and dividing thebasic unit by powers of ten. See also metricsystem.

decision box See flowchart.

decomposition 1. The process of break-ing a fraction up into partial fractions.

database

58

2. Decomposition of a vector v is theprocess of writing it in the form

v = grad(φ) + curl(A)where φ is a scalar and A is a vector. Everyvector may be decomposed in this form.

decreasing function A real function f(x)for which f(x1) ≥ f(x2) for all x1 and x2 inan interval I when x1 < x2. If the strongerinequality f(x1) > f(x2) holds when x1 < x2then f(x) is said to be a strictly decreasingfunction.

decreasing sequence A sequence a1, a2,a3, ... for which ai ≥ ai+1 for all i in the se-quence. If ai > ai+1 for all i in the sequencethe sequence is said to be a strictly decreas-ing sequence.

Dedekind cut /day-dĕ-kint/ A method ofdefining the real numbers, starting fromthe rational numbers. A Dedekind cut is adivision of the rational numbers into twodisjoint sets, A and B, which are nonemptyand which satisfy the conditions: (1) if x∈A and y ∈B then x<y, and (2) A has nolargest member (or, equivalently, B has nosmallest member). The real numbers can bedefined as the set of all Dedekind cuts andcan be shown to have all the requisite prop-erties. The method is named for the Ger-man mathematician Richard Dedekind(1831–1916).

deduction A series of logical steps inwhich a conclusion is reached directly froma set of initial statements (premisses). A de-duction is valid if a sentence or statementthat asserts the premisses and denies theconclusion is a contradiction. Compare in-duction. See contradiction.

definite integral (Riemann integral) Theresult of integrating any function of a sin-gle variable, f(x), between two specifiedvalues of x: x1 and x2. The definite integralof f(x) is written

x2

x1f(x)dx

If the general expression for the integral off(x) (its indefinite integral) is another func-tion of x, g(x), the definite integral is givenby:

g(x1) – g(x2)Compare indefinite integral. See also inte-gration.

definition In a measurement, the ACCU-RACY with which the instrument readingreflects the true value of the quantity beingmeasured.

deformation A geometrical transforma-tion that stretches, shrinks, or twists ashape but does not break up any of its linesor surfaces. It is often called, more pre-cisely, a continuous deformation. See alsotopology; transformation.

degeneracy The occurrence of two differ-ent EIGENFUNCTIONS of an eigenvalue prob-lem that have the same eigenvalue. Animportant example of degeneracy is givenin quantum mechanical systems such asnuclei, atoms and molecules, where degen-eracy occurs when different quantumstates have the same energy. The degener-acy of a system is closely associated with itssymmetry.

degenerate conic See conic.

degenerate conicoid A quadric surfacedescribed by the equation

ax2 + by2 + cz2 + 2fyz + 2gxz + 2hxy +2ux + 2vy +2wz + d = 0

for which ∆ = 0, where ∆ is the determinantdefined byA quadric that is not a degenerate quadricis called a non-degenerate quadric. Thenon-degenerate quadrics can be listed: theellipsoid, the hyperboloid (both of onesheet and two sheets), the elliptic parabo-loid and the hyperbolic paraboloid. Seeconcoid.

degree 1. Symbol: ° A unit of plane angleequal to one ninetieth of a right angle.

59

degree

a h g u

h b f v

g f c w

u v w d

∆ =

2. A unit of temperature. See Celsius de-gree; Fahrenheit degree; kelvin.3. The exponent of a variable. For instance3x3 has a degree of 3. If there are severalterms, the sum of the exponents is used;5xy2z3 has a degree of 6.4. The highest power of a variable in apolynomial. For example in x3 + 2x + 1, thedegree is 3.5. The highest power of an equation. Forexample, the degree of the equation x4 + 2x= 0 is 4.6. The highest power to which the highest-order derivative is raised in a DIFFERENTIAL

EQUATION. For example, the degree of(d2y/dx2)3 + dy/dx = 0

is three. The degree ofd3y/dx3 + 2y(d2y/dx2)2 = 0

is one.

degrees of freedom The number of inde-pendent parameters that are needed tospecify the configuration of a system. Forexample, an atom in space has three inde-pendent coordinates needed to specify itsposition. A molecule with two atoms (e.g.02) has additional degrees of freedom be-cause it can also vibrate and rotate. In fact,a diatomic molecule of this type has six de-grees of freedom. It is usual in physics tointerpret the number of degrees of freedomas the number of independent ways inwhich the system can store energy. See alsophase space.

De l’Hôpital’s rule /dĕ-loh-pee-tahlz/The rule stating that the limit of the ratio oftwo functions of the same variable (x) as xapproaches a value a, is equal to the limitof the ratios of their derivatives with re-spect to x. That is, the limit of f(x)/g(x) asx→a is the limit of f′(x)/g′(x) as x→a.

De l’Hôpital’s rule can be used to findthe limits of f(x)/g(x) at points at whichboth f(x) and g(x) are zero and the ratio istherefore indeterminate. Any function thatgives rise to an indeterminate form andthat can be expressed as a ratio of twofunctions, can be dealt with in this way.For example, in

F(x) = (x2 – 3)/(x –3)writing

f(x) = (x2 – 3)

andg(x) = (x – 3)

givesF(x) = f(x)/g(x)

The limit of F(x) as x→3 is indeterminate(since x – 3 = 0). It can be obtained by usingthe limit of

f′(x)/g′(x) = 2xas x→3. Thus the limit is 6.

If f′(x)/g′(x) also gives an indeterminateform at x = a, De l’Hôpital’s rule can be ap-plied again, differentiating as many timesas is necessary.

The rule is named for the French math-ematician Guillaume François Antoine,Marquis De l’Hôpital (1661–1704).

delta function See Dirac delta function.

De Moivre’s theorem /dĕ-mwahvrz/ Aformula for calculating a power of a com-plex number. If the number is in the polarform

z = r(cosθ + i sinθ)then zn = rn(cosnθ + i sinnθ)

The theorem is named for the Frenchmathematician Abraham De Moivre(1667–1754).

De Morgan’s laws /dĕ-mor-gănz/ Twolaws governing the relation between com-plementation, intersection, and union ofsets. If represents the complement of theset A (i.e. the set of all things not in A) thenDe Morgan’s laws state that:

(1) (A––––∪ B) = A

- ∩ B-

and (2) (A ∩ B) = A

––––∪ B. Analogous laws are true for any finite in-tersection or union of sets. Parallel lawsexist in other areas, e.g. in propositionallogic the equivalences

(p & q) ≡ p ∨ q and (p ∨ q) ≡ p & q

are also known as De Morgan’s laws. Thelaws are named for the English mathemati-cian Augustus De Morgan (1806–71).

denominator The bottom part of a frac-tion. For example, in the fraction ¾, 4 isthe denominator and 3 is the numerator.The denominator is the divisor.

degrees of freedom

60

dense A set S is said to be dense in anotherset T if every point of T either belongs to Sor is a limit point of S. If it is obvious whatthe other set is, then a set is sometimes sim-ply said to be dense; e.g. if we are consid-ering sets of real numbers. For example,the set of rational numbers is dense (on thereal line) because every point of the realline is either a rational number or is thelimit of a sequence of rational numbers.Another way of putting this is to say thatany interval on the real line will alwayscontain infinitely many rationals.

density The amount of matter per unitvolume; mass divided by volume.

denumerable set /di-new-mĕ-ră-băl/ ASET in which the elements can be counted.For example, the set of prime numbers, al-though infinite, can be counted, as can theset of positive even integers. These areknown as denumerably infinite sets. Theset of real numbers, on the other hand, isnot denumerable because between any twoelements there can always be found a third.See countable.

dependent 1. An equation is dependenton a set of equations if every set of valuesof the unknowns that satisfy the set ofequations also satisfies this equation. Oth-erwise the equation is independent of theset of equations.2. Two events are independent if the oc-currence or nonoccurrence of one of themdoes not affect the probability of the oc-currence of the other. Otherwise the eventsare dependent. If A and B are events whoseprobabilities are P(A) and P(B), then A andB are independent if and only if P(A and B)= P(A)P(B). For example, each toss of acoin is an independent event.3. A set of functions f1,…fn are dependentif one can be expressed as a function of theothers or equivalently there exists an ex-pression F(f1,…fn) ≡ 0 with not all ∂F/∂fi =0. Otherwise they are independent. For ex-ample, the functions 2x + y and 4x + 2y +6 are dependent since 4x + 2y + 6 = 2(2x +y) + 6.4. See variable.

5. A set of vectors, matrices, or other ob-jects x1,…xn is said to be linearly depend-ent if there exists a linear relation

a1x1 + a2x2 + … + anxn = 0with at least one of the coefficientsnonzero. A set of objects is linearly inde-pendent if it is not linearly dependent. Itshould be noted that the dependence is rel-ative to the set from which we may pick thecoefficients a1,…an. For example, 2 and πare linearly independent with respect to therational numbers but linearly dependentwith respect to the real numbers. This is thecase since a relation of the form a12 + a2π= 0 does not exist if a1 and a2 are rationalnumbers but if a1 and a2 are allowed to beirrational numbers we may take a1 = π, a2= 2.

dependent variable See variable.

deposit A sum of money paid by a buyer,either to reserve goods or property that hewishes to buy at a later date or as the firstof a series of installments in an installmentplan. If the buyer fails to complete the in-stallments the deposit is normally forfeited.

depth The distance downward from a ref-erence level or backward from a referenceplane. For example, the distance below awater surface and the distance between awall surface and the back of an alcove inthe wall, are both called depths.

derivative The result of DIFFERENTIATION.

derived unit A unit defined in terms ofbase units, and not directly from a stan-dard value of the quantity it measures. Forexample, the newton is a unit of force de-fined as a kilogram meter seconds–2

(kg m s–2). See also SI units.

desktop publishing A system that uses amicrocomputer with word-processing fa-cilities linked to a laser printer to producemultiple copies of a document. The wordprocessor can provide various type fontsand a scanner can be included to add illus-trations (graphics). The result can rival thequality of conventional printing.

61

desktop publishing

determinant /di-ter-mă-nănt/ A functionof a square matrix derived by multiplyingand adding the elements together to obtaina single number. For example, in a 2 × 2matrix the determinant is a1b2 – a2b1. Thisis written as a square array in vertical lines,symbol D2, and is called a second-order de-terminant. Determinants occur in simulta-neous equations. The solution of

a1x + b1y + c1 = 0and

a2x + b2y + c2 = 0is

x = (b1c2 – c1b2)/D2and

y = (c1a2 – a1c2)/D2If a1, a2, b1, b2, c1, and c2 are 1, 2, 3, 4, 5,and 6 respectively, then D2 = –2 and

x = [(3 × 6) – (5 × 4)]/–2 = 1and

y = [(5 × 2) – (1 × 6)]/–2 = –2A third-order determinant has three rowsand columns and arises in a similar way insets of three simultaneous equations inthree variables.

The determinant of a transpose of a ma-trix, |Ã|, is equal to the determinant of thematrix, |A|. If the position of any of therows or columns in the matrix is changed,the determinant remains the same.

determinism The idea that if the presentstate of a system is known exactly and thelaw governing the evolution of that systemwith time is known then the subsequentevolution of the system can be determinedexactly. Classical mechanics is governed bydeterminism. Two developments in thetwentieth century undermined this simplepicture. In quantum mechanics, the uncer-

determinant

62

a1 b1

a2 b2

b2 c2

b3 c3

a2 c2

a3 c3

a2 b2

a3 b3

= a1b2 – a2b1

The second-order determinant of a 2 x 2 matrix.

a1 b1 c1

a2 b2 c2

a3 b3 c3

a1 b1 c1

a2 b2 c2

a3 b3 c3

The third-order determinant of a 3 x 3 matrix.

= a1b2c3 – a1b3c2 + a2b3c1 – a2b1c3

+ a3b1c2 – a3b2c1

– b1+ c1= a1

= a1a1’ – b1b1’ + c1c1’

= a1a1’ – a2a2’ + a3a3’

+ – +

– + –

+ – +

Determinants: A third-order determinant is equal to the sum along any row, or down any column,of the product of each element with its cofactor. The cofactors are given alternate positive andnegative signs in the pattern shown. Fourth- and higher order determinants can be calculated in asimilar way.

tainty principle of the German physicistWerner Heisenberg (1901–76) showedthat it is not possible to know the state of asystem exactly since the position and themomentum of a particle cannot be deter-mined exactly simultaneously. Even withinclassical mechanics itself the developmentof CHAOS THEORY led to the realization thatthe word exactly is very important both inprinciple and in practice because there aremany systems for which slightly differentinitial conditions result in widely differentresults for the state of the system over a pe-riod of time. Consequently, even systemsthat are described in a deterministic waycan, in practice, behave in ways that ap-pear random or chaotic.

developable surface A surface that canbe rolled out flat onto a plane. The lateralsurface of a cone, for example, is devel-opable. A spherical surface is not.

deviation See mean deviation; standarddeviation.

diagonal Joining opposite corners. A di-agonal of a square, for example, cuts it intotwo congruent right angled triangles. In asolid figure, usually a polyhedron, a diago-nal plane is one that passes through twoedges that are not adjacent.

diagonal matrix A square matrix inwhich all the elements are zero exceptthose on the leading diagonal, that is, thefirst element in the first row, the second el-ement in the second row, and so on. Diag-onal matrices, unlike most others, arecommutative in matrix multiplication.

diameter The distance across a plane fig-ure or a solid at its widest point. The di-

ameter of a circle or a sphere is twice theradius.

diametral /dÿ-am-ĕ-trăl/ Denoting a lineor plane that forms a diameter of a figure.For example, a cross-section through thecenter of a sphere is a diametral plane.

dichotomy, principle of In logic, theprinciple that a proposition is either true orfalse, but not both. For example, for twonumbers x and y either x = y or x ≠ y, butnot both.

diffeomorphism /diff-ee-oh-mor-fiz-ăm/A continuous transformation of a spacethat moves the points of a space aroundbut preserves those relationships betweenthem that are used to define which pointsare close to one another. The set of diffeo-morphisms of a space is called the diffeo-morphism group of that space. Thediffeomorphism group underlies discus-sions of invariance in RIEMANNIAN GEOME-TRY and general relativity theory.

difference The result of subtracting onequantity or expression from another.

difference between two squares A spe-cial case of factorization involving twonumbers or expressions that are squares. Ingeneral terms,

x2 – y2 = (x + y)(x – y)

difference equation An equation that ex-presses a relation between finite differences∆, where ∆ is a difference operator whichoperates on the rth term Ur of a sequenceU1, U2, ... Un to produce the (r + 1)th termUr+1. The definition of the difference op-erator means that ∆Ur = Ur+1, ∆U1 =U2–U1, ∆U2 = U3–U2, etc. The expressionsjust given are first finite differences of U1,U2, ... Un since ∆ only operates on themonce. A second finite difference is definedby operating with ∆ a second time. Thisleads to the results ∆2U1 = U3 – 2U2 + U1,∆2U2 = U4 – 2U3 + U2, etc. Higher-orderdifferences ∆3Ur, ∆4Ur, ... can be defined inan analogous way.

The order of a difference equation is theorder of the highest-order finite difference

63

difference equation

a11 0 0

0 a22 0

0 0 a33

A 3 × 3 diagonal matrix.

in it. For example, ∆Ur–aUr = b, where aand b are constants is a first-order differ-ence equation, while ∆2Ur–a∆Ur + bUr =f(r), where a and b are constants and f(r) isa function of r, is a second-order differenceequation. There are analogies between dif-ference equations and differential equa-tions.

difference set The set that is made up ofall the elements of set A that are not el-ements of set B. In terms of a VENN DIA-GRAM the difference set can be shown as ashaded region for two overlapping sets.

differentiable function /dif-ĕ-ren-shă-băl/ A function f for which the derivativeof f exists. More formally, if f is a real func-tion of one variable x then the function isdifferentiable in some interval if the limit of[f(x + δx]/δx exists as δx→0 for all valuesof x in the interval. Thus, a function is dif-ferentiable at a point if the gradient of thegraph y = f(x) can be defined as that point,and therefore that the tangent to the curvecan be defined at that point. In the case ofa function of a complex variable the func-tion f(z) is differentiable at a particularpoint if the limit [f(z + δz)–f(z)]/δz exists asδz→0 for that point and is independent ofthe way in which that point is approached.

differential /dif-ĕ-ren-shăl/ An infinitesi-mal change in a function of one or morevariables, resulting from a small change inthe variables. For example, if f(x) is a func-tion of x, and f changes by ∆f as a result ofa change ∆x in x, the differential df, is de-fined as the limit of ∆f as ∆x becomes infi-nitely small. That is, df = f′(x)dx, wheref′(x) is the derivative of f with respect to x.This is a total differential, because it takesinto account changes in all of the variables,just one in this case.

For a function of two variables, f(x,y)the rate of change of f with respect to x isthe partial derivative ∂f/∂x. The change in fresulting from changing x by dx and keep-ing y constant is the partial differential,(∂f/∂y).dy. For any function, the total dif-ferential is the sum of all the partial differ-entials. For f(x,y):

df = (∂f/∂x).dx + (∂f/∂y).dy

See also differentiation.

differential equation An equation thatcontains derivatives. An example of a sim-ple differential equation is:

dy/dx + 4x + 6 = 0To solve such equations it is necessary touse integration. The equation above can berearranged to give:

dy = –(4x + 6)dxintegrating both sides:

∫dy = ∫–(4x + 6)dxwhich gives:

y = –2x2 – 6x + Cwhere C is a constant of integration. Thevalue of C can be found if particular valuesof x and y are known: for example, if y = 1when x = 0 then C = 1, and the full solutionis

y = –2x2 – 6x + 1Note that the solution to a differentialequation is itself an equation. Differentiat-ing the solution gives the original equation.Equations like that above, which containonly first derivatives (dy/dx) are said to befirst order, if they contain second deriva-tives they are second order; in general, theorder of a differential equation is the high-est derivative in the equation. The degreeof a differential equation is the highestpower of the highest order derivative.

The differential equation in the exam-ple given is a first order and first degreeequation. It is an example of a type ofequation solvable by separating the vari-ables onto both sides of the equation, sothat each can be integrated (the variablesseparable method of solution). Anothertype of first-order first-degree equation isone of the form:

dy/dx = f(y/x)Such equations are known as homoge-neous differential equations. An example isthe equation:

dy/dx = (x2 + y2)/x2

To solve homogeneous equations a substi-tution is made, y = mx, where m is a func-tion of x. Then:

dy/dx = m + xdm/dxand

(x2 + y2)/x2 = (x2 + m2x2)/x2

So the equation becomes:m + xdm/dx = (x2 + m2x2)/x2

difference set

64

or:xdm/dx = 1 + m2 – m

The equation can now be solved by sepa-rating the variables.

An equation of the form:dy/dx + P(x)y = Q(x)

where P(x) and Q(x) are functions of x (butnot y), is a linear differential equation.Equations of this type can be put in a solv-able form by multiplying both sides by theexpression:

exp(∫P(x)dx)This is known as an integrating factor. Forexample, the differential equation

dy/dx + y/x = x2

is a linear first-order differential equation.The function P(x) is 1/x, so the integratingfactor is:

exp(∫dx/x)which is exp(logx); i.e. x. Multiplying bothsides of the equation by x gives:

xdy/dx + y = x3

The left-hand side of the equation is equalto d(xy)/dx, so the equation becomes

d(xy)/dx = x3

Integrating both sides gives:xy = x4/4 + C

where C is a constant.

differential form Any entity that is underan integral sign. For example, the integral∫Adx has a one-form Adx associated withit. Similarly, a volume integral ∫∫∫Bdxdydzhas a three-form Bdxdydz associated with

it. More generally, if an integral is over ndimensions then there is an n-form associ-ated with that integral. There are manyproblems in mathematics and its applica-tions to physical sciences and engineeringthat can be analyzed using differentialforms.

differentiation /dif-ĕ-ren-shee-ay-shŏn/ Aprocess for finding the rate at which onevariable quantity changes with respect toanother. For example, a car might travelalong a road from position x1 to positionx2 in a time interval t1 to t2. Its averagespeed is (x2 – x1)/(t2 – t1), which can bewritten ∆x/∆t, where ∆x represents thechange in x in the time ∆t. However, thecar might accelerate or decelerate in this in-terval and it may be necessary to know thespeed at a particular instant, say t1. In thiscase the time interval ∆t is made infinitelysmall, i.e. t2 can be as close as necessary tot1. The limit of ∆x/∆t as ∆t approaches zerois the instantaneous velocity at t1. The re-sult of differentiation (i.e. the derivative) ofa function y = f(x) is written dy/dx or f′(x).On a graph of f(x), dy/dx at any point is theslope of the tangent to the curve y = f(x) atthat point. See also integration.

differentiator /dif-ĕ-ren-shee-ay-ter/ Ananalog computer device whose output(which is variable) is proportional to the

65

differentiator

y = f(x)

x

xO

yyy

y

Differentiation of a function y = f(x). Thederivative dy/dx is the limit of y/x asy and y become infinitely small.

Differentiation

time differential of the input (also vari-able). See differential.

digit A symbol that forms part of a num-ber. For example, in the number 3121there are four digits. The ordinary (deci-mal) number system has ten digits (0–9),whereas the binary (base two) systemneeds only two, 0 and 1.

digital Using numerical digits. For exam-ple, a digital watch shows the time in num-bers of hours and minutes and not as theposition of hands on a dial. In general, dig-ital devices work by some kind of countingprocess, either mechanical or electronic.The abacus is a very simple example. Earlycalculating machines counted with me-chanical relays. Modern calculators useelectronic switching circuits.

digital/analog converter A device thatconverts digital signals, usually from a dig-ital computer, into continuously varyingelectrical signals for inputting to an ANA-LOG COMPUTER. See computer.

digital computer See computer.

dihedral /dÿ-hee-drăl/ Formed by two in-tersecting planes. Two planes intersectalong a straight line (edge). The dihedralangle (or dihedron) between the planes isthe angle between two lines (one in eachplane) drawn perpendicular to the edgefrom a point on the edge. The dihedralangle of a polyhedron is the angle betweentwo faces.

dihedron /dÿ-hee-drŏn/ See dihedral.

dilatation /dil-ă-tay-shŏn, dÿ-lă-/ A geo-metrical mapping or projection in which afigure is ‘stretched’, not necessarily by thesame amount in each direction. A square,for example, may be mapped into a rectan-gle by dilatation, or a cube into a cuboid.

dimension 1. The number of coordinatesneeded to represent the points on a line,shape, or solid. A plane figure is said to betwo-dimensional; a solid is three-dimen-

sional. In more abstract studies n-dimen-sional spaces can be used.2. The size of a plane figure or solid. Thedimensions of a rectangle are its length andwidth; the dimensions of a rectangular par-allelepiped are its length, width, andheight.3. One of the fundamental physical quanti-ties that can be used to express other quan-tities. Usually, mass [M], length [L], andtime [T] are chosen. Velocity, for example,has dimensions of [L][T]–1 (distance di-vided by time). Force, as defined by theequation:

F = mawhere m is mass and a acceleration, has di-mensions [M][L][T]–2. See also dimen-sional analysis.4. Of a matrix, the number of rows ornumber of columns. A matrix with 4 rowsand 5 columns is 4 × 5 matrix.

dimensional analysis The use of the di-mensions of physical quantities to checkrelationships between them. For instance,Einstein’s equation E = mc2 can bechecked. The dimensions of speed2 are([L][T]–1)2, i.e. [L]2[T]–2, so mc2 has di-mensions of [M][L]2[T]–2. Energy also hasthese dimensions since it is force[M][L][T]–2 multiplied by distance [L]. Di-mensional analysis is also used to obtainthe units of a quantity and to suggest newequations.

Diophantine equation /dÿ-ŏ-fan-tÿn, -teen, -tin/ See indeterminate equation.

Dirac delta function /di-rak/ Symbolδ(x). A mathematical symbol which is usedto represent a sudden pulse. Strictly speak-ing, the Dirac delta function is not a propermathematical function. It can be defined asthe generalization to continuous variablesof the Kronecker delta δij which is definedfor discrete variables i and j by: δij = 1, if i= j, and =0 if i≠j. The Direct delta functioncan also be defined by the properties: δ(x)= 0, if x≠0, ∫ ∞

–∞δ(x)dx = 1, ∫ ∞–∞f(x)δ(x) = f(0).

It is named for the British physicist PaulDirac (1902–84).

digit

66

directed Having a specified positive ornegative sign, or a definite direction. A di-rected number usually has one of the signs+ or – written in front of it. A directedangle is measured from one specified line tothe other. If the direction were reversed,the size of the angle would be a negativenumber.

directed line A straight line in which a di-rection along the line is specified. The di-rection specified is called the positivedirection: the opposite direction is calledthe negative direction. It is also possible toindicate the direction on a directed line byspecifying a point 0 on the line to be theorigin, with the positive direction from thisorigin being directed towards the end x ofthe line. In this notation the directed line isgiven by 0x.

directed line-segment The combinationof a straight line and a specific directionalong that line. For example if P and Q aretwo points on a straight line then PQ is thedirected line-segment from P to Q, and QPis the directed line-segment from Q to P.The concept of a directed line-segment isclosely related to that of a VECTOR.

direction A property of vector quantities,usually defined in reference to a fixed ori-

gin and axes. The direction of a curve at apoint is the angle from the x axis to the tan-gent at the point.

directional derivative The rate ofchange of a function with respect to dis-tance s in a particular direction, or along aspecified curve. Going from a pointP(x,y,z) in the direction that makes anglesα, β, and γ with the x, y, and z axs respec-tively, the directional derivative of a func-tion f(x,y,z) is

df/ds = (∂f/∂x)cosα + (∂f/∂y)cosβ+ (∂f/∂z)cosγ

If there is a direction for which the direc-tional derivative is a maximum, then thisderivative is the gradient of f (grad f) atpoint P. See also grad.

direction angle The angle between a lineand one of the axs in a rectangular Carte-sian coordinate system. In a plane system,it is the angle, α, that the line makes withthe positive direction of the x-axis. In threedimensions, there are three direction an-gles, α, β, and γ, for the x, y, and z axes re-spectively. If two direction angles areknown, the third can be calculated by therelationship:

cos2α + cos2β + cos2γ = 1Cosα, cosβ, and cosγ are called the direc-tion cosines of the line, sometimes given

67

direction angle

z

P (x, y, z )

yO

x

Direction angles: α, β, and γ are made by the line OP with the x-, y-, and z-axes respectively.

the symbols l, m, and n. Any three numbersin the ratio l:m:n are called the directionnumbers or the direction ratio of the line. Ifa line joins the point A (x1,y1,z1) and thepoint B (x2,y2,z2) and the distance betweenA and B is D, then

l = (x2 – x1)/Dm = (y2 – y1)/Dn = (z2 – z1)/D

direction cosines See direction angle.

direction numbers See direction angle.

direction ratio See direction angle.

director circle The locus of points P(x,y)from which two perpendicular tangents aredrawn to an ellipse of the form x2/a2 +y2/b2 = 1 as the points of contact vary. Theequation of a director circle defined in thisway is x2 = a2 + b2, as can be shown by ap-plying coordinate geometry to the equa-tions of the two tangents from P(x,y) to theellipse.

direct proof A logical argument in whichthe theorem or proposition being proved isthe conclusion of a step-by-step processbased on a set of initial statements that areknown or assumed to be true. Compare in-direct proof.

directrix /dă-rek-triks/ 1. A straight lineassociated with a conic, from which theshortest distance to any point on the conicmaintains a constant ratio with the dis-tance from that point to the focus. See alsoconic.2. A plane curve defining the base of a coneor cylinder.

discontinuity See continuous function.

discontinuous See continuous function.

discount 1. The difference between theissue price of a stock or share and its nom-inal value when the issue price is less thanthe nominal value. Compare premium.2. A reduction in the price of an article orcommodity for payment in cash (cash dis-count), or for a large order (bulk discount),or for a retailer who will be selling thegoods on to members of the public (tradediscount).

discrete Denoting a set of events or num-bers in which there are no intermediate lev-els. The set of integers, for example, isdiscrete but the set of rational numbers isnot. Between any two rational numbers, nomatter how close, there can always befound another rational number. The re-sults of tossing dice form a discrete set ofevents, since a die has to land on one of itssix faces. Putting the shot, on the otherhand, does not have a discrete set of out-comes, since it may travel for any distancein a continuous range of lengths.

discriminant /dis-krim-ă-nănt/ The ex-pression (b2 – 4ac) in a QUADRATIC EQUA-TION of the form ax2 + bx + c = 0. If theroots of the equation are equal, the dis-criminant is zero. For example, in

x2 – 4x + 4 = 0b2 – 4ac = 0 and the only root is 2. If thediscriminant is positive, the roots are dif-ferent and real. For example, in

x2 + x – 6 = 0b2 – 4ac = 25 and the roots are 2 and –3. Ifthe discriminant is negative, the roots ofthe equation are complex numbers. For ex-ample, the equation:

x2 + x + 1 = 0has roots [–½ – (√3)/2]i and [–½ + (√3)/2]i.

direction cosines

68

P Q P ∨ Q

T T FT F TF T TF F F

Disjunction (exclusive)

P Q P ∨ Q

T T TT F TF T TF F F

Disjunction (inclusive)

disjoint Two sets are said to be disjoint ifthey have no members in common; i.e. if A∩ B = 0 then A and B are disjoint.

disjunction Symbol: ∨ In logic, the rela-tionship or between two propositions orstatements. Disjunction can be either inclu-sive or exclusive. Inclusive disjunction(sometimes called alternation) is the onemost commonly used in mathematicallogic, and can be interpreted as ‘one or theother or both’. For two propositions P andQ, P ∨ Q is false if P and Q are both false,and true in all other cases. The more rarelyused exclusive disjunction can be inter-preted as ‘either one or the other but notboth’. With this definition P ∨ Q is falsewhen P and Q are both true, as well aswhen they are both false. The truth-tabledefinitions for both types of disjunction areshown in the illustrations. Compare con-junction. See also truth table.

disk A device that is widely used in com-puter systems to store information. It is aflat circular metal plate coated usually onboth sides with a magnetizable substance.Information is stored in the form of smallmagnetized spots, which are closely packedin concentric tracks on the coated surfacesof the disk. The spots are magnetized inone of two directions so that the informa-tion is in binary form. The magnetizationpattern of a group of spots represents a let-ter, digit (0–9), or some other character.One disk can store several million charac-ters. Information can be altered or deletedas necessary by magnetic means. Disks areusually stacked on a common spindle in asingle unit known as a disk pack. Diskpacks storing 200 million characters arecommon.

Information can be recorded on a diskusing a special typewriter; this method isknown as key-to-disk. The information isfed into a computer using a complex devicecalled a disk unit. The disk pack is rotatedat very great speed in the disk unit. Smallelectromagnets, known as read–writeheads, move radially in and out over thesurfaces of the rotating disks. They extract(read) or record (write) items of informa-tion at specified locations on a track, fol-

lowing instructions from the centralprocessor. The time to reach a specified lo-cation is very short. This factor, togetherwith the immense storage capacity, makesthe disk unit a major backing store in acomputer system. Compare drum; mag-netic tape. See also floppy disk; hard disk.

diskette See floppy disk.

dispersion A measure of the extent towhich data are spread about an average.The range, the difference between thelargest and smallest results, is one measure.If Pr is the value below which r% of the re-sults occur, then the range can be writtenas (P100 – P0). The interquartile range is(P75 – P25). The semi-interquartile range is(P75 –P25)/2. The mean deviation of X1, X2,…, Xn measures the spread about the MEAN

X and is

Σn

|xj – x|/nIf values X1, X2, …, Xk occur with fre-quencies f1, f2, …fk it becomes

Σn

fj |xj – x|/Σfj

displacement Symbol: s The vector formof distance, measured in meters (m) and in-volving direction as well as magnitude.

dissipation The removal of energy from asystem to overcome some form of resistiveforce. Without resistance (as in motion in avacuum) there can be no dissipation. Dissi-pated energy normally appears as thermalenergy.

distance Symbol: d The length of the pathbetween two points. The SI unit is themeter (m). Distance may or may not bemeasured in a straight line. It is a scalar;the vector form is displacement.

distance formula The formula for thedistance between two points (x1,y1) and(x2,y2) in Cartesian coordinates. It is:

√[(x1 – x2)2 + (y1 – y2)2]

distance ratio (velocity ratio) For a MA-CHINE, the ratio of the distance moved by

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distance ratio

the effort in a given time to the distancemoved by the load in the same time.

distribution function For a random vari-able x, the function f(x) that is equal to theprobability of each value of x occurring. Ifall values of x between a and b are equallylikely, x has a uniform distribution in thisinterval and a graph of the distributionfunction f(x) against x is a horizontal line.For example, the probability of the results1 to 6 when throwing dice is a uniform dis-tribution. Continuous random variablesusually have a varying distribution func-tion with a maximum value xm and inwhich the probability of x decreases as xmoves away from xm. The cumulative dis-tribution function F(x) is the probability ofa value less than or equal to x. For the diceexample, F(x) is a step function that in-creases from zero to one in six equal steps.For continuous functions, F(x) is often ans-shaped curve. In both cases F(x) is thearea under the curve of f(x) to the left of x.

distributive Denoting an operation that isindependent of being carried out before orafter another operation. For two opera-tions • and °, • is distributive with respectto ° if a•(b°c) = (a•b)°(a•c) for all values ofa, b, and c. In ordinary arithmetic, multi-plication is distributive with respect to ad-dition [a(b + c) = ab + ac] and tosubtraction.

In set theory intersection (∩) is distrib-utive with respect to union (∪):

[A∩(B∪C) = (A∩B)∪(A∩C)]See also associative; commutative.

div /div/ (divergence) Symbol: ∇. A scalaroperator that, for a three-dimensional vec-tor function F(x,y,z), is the sum of thescalar products of the unit vectors and thepartial derivatives in each of the three com-ponent directions. That is:

div F = ∇.F = i.∂F/∂x +j.∂F/∂y + k.∂F/∂z

In physics, div F is used to describe the ex-cess flux leaving an element of volume inspace. This may be a flow of liquid, a flowof heat in a field of varying temperature, oran electric or magnetic flux in an electric ormagnetic field. If there is no source of flux

(heat source, electric charge, etc.) withinthe volume, then div F = 0 and the totalflux entering the volume equals the totalflux leaving. See also grad.

divergence theorem (Gauss’s theorem;Ostrogradsky’s theorem) A basic theoremin vector calculus that states that the inte-gral of the divergence of a vector over avolume enclosed by a surface is equal to theintegral of the normal component of thevector over the closed surface. This can beexpressed by the equation

∫∫F.ndS = ∫∫∫divFdV,where F is the vector, n is the normal to S,and V is the volume. The divergence theo-rem has important physical applications,particularly in electrostatics (where it isknown as Gauss’s law).

divergent sequence A SEQUENCE in whichthe difference between the nth term and theone after it is constant or increases as n in-creases. 1,2,4,8,… is divergent. A diver-gent sequence has no limit. Compareconvergent sequence. See also divergent se-ries; geometric sequence.

divergent series A SERIES in which thesum of all the terms after the nth term doesnot decrease as n increases. A divergent se-ries, unlike a convergent series, has no sumto infinity. An infinite series a1 + a2 + … isdivergent if the partial sums a1 + a2 + … +an tend to infinity as n tends to infinity. Forexample, the series 1 + 2 + 3 + 4 + … is di-vergent. Compare convergent series. Seealso divergent sequence; geometric series.

dividend 1. The number into which an-other number (the divisor) is divided togive a quotient. For example, in 16 ÷ 3, 16is the dividend and 3 is the divisor.2. A share of the profits of a corporationpaid to shareholders. The rate of dividendpaid will depend on profits in the precedingyear. It is expressed as a percentage of thenominal value of the shares. For example,a 10% dividend on a 75c share will pay7.5c per share (independent of the marketprice of the share). See also yield.

distribution function

70

dividers A drawing instrument, similar tocompasses, but with sharp points on bothends. Dividers are used for measuringlengths on a drawing, or for dividingstraight lines.

division Symbol: ÷ The binary operationof finding the quotient of two quantities.Division is the inverse operation to MULTI-PLICATION. In arithmetic, the division oftwo numbers is not commutative (2 ÷ 3 ≠ 3÷ 2), nor associative [(2 ÷ 3) ÷ 4 ≠ 2 ÷ (3 ÷4)]. The identity element for division is oneonly when it comes on the right hand side(5 ÷ 1 = 5 but 1 ÷ 5 ≠ 5).

division of fractions See fractions.

divisor The number by which anothernumber (the dividend) is divided to give aquotient. For example, in 16 ÷ 3, 16 is thedividend and 3 is the divisor. See also fac-tor.

documentation Written instructions andcomments that give a full description of acomputer program. The documentationdescribes the purposes for which the pro-gram can be used, how it operates, theexact form of the inputs and outputs, andhow the computer must be operated. It al-lows the program to be amended whennecessary or to be converted for use on dif-ferent types of machines.

dodecagon /doh-dek-ă-gon/ A plane fig-ure with 12 sides and 12 interior angles.The sum of the interior angles is 1800°.

dodecahedron /doh-dek-ă- hee-drŏn/ Asolid figure with 12 faces. The faces of aregular dodecahedron are all regular pen-tagons.

domain A set of numbers or quantities onwhich a mapping is, or may be, carried out.In algebra, the domain of the function f(x)is the set of values that the independentvariable x can take. If, for example, f(x)represents taking the square root of x, thenthe domain might be defined as all the pos-itive rational numbers. See also range.

D operator The differential operatord/dx. The derivative df/dx of a functionf(x) is often written as Df. This notation isused in solving DIFFERENTIAL EQUATIONS. Asecond derivative, d2f/dx2, is written asD2f, a third derivative, d3f/dx3, as D3f, andso on. In some ways, the D operator can betreated like an ordinary algebraic quantity,despite the fact that it has no numericalvalue. For example, the differential equa-tion

d2y/dx2 + 2xdy/dx + dy/dx + 2x = 0or

D2y + 2xD + D + 2x = 0can be factorized to (D + 2x)(D + 1) = 0. Inthis case, (D + 2x) then operates on thefunction (D + 1).

dot product See scalar product.

double-angle formulae See addition for-mulae.

double integral The result of integratingthe same function twice, first with respectto one variable, holding a second variableconstant, and then with respect to the sec-ond variable, holding the first variable con-stant. For example, if f(x,y) is a function ofthe variables x and y, then the double inte-gral, first with respect to x and then withrespect to y, is:

∫∫f(x,y)dydxThis is equivalent to summing f(x,y) overintervals of both x and y, or to finding thevolume bounded by the surface represent-

71

double integral

Dodecahedron: a regular dodecahedron hasregular pentagonal faces.

ing f(x,y). The integral is not affected bythe order in which the integrations are car-ried out if they are definite integrals. An-other kind of double integral is the result ofintegrating twice with respect to the samevariable. For example, if a car’s accelera-tion a increases with time t in a knownway, then the integral

∫adtis the velocity (v) expressed as a function oftime; the double integral

∫∫adt2 = ∫v.dt = xwhere x is the distance traveled as a func-tion of time.

double point A singular point on a curveat which the curve crosses itself or is tan-

gential to itself. There are several types ofdouble point. At a node the curve crossesover itself forming a loop. In this case it hastwo distinct tangents. At a cusp it doubleback on itself and has only one tangent. Atan acnode two arcs of a curve touch eachother and have the same tangent but, un-like a cusp, the arcs continue through thesingular point to form four arms. An iso-lated double point may also occur. Thissatisfies the equation of the curve but doesnot lie on the main arc of the curve. Seealso isolated point; multiple point.

drum A metal cylinder coated with a mag-netizable substance and used in a computersystem to store information.

double point

72

•O

y

x

node cusp

• x

y

O

•isolated double point

x

y

O

•

y

xO

tacnode

Four types of double point at the origin of a two-dimensional Cartesian coordinate system.

dual See duality.

duality The principle in mathematicswhereby one true theorem can be obtainedfrom another merely by substituting cer-tain words in a systematic way. In a plane,‘point’ and ‘line’ are dual elements and, forexample, drawing a line thrugh a point andmarking a point on a line are dual opera-tions. Theorems that can be obtained fromone another by replacing each element andoperation by its dual are dual theorems.Dual theorems feature prominently inPROJECTIVE GEOMETRY. In logic, ‘implied’and ‘is implied by’ are dual relations andmay be interchanged together with the log-ical connectives ‘and’ and ‘or’. In settheory, the relations ‘is contained in’ and‘contains’ are dual relations and can be in-terchanged along with the ‘union’ and ‘in-tersection’. In general, the principle ofduality is found where the structure underconsideration is a LATTICE.

dummy variable A symbol that can bereplaced by any other symbol withoutchanging the meaning of the expression inwhich it occurs.

duodecahedron /dew-ŏ-dek-ă-hee-drŏn/See dodecahedron.

duodecimal /dew-ŏ-dess-ă-măl/ Denotingor based on twelve. In a duodecimal num-ber system there are twelve different digitsinstead of ten. If, for example, ten andeleven were given the symbols A and B re-

spectively, 12 would be written as 10 and22 as 1A. Duodecimal numbers are of littleuse, but some duodecimal units (1 foot =12 inches) are still in use. Compare binary;decimal; hexadecimal; octal.

duplication of the cube A classic prob-lem of ancient Greek geometry, to find away, using only a straight edge and com-pass, to find the side of a cube the volumeof which is exactly double that of a givencube. It is now known that this cannot bedone. This is the case because it is equiva-lent to the problem of finding the cube rootof 2 using a ruler and compass. This is im-possible because geometrical methods in-volving a ruler and compass can only leadto length in the cases of addition, subtrac-tion, multiplication, division and squareroots but definitely not cube roots.

duty A tax levied on certain kinds oftransactions. Examples include taxes onimporting and exporting goods and the taxlevied on alcohol and tobacco products.

dynamic friction See friction.

dynamics The study of how objects moveunder the action of forces. This includesthe relation between force and motion andthe description of motion itself. See alsomechanics.

dyne /dÿn/ Symbol: dyn The former unitof force used in the c.g.s. system. It is equalto 10–5 N.

73

dyne

e A fundamental mathematical constantdefined by the series:

e = lim(1 + 1/n) n → ∞or by

e = 1 + 1/1! + 1/2! … + 1/n! + …e is the base of natural logarithms. It isboth irrational and transcendental. It is re-lated to π by the formula eiπ = –1. The func-tion ex has the property that it is its ownderivative, that is, dex/dx = ex.

eccentric Denoting intersecting circles,spheres, etc., that do not have the samecenter. Compare concentric.

eccentricity A measure of the shape of aconic. The eccentricity is the ratio of thedistance of a point on the curve from afixed point (the focus) to the distance froma fixed line (the directrix). For a parabola,the eccentricity is 1. For a hyperbola, it isgreater than 1, and for an ellipse, it is be-tween 0 and 1. A circle has an eccentricityof 0.

echelon matrix A matrix in which all therows of a matrix in which all the entries arezero come beneath the rows in which notall the entries are zero and in which thefirst non-zero entry in a row with non-zerovalues is 1, with this entry being in a col-umn which is to the right of the first 1 inthe row above it. It is possible to convertany matrix into an echelon matrix by usingthe technique of GAUSSIAN ELIMINATION.

ecliptic /i-klip-tik/ The apparent pathalong which the Sun moves each year. It isthe great circle formed by the intersectionof the plane of the Earth’s orbit with the ce-lestial sphere.

edge A straight line where two faces of asolid meet. A cube has twelve edges.

efficiency Symbol: η A measure used forprocesses of energy transfer; the ratio ofthe useful energy produced by a system ordevice to the energy input. For example,the efficiency of an electric motor is theratio of its mechanical power output to theelectrical power input. There is no unit ofefficiency; however efficiency is oftenquoted as a percentage. In practical sys-tems some dissipation of energy always oc-curs (by friction, air resistance, etc.) andthe efficiency is less than 1. For a machine,the efficiency is the force ratio divided bythe distance ratio.

effort The force applied to a MACHINE.

eigenfunction /ÿ-gĕn-fung-shŏn/ Seeeigenvalue.

eigenvalue /ÿ-gĕn-val-yoo/ (from Germaneigen = ‘allowed’) An eigenvalue for a lin-ear transformation L on a vector space V isa scalar λ for which there is a nonzero so-lution vector v in V such that Lv = λv andv satisfies any given boundary conditions.The vector v is an eigenvector (or charac-teristic vector) belonging to the eigenvalueλ. An eigenvector for a linear operator ona vector space whose vectors are functionsis also called an eigenfunction. As an ex-ample, consider the equation –y″ = λy withboundary conditions y = 0 when x = a or b.

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E

1 3 8 −2 70 1 5 3 −40 0 0 1 60 0 0 0 0

Echelon matrix

A nonzero solution exists only if λ = n2π2

where n is an integer. The solution then isy = C[sinnπ(x – a)]/(b – a)

where C is an arbitrary constant. Theeigenvalues of the equation are given byn2π2 (where n is an integer). The corre-sponding eigenfunctions (or eigenvectors)are given by

y = C[sin(x – a)]/(b – a)

eigenvector /ÿ-gĕn-vek-ter/ See eigen-value.

elastic collision A collision for which therestitution coefficient is equal to one. Ki-netic energy is conserved during an elasticcollision. In practice, collisions are not per-fectly elastic as some energy is transferredto internal energy of the bodies. See alsorestitution; coefficient of.

electronvolt /i-lek-tron-vohlt/ Symbol: eVA unit of energy equal to 1.602 191 7 ×10–19 joule. It is defined as the energy re-quired to move the charge of an electronacross a potential difference of one volt. Itis normally used only to measure energiesof elementary particles, ions, or states.

element /el-ĕ-mĕnt/ 1. A single item thatbelongs to, or is a member of, a set. ‘Feb-ruary’, for example, is an element of the setmonth in the year. The number 5 is an el-ement of the set of integers between 2 and10. In set notation this is written as

5 ∈ 2,3,4,5,6,7,8,9,102. A line segment forming part of thecurved surface of a surface, as of a cone orcylinder.3. A small part of a line, surface, or volumesummed by integration.4. (of a matrix) See matrix.

elevation A drawing that shows the ap-pearance of a solid object as viewed fromthe front, back or side. See plan. See alsoangle of elevation.

eliminant /i-lim-ă-nănt/ (characteristic; re-sultant) The relationship between coeffi-cients that results from eliminating thevariable from a set of simultaneous equa-tions. For example, in the equations

a1x + b1y + c1 = 0a2x + b2y + c2 = 0a3x + b3y + c3 = 0

the eliminant is given by the matrix of co-efficients:

| a1 b1 c1 || a2 b2 c2 || a3 b3 c3 |

elimination Removing one of the un-knowns in an algebraic equation, for ex-ample, by the substitution of variables orby cancellation.

ellipse A CONIC with an eccentricity be-tween 0 and 1. An ellipse has two foci. Aline through the foci cuts the ellipse at twovertices. The line segment between the ver-tices is the major axis of the ellipse. Thepoint on the major axis mid-way betweenthe vertices is the center of the ellipse. Aline segment through the center perpendic-ular to the major axis is the minor axis. Ei-ther of the chords of the ellipse through afocus parallel to the minor axis is a latusrectum. The area of an ellipse is πab, wherea is half the major axis and b is half theminor axis. (Note that for a circle, in whichthe eccentricity is zero, a = b = r and thearea is πr2.)

The sum property of an ellipse is thatfor any point on the ellipse the sum of thedistances from the point to each focus is aconstant. The ellipse also has a reflectionproperty; for a given point on the ellipsethe two lines from each focus to the pointmake equal angles with a tangent at thatpoint.

In Cartesian coordinates the equation:x2/a2 + y2/b2 = 1

represents an ellipse with its center at theorigin. The major axis is on the x-axis andthe minor axis on the y-axis. The majoraxis is 2a and the minor axis is 2b. The fociof the ellipse are at the points (+ea,0) and(–ea,0), where e is the eccentricity. The twodirectrices are the lines x = a/e and x = –a/e.The length of the latus rectum is 2b2/a.

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ellipse

ellipsoid /i-lip-soid/ A solid body orcurved surface in which every plane cross-section is an ellipse or a circle. An ellipsoidhas three axes of symmetry. In three-di-mensional Cartesian coordinates, the equa-tion of an ellipsoid with its center at theorigin is:

x2/a2 + y2/b2 + z2/c2 = 1where a, b, and c are the points at which itcrosses the x, y, and z axes respectively. Inthis case the axes of symmetry are the co-ordinate axes. A prolate ellipsoid is onegenerated by rotating an ellipse about itsmajor axis. An oblate ellipsoid is generatedby rotation about the minor axis.

elliptic paraboloid A CONICOID that isdescribed by the equation x2/a2 + y2/b2 =2z/c, where a, b and c are constants. The xzand yz planes are planes of reflection sym-metry. Cross-sections of an elliptic parabo-loid formed by planes that satisfy theequation z = k, where k is a non-negativenumber are, in general, ellipses. In thespecial case in which a = b, the ellipsesbecome circles. If k is a negative numberthen the plane does not intersect the ellip-tic paraboloid. Cross-sections of an ellip-tic paraboloid formed by planes that areparallel to either the xz or yz plane areparabolas.

ellipsoid

76

y

x

S

T

P

F1F2 OA B

C

D

dir

ectr

ix

dir

ectr

ix

Ellipse: AB is the major axis and CD is the minor axis. F1 and F2 are the foci. Lines from these toany point P make equal angles with a tangent ST.

prolate ellipsoidoblate ellipsoid

An ellipsoid can be generated by rotating an ellipse about one of its axes. An oblate ellipsoid isgenerated by rotation about the minor axis and a prolate ellipsoid is generated by rotation abouta major axis.

empirical /em-pi-ră-kăl/ Derived directlyfrom experimental results or observations.

empirical probability The probability ofan event occurring as determined empiri-cally by carrying out a large number of tri-als in which the event could take place. Thenumber of times the event occurs iscounted, as is the number of trials. Thevalue of the number of times the event oc-curred divided by the number of trials (n) isthen calculated. The empirical probabilityis defined to be the limiting value of thisratio as n → ∞. In practice, taking n → ∞means that the number of trials has to be-come very large. Exactly how large de-pends on the nature of the problem beingexamined. It can also happen that empiri-cal probability gives a value that is differ-ent from the value theoretically expected.For example, if a coin is tossed one mightexpect that the probability of it landinghead up is exactly 0.5 but it might be thecase that some bias in the coin causes a dif-ferent empirical probability.

empty set Symbol: ∅ (null set) The setthat contains no elements. For example,the set of ‘natural numbers less than 0’ isan empty set. This could be written as m:m∈N; m<0 = ∅.

energy Symbol: W A property of a system– its capacity to do work. Energy and workhave the same unit: the joule (J). It is con-venient to divide energy into KINETIC EN-ERGY (energy of motion) and POTENTIAL

ENERGY (‘stored’ energy). Names are givento many different forms of energy (chemi-cal, electrical, nuclear, etc.); the only realdifference lies in the system under discus-sion. For example, chemical energy is thekinetic and potential energies of electronsin a chemical compound. See also mass–en-ergy equation.

engineering notation See standard form.

enlargement A geometrical PROJECTION

that produces an image larger (smaller ifthe scale factor is less than 1) than, but sim-ilar to, the original shape.

entailment In logic, the relationship thatholds between two (or more) propositionswhen one can be deduced from the other. Ifconclusion C is deducible from premisses Aand B, then A and B are said to entail C.See deduction.

entropy /en-trŏ-pee/ A physical quantitythat originated in thermodynamics and sta-tistical mechanics but can be defined forgeneral dynamical systems and also de-fined in terms of information theory. Thedefinition of entropy in statistical mechan-ics, which provides a molecular picture forentropy in thermodynamics, means thatentropy can be regarded as a measure ofthe disorder of a system. In thermodynam-ics, entropy is a measure of a quantity thatincreases in a system as the ability of theenergy of that system to do work decreases.The entropy S of a system in thermody-namics is defined by dS = dQ/T, where dSis an infinitesimal change in the entropy ofa system, dQ the infinitesimal amount ofheat absorbed by the system and T is thethermodynamic temperature. In statisticalmechanics entropy is given by S = klnW +B, where k is a constant of statistical me-chanics known as the Boltzmann constant,W is the statistical probability for the sys-tem, i.e. the number of distinguishableways the system can be realized, and B isanother constant. Here, lnW is the naturallogarithm of W. The definition of entropyused in general dynamical systems leads toa non-zero value of entropy being associ-ated with chaos. In information theory, en-tropy can be regarded as a measure of theuncertainty in our knowledge about a sys-tem.

enumerable /i-new-mĕ-ră-băl/ See count-able.

envelope Consider a one-parameter fam-ily of curves in three dimensions – i.e. afamily of curves that can be represented interms of a common parameter that is con-stant along each curve, but is changed fromcurve to curve. The envelope of this familyof curves is the surface traced out by thesecurves. This surface is tangent to everycurve of the family. Its equation is obtained

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envelope

by eliminating the parameter between theequation of the curve and the partial deriv-ative of this equation with respect to theparameter. For example, the envelope ofthe family of paraboloids given by x2 + y2

= 4a(z – a) is the equation obtained byeliminating a from the equations x2 + y2 =4a(z – a) and z – 2a = 0, i.e. the circularcone x2 + y2 = z2.

epicycle /ep-ă-sÿ-kăl/ A circle that rollsaround the circumference of another, trac-ing out an EPICYCLOID.

epicycloid /ep-ă-sÿ-kloid/ The planecurve traced out by a point on a circle orepicycle rolling along the outside of an-other fixed circle. For example, if a smallcog wheel turns on a larger stationarywheel, then a point on the rim of thesmaller wheel traces out an epicycloid. In atwo-dimensional Cartesian-coordinate sys-tem that has a fixed circle of radius a cen-tred at the origin and another of radius brolling around the circumference, theepicycloid is a series of continuous arcsthat move away from the first circle to adistance 2b and then return to touch itagain at a cusp point where the next arcbegins. The epicycloid has only one arc if a= b, two if a = b/2, and so on. If the anglebetween the radius from the origin to themoving point of contact between the two

circles is θ, the epicycloid is defined by theparametric equations:x = (a + b) cosθ – a cos[(a + b)θ/a]y = (a + b) sinθ – a sin[(a + b)θ/a] See alsohypocycloid.

equal Describing quantities that are thesame. For example, the quantities on theleft-hand side of an equation are equal tothe quantities on the right-hand side of theequation; two sides of an isosceles triangleare equal in length. Equality is often de-noted by the equals sign =.

equality Symbol: = The relationship be-tween two quantities that have the samevalue or values. If two quantities are notequal, the symbol ≠ is used. For example, x≠ 0 means that the variable x cannot takethe value zero. When the equality is onlyapproximate, the symbol ≅ is used. For ex-ample, if ∆x is small compared to x then x+ (∆x)2 ≅ x. When two expressions are ex-actly equivalent the symbol ≡ is used. Forexample sin2α ≡ 1 – cos2α because it ap-plies for all values of the variable α. Seealso equation; inequality.

equation A mathematical statement thatone expression is equal to another, that is,two quantities joined by an equals sign. Analgebraic equation contains unknown orvariable quantities. It may state that twoquantities are identical for all values of the

epicycle

78

y

xO

Pb

a

Epicycloid: the epicycloid traced out by a point P on a circle of radius b that rolls round a large circle of radius a.

variables, and in this case the identity sym-bol ≡ is normally used. For example:

x2 – 4 ≡ (x – 2)(x + 2)is an identity because it is true for all valuesthat x might have. The other kind of alge-braic equation is a conditional equation,which is true only for certain values of thevariables. To solve such an equation, thatis, to find the values of the variables atwhich it is valid, it often has to be re-arranged into a simpler form. In simplify-ing an equation, the same operation can becarried out on the expressions on bothsides of the equals sign. For example,

2x – 3 = 4x + 2can be simplified by adding 3 to both sidesto give

2x = 4x + 5then subtracting 4x from both sides to give

–2x = 5and finally dividing both sides by –2 to ob-tain the solution x = –5/2.

This kind of equation is called a linearequation, because the highest power of thevariable x is one. It could also be written inthe form –2x – 5 = 0. On a Cartesian coor-dinate graph,

y = –2x – 5is a straight line that crosses the x-axis at x= –5/2.

Performing the same operation on bothsides of an equation does not necessarilygive an equation exactly equivalent to theoriginal. For example, starting with x = yand squaring both sides gives x2 = y2,which means that x = y or x = –y. In thiscase the symbol ⇒ is used between theequations, meaning that the first impliesthe second, but the second does not implythe first. That is,

x = y ⇒ x2 = y2

Where the two equations are equiva-lent, the symbol ⇔ is used, for example,

2x = 2 ⇔ x = 1

equations of motion Equations that de-scribe the motion of an object with con-stant accleration (a). They relate thevelocity v1 of the object at the start of tim-ing to its velocity v2 at some later time tand to the object’s displacement s. Theyare:

v2 = v1 + at2

s = (v1 + v2)t/2s = v1t + at2/2s = v2t – at2/2v2

2 = v12 + 2as

equator On the Earth’s surface, the circleformed by the plane cross-section that per-pendicularly bisects the axis of rotation.The plane in which the circle lies is calledthe equatorial plane. A similar circle onany sphere with a defined axis is also calledan equator, or equatorial circle.

equatorial See equator.

equiangular /ee-kwee-ang-gyŭ-ler, ek-wee-/ Having equal angles.

equidistant /ee-kwă-dis-tănt, ek-wă-/ Atthe same distance. For example, all pointson the circumference of a circle are equidis-tant from the center.

equilateral /ee-kwă-lat-ĕ-răl, ek-wă-/Having sides of equal length. For example,an equilateral triangle has three sides ofequal length (and equal interior angles of60°).

equilibrant /i-kwil-ă-brănt/ A single forcethat is able to balance a given set of forcesand thus cause equilibrium. It is equal andopposite to the resultant of the givenforces.

equilibrium /ee-kwă-lib-ree-ŭm, ek-wă-/A state of constant momentum. An objectis in equilibrium if: 1. its linear momentumdoes not change (it moves in a straight lineat constant speed and has constant mass,or is at rest);2. its angular momentum does not change(its rotation is zero or constant).For these conditions to be met: 1. the re-sultant of all outside forces acting on theobject must be zero (or there are no outsideforces);2. there is no resultant turning-effect (mo-ment).

An object is not in equilibrium if any ofthe following are true: 1. its mass is chang-ing;

79

equilibrium

2. its speed is changing;3. its direction is changing;4. its rotational speed is changing.

See also stability.

equivalence /i-kwiv-ă-lĕns/ See bicondi-tional.

equivalence class The set of elements [x]in a set S that is equivalent to x by anEQUIVALENCE RELATION on that set. The dis-tinct equivalence classes of a set are said tobe a partition of that set, with each elementof the set belonging to only one of theequivalence classes.

equivalence principle See relativity;theory of.

equivalence relation A binary relation Rdefined on a set S is said to be an equiva-lence relation if it satisfies the followingthree properties: (1) xRx for every x in S –this is the property of reflexivity, (2) if xRythen yRx – this is the property of symme-try, and (3) if xRy and yRz then xRz – thisis the property of transitivity. Such rela-tions are especially important because theypartition the set on which they are definedinto disjoint classes, known as equivalenceclasses.

equivalent fractions /i-kwiv-ă-lĕnt/ Twoor more fractions that represent the samerational number; they can be canceled untilthey are the same fraction (see cancella-tion). For example, 10/16, 15/24, 45/72and 65/104 are equivalent fractions (equalto 5/8).

erase head The part of any magneticrecording device – tape or disk – that erasesdata (recorded signals) before the writehead records new data. See write head.

Eratosthenes, sieve of /e-ră-tos-th’ĕ-neez/A method of finding prime numbers. Tofind all the prime numbers less than a givennumber n one first goes through all thenumbers from 2 to n removing all thosethat are multiples of 2. Then all those after3 are examined and all the multiples of 3are removed. One proceeds in this way

with all the numbers less than or equal to√n. Only prime numbers will remain. Thismethod in number theory is named for theGreek astronomer Eratosthenes of Cyrene(c. 276 BC–c. 194 BC).

erg A former unit of energy used in thec.g.s. system. It is equal to 10–7 joule.

Erlangen program /er-lang-ĕn/ A view ofgeometry in which each type of geometry isconsidered to be characterized by a groupof transformations and the invariants ofsuch transformations. For example, thegeometry of a crystal lattice is character-ized by the SPACE GROUP of that lattice. TheErlangen program is so called because theeminent German mathematician FelixKlein put forward this idea when he be-came a member of the faculty of ErlangenUniversity in 1872. Not all types of geom-etry can be incorporated into the Erlangenprogram. For example, RIEMANNIAN GEOM-ETRY cannot be incorporated directly, al-though there are some ways of generalizingthe Erlangen program that can include Rie-mannian geometry.

error The uncertainty in a measurementor estimate of a quantity. For example, ona mercury thermometer, it is often possibleto read temperature only to the nearest de-gree Celsius. A temperature of 20°C shouldthen be written as (20 ± 0.5)°C because itreally means ‘between 19.5°C and 20.5°C’.There are two basic types of error. Ran-dom error occurs in any direction, cannotbe predicted, and cannot be compensatedfor. It includes the limitations in the accu-racy of the measuring instrument and thelimitations in reading it. Systematic errorarises from faults or changes in conditionsthat can be corrected for. For example, if a1 gram weight used on a balance is 2 mil-ligrams underweight, every measurementtaken with it will be 2 milligrams less thanthe correct value.

escape speed (escape velocity) The mini-mum initial speed (velocity) that an objectmust have in order to escape from the sur-face of a planet (or moon) against the grav-itational attraction. The escape speed is

equivalence

80

equal to √(2GM/r), where G is the gravita-tional constant, M is the mass of the planet,and r is the radius of the planet. The con-cept also applies to the escape of the objectfrom a distant orbit.

estimate A rough calculation, usually in-volving one or more approximations,made to give a preliminary answer to aproblem.

ether /ee-th’er/ (aether) A hypotheticalfluid, formerly thought to permeate allspace and to be the medium through whichelectromagnetic waves were propagated.See relativity; theory of.

Euclidean algorithm /yoo-klid-ee-ăn/ Amethod of finding the highest common fac-tor of two positive integers. The smallernumber is divided into the larger. The re-mainder is then divided into the smallernumber, obtaining a second remainder.This second remainder is then divided intothe first remainder, to give a third remain-der. This is divided into the second, and soon, until a zero remainder is obtained. Theremainder preceding this is the highestcommon factor of the two numbers. Forexample, the numbers 54 and 930. Divid-ing 54 into 930 gives 17 with a remainderof 12. Dividing 12 into 54 gives 4 with a re-mainder of 6. Dividing 6 into 12 gives 2with a remainder of 0. Thus 6 is the high-est common factor of 54 and 930. The al-gorithm is named for the Greekmathematician Euclid (c. 330 BC–c. 260BC).

Euclidean geometry A system of geome-try described by the Greek mathematicianEuclid in his book Elements (c. 300 BC). Itis based on a number of definitions – point,line, etc. – together with a number of basicassumptions. These were axioms or ‘com-mon notions’ – for example, that the wholeis greater than the part – and postulatesabout geometric properties – for example,that a straight line is determined by twopoints. Using these basic ideas a large num-ber of theorems were proved using formaldeductive arguments. The basic assump-tions of Euclid have been modified, but the

system is essentially that used today for‘pure’ geometry.

One important postulate in Euclid’ssystem is that concerned with parallel lines(the parallel postulate). Its modern form isthat if a point lies outside a straight lineonly one straight line can be drawnthrough that point parallel to the otherline. See non-Euclidean geometry.

Euler characteristic /oi-ler/ A topologicalproperty of a curve or surface. For a curve,the Euler characteristic is the number ofvertices minus the number of closed con-tinuous line segments between. For exam-ple, any polygon has an Eulercharacteristic of zero. For a surface, theEuler characteristic is equal to the numberof vertices plus the number of faces minusthe number of edges. For example, a cubehas an Euler characteristic of 2, and acylinder, a Möbius strip, and a Klein bottlehave an Euler characteristic of zero. TheEuler characteristic is named for the Swissmathematician Leonhard Euler (1707–83).

Euler line A straight line on which theCENTROID G, the CIRCUMCENTER C and theORTHOCENTER O of a triangle all lie. Theratio of CG to OG on this line is always1/2.

Euler’s constant Symbol: γ A fundamen-tal mathematical constant defined by thelimit of:

1 + 1/2 + 1/3 + … + 1/n – log nas n → ∞. To six figures its value is0.577 216. It is not known whether Euler’sconstant is irrational or not.

Euler’s formula 1. (for polyhedra) Theformula that relates the number of verticesv, faces f, and edges e in a polyhedron, thatis:

v + f – e = 2For example, a cube has eight vertices

six faces and twelve edges:8 + 6 – 12 = 2

Using the theorem it can be shown thatthere are only five regular polyhedrons.2. The definition of the function eiθ for anyreal value of θ, where i is the square root of–1, is

81

Euler’s formula

eiθ = cosθ + i sinθAny complex number z = x + iy can be

written in this form. x = rcosθ and y = rsinθare real, with r and θ representing z on anArgand diagram. Note that putting θ = πgives eiπ = –1 and θ = 2π gives e2πi = 1.

even Divisible by two. The set of evennumbers is 2,4,6,8,…. Compare odd.

even function A function f(x) of a vari-able x, for which f(–x) = f(x). For example,cosx and x2 are even functions of x. Com-pare odd function.

event In probability, an event is any sub-set of the possible outcomes of an experi-ment. The event is said to occur if theoutcome is a member of the subset. For ex-ample, if two dice are thrown, an event is asubset of all ordered pairs (m,n) where mand n are each one of the integers 1, 2, 3,4, 5, 6. Thus (1,3),(2,2),(3,1) is an event,which may also be described as ‘obtaininga sum of four’. See also dependent.

evolute /ev-ŏ-loot/ The evolute of a givencurve is the locus of the centers of curva-ture of all the points on the curve. The evo-lute of a surface is another surface formedby the locus of all the centers of curvatureof the first surface.

exact differential equation A differen-tial equation of the form P(x,y)dx +Q(x,y)dy = 0, which can be matched to thedifferential dφ, where dφ = (dφ/dx)dx +(dφ/dy)dy, where φ(x,y) is a function of xand y. If this is the case, then φ(x,y) = C,where C is a constant and dφ = 0. If thisfunction φ(x,y) exists then: P(x,y)dx +Q(x,y) = (dφ/dx)dx + (dφ/dy)dy, and P(x,y)= dφ/dx, Q(x,y) = dφ/dy. The properties ofan exact differential equation can be usedto find solutions of that equation. For a dif-ferential equation to be exact it is necessaryand sufficient that the second mixed partialderivatives of φ(x,y) do not depend on theorder of differentiation, i.e. ∂p(x,y)/∂y =∂2φ/(∂y∂x) = ∂2φ/(∂x∂y) = ∂Q(x,y)/∂x. Forexample, the equation (x + 2y)dx + (2x +y)dy = 0 is an exact differential equation.

excluded middle, law of See laws ofthought.

exclusive disjunction (exclusive OR) Seedisjunction.

exclusive OR gate See logic gate.

existence theorem A theorem thatproves that one or more mathematical en-tities of a certain kind exists; e.g. that afunction has a zero or a fixed point. An ex-istence proof may be indirect and showthat a certain entity must exist without giv-ing any information about it or how to findit.

existential quantifier In mathematicallogic, a symbol meaning ‘there is (are)’. It isusually written ∃. The quantifier is fol-lowed by a variable that it is said to bind.Thus (∃x)F(x) means ‘There is somethingthat has property F’.

expansion A quantity expressed as a sumof a series of terms. For instance, the ex-pression:

(x + 1)(x + 2)can be expanded to:

x2 + 3x + 2Often a function can be written as an

infinite series that is convergent. The func-tion can then be approximated to any re-quired accuracy by taking the sum of asufficient number of terms at the beginningof the series. There are general formulaefor expanding some types of expression.For example, the expansion of (1 + x)n is1 + nx + [n(n – 1)/2!]x2 +[n(n – 1)(n – 2)/3!] x3 + …where x is a variable between –1 and +1,and n is an integer. See binomial expan-sion; determinant; Fourier series; Taylorseries.

expectation See expected value.

expected value (expectation) The valueof a variable quantity that is calculated tobe most likely to occur. If x can take any ofthe set of discrete values

x1,x2,…xn

even

82

which have corresponding probabilitiesp1,p2,…pn, then the expected value is

E(x) = x1p1 + x2p2 + … + xnpnIf x is a continuous variable with a

probability density function f(x), then

E(x) = ∞

–∞ xf(x)dx

explicit Denoting a function that containsno dependent variables. Compare implicit.

exponent /eks-poh-nĕnt/ A number orsymbol placed as a superscript after an ex-pression to indicate the power to which itis raised. For example, x is an exponent inyx and in (ay + b)x.

The laws of exponents are used forcombining exponents of numbers as fol-lows:

Multiplication:xaxb = xa+b

Division:xa/xb = xa–b

Power of a power:(xa)b = xab

Negative exponent:x–a = 1/xa

Fractional exponent:xa/b = b√xa

A number raised to the power zero isequal to 1; i.e. x0 = 1.

exponential /eks-pŏ-nen-shăl/ A functionor quantity that varies as the power of an-other quantity. In y = 4x, y is said to varyexponentially with respect to x. The func-tion ex (or expx), where e is the base of nat-ural logarithms, is the exponential of x.

The infinite series1 + x + x2/2! + x3/3! + … + xn/n! + …is equal to ex and is known as the expo-nential series. The exponential form of acomplex number is

reiθ = r(cosθ + i sinθ)See also complex number; Euler’s for-

mula; Taylor series.

exponential series The infinite power se-ries that is the expansion of the function ex,namely:1 + x + x2/2! + x3/3! + …

+ xn/n! + …

This series is convergent for all real-number values of the variable x.

Replacing x by –x gives an alternatingseries for e–x:1 – x + x2/2! – x3/3! …

Series for sinhx and coshx can be ob-tained by combining series for ex and e–x.

expression A combination of symbols(representing numbers of other mathemat-ical entities) and operations; e.g. 3x2, √(x2

+ 2), ex – 1.

exterior angle The angle formed on theoutside of a plane figure between the ex-tension of one straight edge beyond a ver-tex, and the outer side of the other straightedge at that vertex. In a triangle, the exte-rior angle at one vertex equals the sum ofthe angles on the insides of the other twovertices, i.e. the sum of the interior oppo-site angles. Compare interior angle.

extraction The process of finding a rootof a number.

extrapolation The process of estimatinga value outside a known range of values.For example, if the speed of an engine iscontrolled by a lever, and depressing thelever by two, four, and six centimetersgives speeds of 20, 30, and 40 revolutionsper second respectively, then one can ex-trapolate from this information and as-sume that depressing it by a further twocentimeters will increase the speed to 50revolutions per second. Extrapolation canbe carried out graphically; for example, agraph can be drawn over a known range ofvalues and the resulting curve extended.The further from the known range, thegreater will be the uncertainty in the ex-trapolation. The case in which the graph ofthe behavior is a straight line is a linear ex-trapolation. Compare interpolation.

83

extrapolation

β

α

γ δ

The exterior angle δ = 180° – γ = α + β.

face A flat surface on the outside of a solidfigure. A cube has six identical faces.

facet A FACE, or flat side, of a many-sidedobject.

factor (divisor) A number by which an-other number is divided. See also commonfactor.

factorial The product of all the wholenumbers less than or equal to a number.For example, factorial 7, written 7!, isequal to 7× 6 × 5 × 4 × 3 × 2 × 1. Factorialzero is defined as 1.

factorization The process of changing al-gebraic or numerical expressions from asum of terms into a product. For example,the left side of the equation 4x2 – 4x – 8 =0 can be factorized to (2x + 2)(2x – 4) mak-ing it easy to solve for x. As the product ofthe two factors is 0 when either of the fac-tors is 0, it follows that (2x + 2) = 0 and (2x– 4) = 0 will provide solutions, i.e. x = –1and x = 2.

factor theorem The condition that (x – a)is a factor of a polynomial f(x) in a variablex if and only if f(a) = 0. For example, if f(x)= x2 + x – 6, f(2) = 4 + 2 – 6 = 0 and f(–3)= 9 – 3 – 6 = 0, so the factors of f(x) are (x– 2) and (x + 3). The factor theorem is de-rived from the remainder theorem.

Fahrenheit degree /fa-rĕn-hÿt/ Symbol:°F A unit of temperature difference equalto one hundred and eightieth of the differ-ence between the temperatures of freezingand boiling water. On the Fahrenheit scalewater freezes at 32°F and boils at 212°F.To convert from a temperature on the

Fahrenheit scale (TF) to a temperature onthe Celsius scale (TC) the following for-mula is used: TF = 9TC/5 + 32. The scale isnamed for the German physicist (Gabriel)Daniel Fahrenheit (1686–1736).

fallacy See logic.

family A set of related curves or figures.For example, the equation y = 3x + c rep-resents a family of parallel straight lines.

farad /fa-răd, -rad/ Symbol: F The SI unitof capacitance. When the plates of a capac-itor are charged by one coulomb and thereis a potential difference of one volt betweenthem, then the capacitor has a capacitanceof one farad. 1 F = 1 CV–1, 1 farad = 1coulomb per volt. The unit is named for theBritish physicist and chemist Michael Fara-day (1791–1867).

Farey sequence /fair-ee/ The sequence ofall fractions in lowest terms whose denom-inators are less than n, where n is the orderof the sequence, listed in increasing size.For example, the Farey sequence of order 5is 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5.The sequence is named for the Englishmathematician John Farey (1766–1826).

fathom A unit of length used to measuredepth of water. It is equal to 6 feet(1.8288 m).

F distribution The statistical distributionfollowed by the ratio of variances, s1

2/s22,

of pairs of random samples, size n1 and n2,taken from a normal distribution. It is usedto compare different estimates of the samevariance.

feedback See cybernetics.

84

F

femto- Symbol: f A prefix denoting 10–15.For example, 1 femtometer (fm) = 10–15

meter (m).

Fermat’s last theorem /fair-mahz/ Thetheorem that the equation

xn + yn = zn

where n is an integer greater than 2, canhave no solution for x, y, and z. Fermatwrote in the margin of a book on equationsthat he had discovered a ‘truly wonderful’proof of the theorem but that the marginwas too small to write it down. It is nowgenerally thought that he was deludinghimself; a proof was finally found in 1995using advanced mathematics which did notexist at the time of Fermat (the seventeenthcentury). The theorem is named for theFrench mathematician and physicist Pierrede Fermat (1601–65).

Fermat’s principle A fundamental prin-ciple of optics that states that the path alight ray takes is always the path that takesthe least time. Fermat’s principle can beused to derive the laws describing the re-flection and refraction of light. It is an ex-ample of a VARIATIONAL PRINCIPLE.

fermi /fer-mee/ A unit of length equal to10–15 meter. It was formerly used in atomicand nuclear physics. The unit is named forthe Italian–American physicist EnricoFermi (1901–54).

Fibonacci numbers /fee-bŏ-nah-chee/The infinite sequence in which successivenumbers are formed by adding the two pre-vious numbers, that is:

1, 1, 2, 3, 5, 8, 13, 21, …The Fibonacci sequence of numbers is

named for the Italian mathematicianLeonardo Fibonacci (c. 1170–c. 1250).

fictitious force A force in a system thatarises because of the frame of reference ofthe observer. Such ‘forces’ are said to be‘fictitious’ because they do not actuallyexist; they can be removed by transfer to adifferent frame of refernce. Examples arecentrifugal force and Coriolis force.

field 1. A set of entities with two opera-tions, called addition and multiplication.The entities form a commutative groupunder addition with 0 as the identity el-ement. If 0 is omitted, the entities form acommutative group under multiplication.Also the distribution law, a(b + c) = ab +ac, applies for all a, b, and c. An exampleof a field is the set of rational numbers.2. A region in which a particle or body ex-erts a force on another particle or bodythrough space. In a gravitational field amass is supposed to affect the properties ofthe surrounding space so that another massin this region experiences a force. The re-gion is thus called a ‘field of force’. Electric,magnetic, and electromagnetic fields canbe similarly described. The concept of afield was introduced to explain action at adistance.

figure A shape formed by a combinationof points, lines, curves, or surfaces. Circles,squares, and triangles are plane figures.Spheres, cubes, and pyramids are solid fig-ures.

finite decimal See decimal.

finite sequence See sequence.

finite series See series.

finite set A set that has a fixed countablenumber of elements. For example, the setof ‘months in the year’ has 12 membersand is therefore a finite set. Compare infi-nite set.

first moment For an area A about anaxis, the product of A and the distance ofC, the centroid of A, from the axis. Thus, ifA is in the xy plane and the centroid C ofthe area has coordinates which are denotedby (x,y) then the first moment of A aboutthe x-axis is Ay and the first moment of Aabout the y-axis is Ax. For a volume Vabout the axis of rotation of a volume ofrevolution, the product of V and the dis-tance of C, the centroid of the volume,from the axis. Thus, if a volume V which isgenerated by rotating an area about the x-axis has a centroid C which has coordi-

85

first moment

nates denoted by (x,0) then the first mo-ment of V about the x-axis is zero and thefirst moment of V about the y-axis is Vx.

first-order differential equation A DIF-FERENTIAL EQUATION in which the highestderivative of the dependent variable is afirst derivative.

fixed-point iteration An approximatemethod for finding the root of an equationf(x) = 0. The first step in the procedure is towrite the equation as x = g(x). The nextstep is to take a value x0 to be an approxi-mation to a true root. Subsequent approx-imate values of the root are found by usingthe equation xi+1 = g(xi). If the values of xitend to a limiting value a as i increases thena = g(a). This means that a is a root of theequation f(x) = 0.

fixed-point notation See floating point.

fixed-point theorem A theorem thatdemonstrates that a function leaves onepoint in its domain unchanged, i.e. forwhich f(x) = x. One celebrated example isBrouwer’s fixed point theorem, whichstates that any continuous transformationof a circular disk onto itself must have afixed point.

flip-flop See bistable circuit.

floating objects, law of See flotation;law of.

floating-point A notation used to de-scribe real numbers, particularly in com-puters. In this notation, a number iswritten in the form a × bn, where a is anumber between 0.1 and 1, b is the numberbase being used (usually 10 or 2) and n isan integer. The numbers a, b and n arecalled the mantissa, the base, and the ex-ponent respectively. For example, the num-ber 2538.9 can be written as 0.25389 ×104. Floating-point notation is contrastedwith fixed-point notation, in which allnumbers have both a fixed number of dig-its and a fixed number of digits after thedecimal point. If the number in the exam-ple given is expressed by six digits, with

two of these digits being after the decimalpoint, it is written as 2538.90.

floppy disk (diskette) A device that canbe used to store data in electronic form,consisting of a flexible plastic DISK with amagnetic coating on one or both sides. It ispermanently encased in a stiff envelope in-side which it can be made to rotate. Aread–write head operates through a slot inthe envelope.

flotation, law of An object floating in afluid displaces its own weight of fluid. Thisfollows from Archimedes’ principle for thespecial case of floating objects. (A floatingobject is in equilibrium, its only supportcoming from the fluid. It may be totally orpartly submerged.)

flowchart A diagram on which can berepresented the major steps in a processused, say, in industry, or a problem to beinvestigated, or a task to be performed. Aflowchart is built up from a number ofboxes connected by arrowed lines. Theboxes, of various shapes, have a label at-tached showing for example the operationor calculation to be done at each step. At adecision box a question is asked. The an-swer, either yes or no, determines which oftwo possible paths to take. Computer pro-grams are often written by first drawing aflowchart of the problem or task in hand.See also program.

fluctuation A deviation from the averagevalue of some quantity. If there are manyparticles in a system the fluctuations in thevalue of a quantity Q can usually be de-scribed by the equation

<Q2> – <Q>2 = 1/<Q>where <Q> denotes the average value of a quantity. If it is proportional to the num-ber of particles in the system then the fluc-tuations are usually very small. However,there are many problems in which fluc-tuations are important, a notable ex-ample being a system near to a phase tran-sition.

fluid ounce See ounce.

first-order differential equation

86

flywheel A large heavy wheel (with alarge moment of inertia) used in mechani-cal devices. Energy is used to make thewheel rotate at high speed; the inertia ofthe wheel keeps the device moving at con-stant speed, even though there may be fluc-tuations in the torque. A flywheel thus actsas an ‘energy-storage’ device.

focal chord A chord of a conic that passesthrough a focus.

focal radius A line from the focus of aconic to a point on the conic.

focus (pl. focuses or foci) A point associ-ated with a CONIC. The distance betweenthe focus and any point on the curve is in afixed ratio (the eccentricity) to the distancebetween that point and a line (the direc-trix). An ellipse has two foci. The sum ofthe distances to each focus is the same forall points on the ellipse.

87

focus

set F at Aset B at end of Z

open bookbetween F and B

doesLL follow

word?

NO doesRR precede

word?

NO

NO

YES

word onleft-hand

page

word onright-hand

page

doesLR follow

word?

put B onpreceding

page

put F onfollowing

page

YES YES

A flowchart for finding which page a word is on in this dictionary (assuming that it is in). F is afront marker, B a back marker, LL is the first word on a left-hand page, RR the last word on a right-hand page, and LR the first word on the right-hand page.

foot Symbol: ft The unit of length in thef.p.s. system (one third of a yard). It isequal to 0.304 8 meter.

force Symbol: F That which tends tochange an object’s momentum. Force is avector; the unit is the newton (N). In SI,this unit is so defined that:

F = d(mv)/dtfrom Newton’s second law.

forced oscillation (forced vibration) Theoscillation of a system or object at a fre-quency other than its natural frequency.Forced oscillation must be induced by anexternal periodic force. Compare free os-cillation. See also resonance.

force ratio (mechanical advantage) For aMACHINE, the ratio of the output force(load) to the input force (effort). There isno unit; the ratio is, however, sometimesgiven as a percentage. It is quite possiblefor force ratios far greater than one to beobtained. Indeed many machines are de-signed for this so that a small effort canovercome a large load. However the effi-ciency cannot be greater than one and alarge force ratio implies a large distanceratio.

forces, parallelogram (law) of See par-allelogram of vectors.

forces, triangle (law) of See triangle ofvectors.

formalism A program for studying thefoundations of mathematics in which thecompleteness and consistency of mathe-matical systems is examined. This programwas dealt a fatal blow by the discovery ofGÖDEL’S INCOMPLETENESS THEOREM.

formal logic See symbolic logic.

format The arrangement of informationon a printed page, on a punched card, in acomputer storage device, etc., that must orshould be used to meet with certain re-quirements.

formula (pl. formulas or formulae) A gen-

eral expression that can be applied to sev-eral different values of the quantities inquestion. For example, the formula for thearea of a circle is πr2, where r is the radius.

Foucault pendulum /foo-koh/ A simplependulum consisting of a heavy bob on along string. The period is large and theplane of vibration rotates slowly over a pe-riod of time as a result of the rotation ofthe Earth below it. The apparent forcecausing this movement is the Coriolisforce. The pendulum is named for theFrench physicist Jean Bernard Léon Fou-cault (1819–68).

four-color problem A problem in topol-ogy concerning the division of the surfaceof a sphere into regions. The name comesfrom the coloring of maps. It appears thatin coloring a map it is not necessary to usemore than four colors to distinguish re-gions from each other. Two regions with acommon line boundary between them needdifferent colors, but two regions meeting ata point do not. This was proved by Appeland Haken in 1976 with the extensive useof computers. On the surface of a torusonly seven colors are necessary to distin-guish regions.

Fourier series /foo-ree-ay, -er/ A methodof expanding a function by expressing it asan infinite series of periodic functions(sines and cosines). The general mathemat-ical form of the Fourier series is:

f(x) = a0/2 + (a1cosx + b1sinx) +(a2cos2x + b2sin2x) +(a3cos3x + b3sin3x) +

(ancosnx + bnsinnx) + …The constants a0, a1, b1, etc., called Fouriercoefficients, are obtained by the formulae:

a0 = (1/π)-π

πf(x)dx

an = (1/π)-π

πf(x)cosnxdx

bn = (1/π)-π

πf(x)sinnxdx

The series is named for the French mathe-matician Baron (Jean Baptiste) JosephFourier (1768–1830).

foot

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fourth root of unity A complex numberz which satisfies the relation z4 = 1. This re-lation is satisfied by four values of z,namely 1, –1, i, and –i.

f.p.s. system A system of units that usesthe foot, the pound, and the second as thebase units. It has now been largely replacedby SI units for scientific and technical workalthough it is still used to some extent inthe USA.

fractal /frak-tăl/ A curve or surface thathas a fractional dimension and is formedby the limit of a series of successive opera-tions. A typical example of a fractal curveis the snowflake curve (also known as theKoch curve and named for the Swedishmathematician Helge von Koch (1870–1924)). This is generated by starting withan equilateral triangle and dividing eachside into three equal parts. The center partof each of these sides is then used as thebase of three smaller equilateral triangleserected on the original sides. If the centerparts are removed, the result is a star-shaped figure with 12 sides. The next stageis to divide each of the 12 sides into threeand generate more triangles. The process iscontinued indefinitely with the resultinggeneration of a snowflake-shaped curve. A

curve of this type in the limit has a dimen-sion that lies between 1 (a line) and 2 (asurface). The snowflake curve actually hasa dimension of 1.26.

One important aspect of fractals is thatthey are generated by an iterative processand that a small part of the figure containsthe information that could produce thewhole figure. In this sense, fractals are saidto be ‘self-similar’. The study of fractalshas applications in chaos theory and in cer-tain scientific fields (e.g. the growth ofcrystals). It is also important in computergraphics, both as a method of generatingstriking abstract images and, because ofthe self-similarity property, as a method ofcompressing large graphics files.

See also Mandelbrot set.

fraction A number written as a quotient;i.e. as one number divided by another. Forexample, in the fraction ⅔, 2 is known asthe numerator and 3 as the denominator.When both numerator and denominatorare integers, the fraction is known as a sim-ple, common, or vulgar fraction. A com-plex fraction has another fraction asnumerator or denominator, for example(2/3)/(5/7) is a complex fraction. A unitfraction is a fraction that has 1 as the nu-merator. If the numerator of a fraction is

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fraction

Fractal: generation of the snowflake curve.

less than the denominator, the fraction isknown as a proper fraction. If not, it is animproper fraction. For example, 5/2 is animproper fraction and can be written as2½. In this form it is called a mixed num-ber.

In adding or subtracting fractions, thefractions are put in terms of their lowestcommon denominator. For example,

1/2 + 1/3 = 3/6 + 2/6 = 5/6In multiplying fractions, the numeratorsare multiplied and the denominators multi-plied. For example:

2/3 × 5/7 = (2 × 5)/(3 × 7) = 10/21In dividing fractions, one fraction is in-verted thus:

2/3 ÷ 1/2 = 2/3 × 2/1 = 4/3See also ratio.

frame of reference A set of coordinateaxes with which the position of any objectmay be specified as it changes with time.The origin of the axes and their spatial di-rections must be specified at every instantof time for the frame to be fully deter-mined.

framework A collection of light rods thatare joined together at their ends to give arigid structure. In a framework the rods aresaid to be light if their weights are verymuch smaller than the loads they are bear-ing. When a framework has external forcesacting on it each rod in the framework caneither prevent the structure from collapsingor stop the joints connecting the rods frombecoming separated. If a rod is stopping acollapse it exerts a push at both ends and issaid to be in compression or in thrust. If arod is stopping the structure from becom-ing separated it exerts a pull at both endsand is said to be in tension. If the whole ofthe framework is in equilibrium then theforces at each joint have to be in equilib-rium. This means that in equilibrium theexternal forces on the framework are inequilibrium with the internal forces sincethe forces in the rods occur as equal andopposite forces.

freedom, degrees of The number of in-dependent quantities that are necessary todetermine an object or system. The number

of degrees of freedom is reduced by con-straints on the system since the number ofindependent quantities necessary to deter-mine the system is also reduced. For exam-ple, a point in space has three degrees offreedom since three coordinates are neededto determine its postion. If the point is con-strained to lie on a curve in space, it hasthen only one degree of freedom, since onlyone parameter is needed to specify its posi-tion on the curve.

free oscillation (free vibration) An oscil-lation at the natural frequency of the sys-tem or object. For example, a pendulumcan be forced to swing at any frequency byapplying a periodic external force, but itwill swing freely at only one frequency,which depends on its length. Compareforced oscillation. See also resonance.

free variable In mathematical logic, avariable that is not within the scope of anyquantifier. (If it is within the scope of aquantifier it is bound.) In the formula F(y)→ (∃x)G(x), y is a free variable.

French curve A drawing instrument con-sisting of a rigid piece of plastic or metalsheeting with curved edges. The curvatureof the edges varies from being almoststraight to very tight curves, so that a partof the edge can be chosen to guide a pen orpencil along any desired curvature. An-other instrument that serves the same pur-pose consists of a deformable strip of leadbar cut into short sections and surroundedby a thick layer of plastic. This is bent toform any required curvature.

frequency Symbol: f, ν The number of cy-cles per unit time of an oscillation (e.g. a

frame of reference

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French curve

pendulum, vibrating system, wave, alter-nating current, etc.). The unit is the hertz(Hz). The symbol f is used for frequency,although ν is often employed for the fre-quency of light or other electromagneticradiation.

Angular frequency (ω) is related to fre-quency by ω = 2πf.

The frequency of an event is the numberof times that it has occurred, as recorded ina FREQUENCY TABLE.

frequency curve A smoothed FREQUENCY

POLYGON for data that can take a continu-ous set of values. As the amount of data isincreased and the size of class interval de-creased, the frequency polygon moreclosely approximates a smooth curve. Rel-ative frequency curves are smoothed rela-tive frequency polygons. See also skewness.

frequency function 1. The function thatgives the values of the frequency of each re-sult or observation in an experiment. For alarge sample that is representative of thewhole population, the observed frequencyfunction will be the same as the probabilityDISTRIBUTION FUNCTION f(x) of a populationvariable x.2. See random variable.

frequency polygon The graph obtainedwhen the mid-points of the tops of the rec-tangles in a HISTOGRAM with equal class in-tervals are joined by line segments. Thearea under the polygon is equal to the totalarea of the rectangles.

frequency table A table showing howoften each type (class) of result occurs in asample or experiment. For example, thedaily wages received by 100 employes in acompany could be shown as the number ineach range from $50.00 to $74.99, $75.00to $99.99, and so on. In this case the rep-resentative value of each class (the classmark) is $(50 + 74.99)/2, etc. See also his-togram.

Fresnel integrals /fray-nel/ Two integralsC(x) and S(x) defined by:

C(x) = ∫x0 cos(πu2/2)duS(x) = ∫x0 sin(πu2/2)du.

These integrals occur in the theory of dif-fraction and have been extensively tabu-lated. The Fresnel integrals can becombined to give:

C(x) – i S(x) = ∫x0 exp[–i(πu2/2)] du,C(x) + i S(x) = ∫x0 exp[i(πu2/2)] du.

friction A force opposing the relative mo-tion of two surfaces in contact. In fact,each surface applies a force on the other inthe opposite direction to the relative mo-tion; the forces are parallel to the line ofcontact. The exact causes of friction arestill not fully understood. It probably re-sults from minute surface roughness, evenon apparently ‘smooth’ surfaces. Frictionalforces do not depend on the area of con-tact. Presumably lubricants act by separat-ing the surfaces. For friction between twosolid surfaces, sliding friction (or kineticfriction) opposes friction between twomoving surfaces. It is less than the force ofstatic (or limiting) friction, which opposesslip between surfaces that are at rest.Rolling friction occurs when a body isrolling on a surface: here the surface incontact is constantly changing. Frictionalforce (F) is proportional to the force hold-ing the bodies together (the ‘normal reac-tion’ R). The constants of proportionality(for different cases) are called coefficientsof friction (symbol: µ):

µ = F/RTwo laws of friction are sometimes stated:1. The frictional force is independent of thearea of contact (for the same force holdingthe surfaces together).2. The frictional force is proportional tothe force holding the surfaces together. Insliding friction it is independent of the rel-ative velocities of the surfaces.

frustum /frus-tŭm/ (pl. frustums or frusta)A geometric solid produced by two parallelplanes cutting a solid, or by one plane par-allel to the base.

fulcrum /fûl-krŭm/ (pl. fulcrums or ful-cra) The point about which a lever turns.

function (mapping) Any defined pro-cedure that relates one number, quantity,etc., to another or others. In algebra, a

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function

function of a variable x is often written asf(x). If two variable quantities, x and y, arerelated by the equation y = x2 + 2, for ex-ample, then y is a function of x or y = f(x)= x2 + 2. The function here means ‘squarethe number and add two’. x is the indepen-dent variable and y is the dependent vari-able. The inverse function – the one thatexpresses x in terms of y in this case –would be x = ±√(y –2), which might bewritten as x = g(y).

A function can be regarded as a rela-tionship between the elements of one set(the range) and those of another set (thedomain). For each element of the first setthere is a corresponding element of the sec-ond set into which it is ‘mapped’ by thefunction. For example, the set of numbers1,2,3,4 is mapped into the set 1,8,27,64by taking the cube of each element. A func-tion may also map elements of a set intoothers in the same set. Within the set allwomen, there are two subsets mothersand daughters. The mapping betweenthem is ‘is the mother of’ and the inverse is‘is the daughter of’.

functional A function in which both thedomain and range can be sets of functions.Roughly speaking, a functional can be con-sidered to be a function of a function.Functionals are used extensively in analysisand physics, particularly for problems in-volving many degrees of freedom. A func-tional F of a function f is denoted by F[f].See also functional analysis.

functional analysis A branch of analysisthat deals with mappings between classesof functions and the OPERATORS that bringabout such mappings. In functional analy-sis a function can be regarded as a point inan abstract space. Functional analysis was

extensively investigated in the twentiethcentury, with applications to differentialequations, integral equations, and quan-tum mechanics. These enabled the founda-tions of these subjects to be laid on a soundaxiomatic basis.

fundamental The simplest way (mode) inwhich an object can vibrate. The funda-mental frequency is the frequency of thisvibration. The less simple modes of vibra-tion are the higher harmonics; their fre-quencies are higher than that of thefundamental.

fundamental theorem of algebra Everypolynomial equation of the form:

a0zn + a1zn–1 + a2zn–2 + …an–1z + an = 0

in which a0, a1, a2, etc., are complex num-bers, has at least one complex root. Seealso polynomial.

fundamental theorem of calculus Thetheorem used in calculating the value of aDEFINITE INTEGRAL. If f(x) is a continuousfunction of x in the interval a ≤ x ≤ b, andif g(x) is any INDEFINITE INTEGRAL of f(x),then:

b

af(x)dx = [g(x)b

a] = g(b) – g(a)

fundamental units The units of length,mass, and time that form the basis of mostsystems of units. In SI, the fundamentalunits are the meter, the kilogram, and thesecond. See also base unit.

furlong A unit of length equal to oneeighth of a mile. It is equivalent to 201.168m.

functional

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gallon A unit of capacity usually used tomeasure volumes of liquids. In the USA it isdefined as 231 cubic inches and is equal to3.785 4 × 10–3 m3. In the UK it is definedas the space occupied by 10 pounds of purewater and is equal to 4.546 1 × 10–3 m3. 1UK gallon is equal to 1.2 US gallons.

game theory A mathematical theory ofthe optimal behavior in competitive situa-tions in which the outcomes depend notonly on the participants’ choices but alsoon chance and the choices of others. Agame may be defined as a set of rules de-scribing a competitive situation involving anumber of competing individuals orgroups of individuals. These rules givetheir permissible actions at each stage ofthe game, the amount of information avail-able, the probabilities associated with thechance events that might occur, the cir-cumstances under which the competitionends, and a pay-off scheme specifying theamount each player pays or receives atsuch a conclusion. It is assumed that theplayers are rational in the sense that theyprefer better rather than worse outcomesand are able to place the possible outcomesin order of merit. Game theory has appli-cations in military science, economics, pol-itics, and many other fields.

gamma function The integral function

Γ(x) = ∞

0tx–1e–tdt

If x is a positive integer n, then Γ(n) = n! Ifx is an integral multiple of ½, the functionis a multiple of √π.

Γ(½) = √πΓ(3/2) = (½)√π

etc.

gate See logic gate.

gauss /gows/ Symbol: G The unit of mag-netic flux density in the c.g.s. system. It isequal to 10–4 tesla.

Gaussian distribution /gow-see-ăn/ Seenormal distribution.

Gaussian elimination A technique usedin solving a set of linear equations for sev-eral unknown quantities. The set of equa-tions is expressed in terms of a matrix thatis formed from the coefficients and con-stants of the equations then converted intoECHELON FORM by elementary row opera-tions, i.e. by multiplying a row by a num-ber, adding a row which has beenmultiplied by a number to another row orswapping two rows. The set of solutionscorresponding to an equation that has beentransformed so as to give values of the un-known quantities directly is the same set ofsolutions for the original untransformedequations.

Gauss’s theorem See divergence theo-rem.

general conic See conic.

general form (of an equation) A formulathat defines a type of relationship betweenvariables but does not specify values forconstants. For example, the general formof a polynomial equation in x is

axn + bxn–1 + cxn–2 + … = 0a, b, c, etc., are constants and n is the high-est integer power of x, called the degree ofthe polynomial. Similarly, the general formof a quadratic equation is

ax2 + bx + c = 0See also equation; polynomial.

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G

general theory See relativity; theory of.

generator A line that generates a surface;for example, in a cone, cylinder, or solid ofrevolution.

geodesic /jee-ŏ-dess-ik/ A line on a surfacebetween two points that is the shortest dis-tance between the points. On a plane a ge-odesic is a straight line. On a sphericalsurface it is part of a great circle of thesphere.

geometric distribution The distributionof the number of independent Bernoulli tri-als before a successful result is obtained;for example, the distribution of the num-ber of times a coin has to be tossed beforea head comes up. The probability that thenumber of trials (x) is k is

P(x=k) = qk–1pThe mean and variance are 1/p and q/p2 re-spectively. The moment generating func-tion is etp/(1 – qet).

geometric mean See mean.

geometric progression See geometric se-quence.

geometric sequence (geometric progres-sion) A SEQUENCE in which the ratio ofeach term to the one after it is constant, forexample, 1, 3, 9, 27, … The general for-mula for the nth term of a geometric se-quence is un = arn. The ratio is called thecommon ratio. In the example the firstterm, a, is 1, the common ratio, r, is 3, andso un equals 3n. If a geometric sequence isconvergent, r lies between 1 and –1 (exclu-sive) and the limit of the sequence is 0.That is, un approaches zero as n becomesinfinitely large. Compare arithmetic se-quence. See also convergent sequence; di-vergent sequence; geometric series.

geometric series A SERIES in which theratio of each term to the one after it is con-stant, for example, 1 + 2 + 4 + 8 + 16 + …. The general formula for a geometric seriesis

Sn = a + ar + ar2 + … + arn =a(rn – 1)/(r – 1)

In the example, the first term, a, is 1, thecommon ratio, r, is 2, and so the nth termarn equals 2n. If r is greater than 1, the se-ries will not be convergent. If –1 < r < 1 andthe sum of all the terms after the nth termcan be made as small as required by mak-ing n large enough, then the series is con-vergent. This means that there is a finitesum even when n is infinitely large. Thesum to infinity of a convergent geometricseries is a/(1 – r). Compare arithmetic se-ries. See also convergent series; divergentseries; geometric sequence.

geometry The branch of mathematicsconcerned with points, lines, curves, andsurfaces – their measurement, relation-ships, and properties that are invariantunder a given group of transformations.For example, geometry deals with themeasurement or calculation of angles be-tween straight lines, the basic properties ofcircles, and the relationship between linesand points on a surface. See analyticalgeometry; Euclidean geometry; non-Eu-clidean geometry; topology.

giga- Symbol: G A prefix denoting 109.For example, 1 gigahertz (GHz) = 109 hertz(Hz).

gill /gil/ A unit of capacity equal to onequarter of a pint. A US gill is equivalent to1.182 9 × 10–4 m3 and a UK gill is equiva-lent to 1.420 × 10–4 m3. See pint.

glide A symmetry that can occur in crys-tals. It consists of the combination of a re-flection and a translation. See also spacegroup.

Gödel’s incompleteness theorem /goh-dĕlz/ A fundamental result of mathemati-cal logic showing that any formal systempowerful enough to express the truths ofarithmetic must be incomplete; that is thatit will contain statements that are true butcannot be proved using the system itself.The incompleteness theorem is named forthe Austrian–American mathematicianKurt Gödel (1906–78).

general theory

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Goldbach conjecture /golt-bahkh/ Theconjecture that every even number otherthan 2 is the sum of two prime numbers; sofar, unproved. The conjecture is named forthe Prussian mathematician and historianChristian Goldbach (1690–1764).

golden rectangle A rectangle in whichthe adjacent sides are in the ratio (1 +√5)/2.

golden section The division of a line oflength l into two lengths a and b so that l/a= a/b, that is, a/b = (1 + √5)/2. Proportionsbased on the golden section are particu-larly pleasing to the eye and occur in manypaintings, buildings, designs, etc.

goniometer /gon-ee-om-ĕ-ter/ An instru-ment for measuring the angles between ad-joining flat surfaces, such as the faces of acrystal.

governor A mechanical device to controlthe speed of a machine. One type of simplegovernor consists of two loads attached toa shaft so that as the speed of rotation ofthe shaft increases, the loads move fartheroutward from the center of rotation, whilestill remaining attached to the shaft. Asthey move outward they operate a controlthat reduces the rate of fuel or energy inputto the machine. As they reduce speed andmove inward they increase the fuel or en-ergy input. Thus, on the principle of nega-tive feedback, the speed of the machine iskept fairly constant under varying condi-tions of load.

grad (gradient) Symbol: ∇ A vector oper-ator that, for any scalar function f(x,y,z),has components in the x, y, and z direc-tions equal to the PARTIAL DERIVATIVES withrespect to x, y, and z in that order. It is de-fined as:

grad f = ∇f = i∂f/∂x + j∂f/∂y + k∂f/∂zwhere i, j, and k are the unit vectors in thex, y, and z directions. In physics, ∇F isoften used to describe the spatial variationin the magnitude of a force F in, for exam-ple, a magnetic or gravitational field. It is avector with the direction in which the rateof change of F is a maximum, if such a

maximum exists. In the Earth’s gravita-tional field this would be radially towardthe center of the Earth (downward). In amagnetic field, ∇F would point along thelines of force.

grade Symbol: g A unit of plane angleequal to one hundredth of a right angle. 1g

is equal to 0.9°.

gradient 1. (slope) In rectangular Carte-sian coordinates, the rate at which the y-coordinate of a curve or a straight linechanges with respect to the x-coordinate.The straight line y = 2x + 4 has a gradientof +2; y increases by two for every unit in-crease in x. The general equation of astraight line is y = mx + c, where m is thegradient and c is a constant. (0,c) is thepoint at which the line cuts the y-axis, i.e.the intercept. If m is negative, y decreasesas x increases.

For a curve, the gradient changes con-tinuously; the gradient at a point is the gra-dient of the straight line that is a tangent tothe curve at that point. For the curve y =f(x), the gradient is the DERIVATIVE dy/dx.For example, the curve y = x2 has a gradi-ent given by dy/dx = 2x, at any particularvalue of x.2. See grad.

gram Symbol: g A unit of mass defined as10–3 kilogram.

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gram

6 –

5 –

4 –

3 –

2 –

1 –

– – – – – –

1 2 3 4 5 6 x

y

•

•

The gradient of the curve at the point (2,2) is 2,and at the point (5,5) is 1/2.

gram-atom See mole.

gramme An alternative spelling of gram.

gram-molecule See mole.

graph 1. A drawing that shows the rela-tionship between numbers or quantities.Graphs are usually drawn with coordinateaxes at right angles. For example, theheights of children of different ages can beshown by making the distance along a hor-izontal line represent the age in years andthe distance up a vertical line represent theheight in meters. A point marked on thegraph ten units along and 1.5 units up rep-resents a ten-year-old who is 1.5 meterstall. Similarly, graphs are used to give ageometric representation of equations. Thegraph of y = x2 is a parabola. The graph ofy = 3x + 10 is a straight line. Simultaneousequations can be solved by drawing thegraphs of the equations, and finding thepoints where they cross. For the two equa-tions above, the graphs cross at two points:x = –2, y = 4 and x = 5, y = 25.

There are various types of graph. Some,such as the HISTOGRAM and the PIE CHART,are used to display numerical informationin a form that is simple and quickly under-stood. Some, such as CONVERSION GRAPHS,are used as part of a calculation. Others,such as SCATTER DIAGRAMS, may be used inanalysing the results of a scientific experi-ment. See also bar chart.2. (topology) A network of lines and ver-tices. See Königsberg bridge problem.

graphics display (graphical display unit)See visual display unit.

gravitation The concept originated byIsaac Newton around 1666 to account forthe apparent motion of the Moon aroundthe Earth, the essence being a force of at-traction, called gravity, between the Moonand the Earth. Newton used this theory ofgravitation to give the first satisfactory ex-planations of many facts, such as Kepler’slaws of planetary motion, the ocean tides,and the precession of the equinoxes. Seealso Newton’s law of universal gravitation.

gravitational constant Symbol: G Theconstant of proportionality in the equationthat expresses NEWTON’S LAW OF UNIVERSAL

GRAVITATION:F = Gm1m2/r2

where F is the gravitational attraction be-tween two point masses m1 and m2 sepa-rated by a distance r. The value of G is 6.67× 10–11 N m2 kg–2. It is regarded as a uni-versal constant, although it has been sug-gested that the value of G may be changingslowly owing to the expansion of the Uni-verse. (There is no current evidence that Ghas changed with time.)

gravitational field The region of space inwhich one body attracts other bodies as aresult of their mass. To escape from thisfield a body has to be projected outwardwith a certain speed (the escape speed). Thestrength of the gravitational field at a pointis given by the ratio force/mass, which isequivalent to the acceleration of free fall, g.This may be defined as GM/r2, where G isthe gravitational constant, M the mass ofthe object at the center of the field, and rthe distance between the object and thepoint in question. The standard value ofthe acceleration of free fall at the Earth’ssurface is 9.8 m s–2, but it varies with alti-tude (i.e. with r2).

gravitational mass The MASS of a body asmeasured by the force of attraction be-tween masses. The value is given by New-ton’s law of universal gravitation. Inertialand gravitational masses appear to beequal in a uniform gravitational field. Seealso inertial mass.

gravity The gravitational pull of the Earth(or other celestial body) on an object. Theforce of gravity on an object causes itsweight.

gravity, center of See center of mass.

great circle A circle on the surface of asphere that has the same radius as thesphere. A great circle is formed by a cross-section by any plane that passes throughthe center of the sphere. Compare smallcircle.

gram-atom

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greatest upper bound See bound.

Green’s function A solution for certaintypes of partial differential equation. Themethod of Green’s functions is used exten-sively in many branches of theoreticalphysics, particularly mechanics, electrody-namics, acoustics, the many-body problemin quantum mechanics and quantum fieldtheory. In these physical applications theGreen’s function can be considered to be aresponse to an impulse, with the Green’sfunction method for finding the solution toa partial differential equation being thesum or integral of the responses to im-pulses. An example of a Green’s function isthe electric potential due to a unit pointsource of charge.

Green’s theorem A result in VECTOR CAL-CULUS that is a corollary of the DIVERGENCE

THEOREM (Gauss theorem). If u and v arescalar functions, S indicates a surface inte-gral and V a volume integral, Green’s the-orem states that

∫v(U∇.∇v – v∇.∇u)dV =∫S(U∇v–v∇u).dS.

An alternative form of Green’s theorem is∫Su∇v.dS = ∫vu∇.∇v + ∫V∇u.∇vdV.

Green’s theorem is used extensively inphysical applications of vector calculussuch as problems in electrodynamics.

gross 1. Denoting a weight of goods in-cluding the weight of the container orpacking.2. Denoting a profit calculated before de-ducting overhead costs, expenses, and(usually) taxes.

Compare net.

group A set having certain additionalproperties: 1. In a group there is a binaryoperation for which the elements of the setcan be related in pairs, giving results thatare also members of the group (the prop-erty of closure). For example, the set of allpositive and negative numbers and zeroform a group under the operation of addi-tion. Adding any member to any othergives an element that is also a member ofthe group; e.g. 3 + (–2) = 1, etc.

2. There is an identity element for the op-eration – i.e. an element that, combinedwith another, leaves it unchanged. In theexample, the identity element is zero:adding zero to any member leaves it un-changed; 3 + 0 = 3, etc.3. For each element of the group there isanother element – its inverse. Combiningan element with its inverse leads to theidentity element. In the example, the num-ber +3 has an inverse –3 (and vice versa);thus +3 + (–3) = 0.4. The associative law holds for the mem-bers of the group. In the example:

2 + (3 + 5) = (2 + 3) + 5Any set of elements obeying the above rulesforms a group. Note that the binary opera-tion need not be addition. Group theory isimportant in many branches of mathemat-ics – for instance, in the theory of roots ofequations. It is also very useful in diversebranches of science. In chemistry, grouptheory is used in describing symmetries ofatoms and molecules to determine their en-ergy levels and explain their spectra. Inphysics, certain elementary particles can beclassified into mathematical groups on thebasis of their quantum numbers (this led tothe discovery of the omega-minus particleas a missing member of a group). Grouptheory has also been applied to other sub-jects, such as linguistics.

See also Abelian group; cyclic group.

group speed If a wave motion has a phasespeed that depends on wavelength, the dis-turbance of a progressive wave travels witha different speed than the phase speed. Thisis called the group speed. It is the speedwith which the group of waves travels, andis given by:

U = c – λdc/dλwhere c is the phase speed. The groupspeed is the one that is usually obtained bymeasurement. If there is no dispersion ofthe wave motion, as for electromagneticradiation in free space, the group andphase speeds are equal.

Guldinus theorem See Pappus’ theo-rems.

gyroscope /jÿ-rŏ-skohp/ A rotating object

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gyroscope

that tends to maintain a fixed orientationin space. For example, the axis of the ro-tating Earth always points in the same di-rection toward the Pole Star (except for asmall PRECESSION). A spinning top or a cy-

clist are stable when moving at speed be-cause of the gyroscopic effect. Practicalapplications are the navigational gyro-compass and automatic stabilizers in shipsand aircraft.

gyroscope

98

half-angle formulae See addition formu-lae.

half-plane A set of points that lie on oneside of a straight line. If the line l is denotedby ax + by + c = 0, where a, b, and c areconstants, then the open half-plane on oneside of the line is given by the set of points(x,y) that satisfy the inequality ax + by + c> 0. The open half-plane on the other sideof the plane is given by the set of points(x,y) that satisfy the inequality ax + by + c< 0. The inequalities ax + by + c > 0 and ax+ by + c ≤ 0 define closed half planes onopposite sides of the plane. Half planes(both open and closed) are used extensivelyin linear programming.

Hamiltonian /ham-ăl-toh-nee-ăn/ In clas-sical mechanics, a function of the coordi-nates qi, i = 1, 2, … n, and momenta, pi, i =1, 2, … n, generally denoted by H and de-fined by

H = ∑piqi – Lwhere qi denotes the derivative with re-spect to time and L is the Lagrangian func-tion of the system expressed as a functionof the coordinates momenta and time. Ifthe Lagrangian function does not dependexplicitly on time, the system is said to beconservative and H is the total energy ofthe system. The Hamiltonian function isnamed for the Irish mathematician SirWilliam Rowan Hamilton (1805–65).

Hamiltonian graph A graph that has aHamiltonian cycle, i.e. a cycle containseach vertex, where a cycle of a graph is asequence of alternating vertices and edgeswhich can be written as v0, l1, v1, l2, v2 ...,ll, vl, where li is an edge which connects thevertices vi–1 and vi. In this definition all theedges are different and all the vertices are

different, with the important exceptionthat v0 = vl. The Hamiltonian graph is ofinterest in graph theory when the problemof going around the graph along edges soas to visit each vertex only once is consid-ered.

ham-sandwich theorem 1. A ham sand-wich can be cut with one stroke of a knifeso that the ham and each slice of bread areexactly cut in half. More formally, if A, B,and C are bounded connected sets in space,then there is a plane that cuts each set intotwo sets with equal volume.2. If f(x) ≤ g(x) ≤ h(x) for all x and the func-tions f and h have the same limit, then galso has this limit.

hard copy In computer science, a docu-ment that can be read, such as a computerprintout in plain language. See printer.

hard disk A rigid magnetic disk thatstores programs and data in a computer.They are usually fixed in the machine andcannot be removed. See also disk; floppydisk.

hardware The physical embodiment of acomputer system, i.e. its electronic cir-cuitry, disk and magnetic tape units, lineprinters, cabinets, etc. Compare software.

harmonic analysis The use of trigono-metric series to study mathematical func-tions. See Fourier series.

harmonic mean See mean.

harmonic motion A regularly repeatedsequence that can be expressed as the sumof a set of sine waves. Each component sinewave represents a possible simple har-

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H

harmonic progression

100

monic motion. The complex vibration ofsound sources (with fundamental andovertones), for instance, is a harmonic mo-tion, as is the sound wave produced. Seealso simple harmonic motion.

harmonic progression See harmonicsequence.

harmonic sequence (harmonic progres-sion) An ordered set of numbers, the reci-procals of which have a constant differencebetween them; for example, 1, ½, ⅓, ¼, …1/n. In this example 1, 2, 3, 4, … n havea constant difference – i.e. they form anARITHMETIC SEQUENCE. The reciprocals ofthe terms in a harmonic sequence form anarithmetic sequence, and vice versa.

harmonic series The sum of the terms ina harmonic sequence; for example: 1 + ½ +⅓ + ¼ + … The harmonic series is a diver-gent series.

HCF Highest common factor. See com-mon factor.

head An input/output device in a com-puter that can read, write, or erase signalsonto or from magnetic tape or disk. Seeerase head; write head.

heat equation An equation which de-scribes the flow of heat. In three spatial di-mensions it states that the rate of change ofthe absolute temperature T with respect totime t is proportional to the LAPLACIAN ∇2

of T: ∂T/∂t = c∇2T, where c is a constant.In the case of one space dimension with co-ordinate x this equation becomes ∂T/∂t =c∂2T/∂x2. The heat equation is differentfrom the wave equation since the heatequation has a first derivative with respectto time whereas the wave equation has asecond derivative with respect to time.Physically, this corresponds to heat con-duction being an irreversible process, likefriction, whereas wave motion is re-versible. The heat equation has the sameform as the general equation for diffusion.

hectare /hek-tair/ Symbol ha. A unit ofarea equal to 10 000 square metres. tHE-hectare is most often used as a convenientunit of land.

hecto- Symbol: h A prefix denoting 102.For example, 1 hectometer (hm) = 102

meters (m).

height A vertical distance, usually up-ward, from a base line or plane. For exam-ple, the perpendicular distance from thebase of a triangle to the vertex opposite,and the distance between the uppermostand the base planes of a cuboid, are bothknown as the height of the figure.

helix /hee-liks/ A spiral-shaped spacecurve. A cylindrical helix lies on a cylinder.A conical helix lies on a cone. For example,the shape of the thread on a screw is ahelix. In a straight screw, it is a cylindricalhelix and in a conically tapered screw it isa conical helix.

Helmholtz’s theorem /helm-holts/ A the-orem in VECTOR CALCULUS that states that ifa vector V satisfies certain general mathe-matical conditions then V can be written asthe sum of an IRROTATIONAL VECTOR and aSOLENOIDAL VECTOR. This theorem is im-portant in electrodynamics. It is named forthe German physicist Hermann von Helm-holtz (1821–94).

hemisphere The surface bounded by halfof a SPHERE and a plane through the centerof the sphere.

cylindricalhelix

conicalhelix

Types of helix

henry /hen-ree/ Symbol: H The SI unit ofinductance, equal to the inductance of aclosed circuit that has a magnetic flux ofone weber per ampere of a current in thecircuit. 1 H = 1 Wb A–1. The unit is namedfor the American physicist Joseph Henry(1797–1878).

heptagon /hep-tă-gon/ A plane figure withseven straight sides. A regular heptagonhas seven equal sides and seven equal an-gles.

Hermitian matrix /her-mee-shăn, -mish-ăn, air-mee-shăn/ The Hermitian conjugateof a matrix is the transpose of the complexconjugate of the matrix, where the com-plex conjugate of a matrix is the matrixwhose elements are the complex conju-gates of the corresponding elements of thegiven matrix (see conjugate complex num-bers). A Hermitian matrix is a matrix thatis its own Hermitian conjugate; i.e. asquare matrix such that aij is the complexconjugate of aij for all i and j where aij is theelement in the ith row and jth column. Thematrix is named for the French mathemati-cian Charles Hermite (1822–1901).

Hero’s formula A formula for the area ofa triangle with sides a, b, and c:

A = √[s(s – a)(s – b)(s – c)]where s is half the perimeter; i.e. ½(a + b +c). The formula is named for the Greekmathematician and inventor Hero ofAlexandria (fl. AD 62).

hertz /herts/ Symbol: Hz The SI unit of fre-quency, defined as one cycle per second(s–1). Note that the hertz is used for regu-larly repeated processes, such as vibrationor wave motion. An irregular process, suchas radioactive decay, would have units ex-pressed as s–1 (per second). The unit isnamed for the German physicist HeinrichRudolf Hertz (1857–94).

heuristic /hyû-ris-tik/ Based on trial anderror, as for example some techniques in it-erative calculations. See also iteration.

hexadecimal /heks-ă-dess-ă-măl/ Denot-ing or based on the number sixteen. A

hexadecimal number is made up with six-teen different digits instead of the ten in thedecimal system. Normally these are 0, 1, 2,3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. For ex-ample, 16 is written as 10, 21 is written as15 (16 + 5), 59 is written as 3B [(3 × 16) +11]. Hexadecimal numbers are sometimesused in computer systems because they aremuch shorter than the long strings of bi-nary digits that the machine normally uses.Binary numbers are easily converted intohexadecimal numbers by grouping the dig-its in fours. Compare binary; decimal;duodecimal; octal.

hexagon /heks-ă-gon/ A plane figure withsix straight sides. A regular hexagon is onewith all six sides and all six angles equal,the angles all being 120°. Congruent regu-lar hexagons can be fitted together to covercompletely a plane surface. Apart fromsquares and equilateral triangles, they arethe only regular polygons with this prop-erty.

hexahedron /heks-ă- hee-drŏn/ A POLYHE-DRON that has six faces. For example, thecube, the cuboid, and the rhombohedronare all hexahedrons. The cube is a regularhexahedron; all six faces are congruentsquares.

higher derivative The n-th derivative ofa function f of x, where n is greater than orequal to two. If one writes y = f(x) thehigher derivatives are denoted by dny/dxn

or y(n). The second derivative of y with re-spect to x is denoted by d2y/dx2 or y″. It isfound by differentiating dy/dx with respectto x. Similarly, the third derivative of ywith respect to x is denoted by d3y/dx3 ory′″. For example, if y = x4 + x3 + 1, dy/dx =4x3 + 3x2, d2y/dx2 = 12x2 + 6x, d3y/dx3 =24x + 6.

highest common factor See commonfactor.

high-level language See program.

Hilbert’s problems /hil-berts/ A set of 23important mathematical problems posedby the German mathematician David

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Hilbert’s problems

Hilbert (1862–1941) at the InternationalCongress of Mathematics in 1900. This setof problems has stimulated a great deal ofimportant work in mathematics since1900. Some of Hilbert’s problems havebeen solved but, at the time of writing,some of them remain unsolved.

histogram /his-tŏ-gram/ A statisticalgraph that represents, by the length of arectangular column, the number of timesthat each class of result occurs in a sampleor experiment. See also frequency polygon.

holomorphic /hol-ŏ-mor-fik, hoh-lŏ-/ Seeanalytic.

homeomorphism /hoh-mee-ŏ-mor-fiz-ăm, hom-ee-/ A one-one transformationbetween two topological spaces that is con-tinuous in both directions. What thismeans is that if two figures are homeomor-phic one can be continuously deformedinto the other without tearing. For exam-ple, any two spheres of any size are home-omorphic. But a sphere and a torus are not.

homogeneous /hoh-mŏ-jee-nee-ŭs, hom-ŏ-/ 1. (in a function) Having all the termsto the same degree in the variables. For ahomogeneous function f(x,y,z,…) of de-gree n

F(kx,ky,kz,…) = knf(x,y,z)for all values of k. For example, x2 + xy +y2 is a homogeneous function of degree 2and

(kx)2 + kx.ky + (ky)2 =k2(x2 + xy + y2)

2. Describing a substance or object inwhich the properties do not vary with po-sition; in particular, the density is constantthroughout.

homomorphism /hoh-mŏ-mor-fiz-ăm,hom-ŏ-/ If S and T are sets on which binaryrelations * and • are defined respectively, amapping h from S and T is a homomor-phism if it satisfies the condition h(x*y) =h(x)•h(y) for all x and y in S, i.e. it pre-serves structure. If the mapping is one-to-one it is called an isomorphism and the setsS and T are isomorphic.

Hooke’s law For an elastic materialbelow its elastic limit, the extension result-ing from the application of a load (force) isproportional to the load. The law is namedfor the English physicist Robert Hooke(1635–1703).

horizontal Describing a line that is paral-lel with the horizon. It is at right angles tothe vertical.

horsepower Symbol: HP A unit of powerequal to 550 foot-pounds per second. It isequivalent to 746 W.

hour A unit of time equal to 60 minutes or1/24 of a day.

hundredweight Symbol: cwt In the UK, aunit of mass equal to 112 pounds. It isequivalent to 50.802 3 kg. In the USA ahundredweight is equal to 100 pounds, butthis unit is rarely used.

Huygens’ principle /hÿ-gĕnz/ (alsoknown as HUYGENS’ CONSTRUCTION) A re-sult in the theory of waves which statesthat each point of a wave-front which ispropagating serves as the source of sec-ondary waves, with the secondary waveshaving the same frequency and speed as theoriginal wave. Huygens’ principle is usedextensively in optics to analyze light waves.The modification of Huygens’ principle toincorporate diffraction and interference issometimes called the Huygens–Fresnelprinciple. Both Huygens’ principle and theHuygens–Fresnel principle can be derivedas mathematical consequence of the WAVE

EQUATION. The principle is named for theDutch astronomer and physicist ChristiaanHuygens (1629–95).

hybrid computer A computer systemcontaining both analog and digital devicesso that the properties or each can be usedto the greatest advantage. For instance, adigital and an analog computer can be in-terconnected so that data can be trans-ferred between them. This is achieved bymeans of a hybrid interface. Hybrid com-puters are designed for specific tasks andhave a variety of uses, mainly in scientific

histogram

102

and technical fields. See also analog com-puter; computer.

hydraulic press A MACHINE in whichforces are transferred by way of pressure ina fluid. In a hydraulic press the effort F1 isapplied over a small area A1 and the loadF2 exerted over a larger area A2. Since thepressure is the same, F1/A1 = F2/A2. Theforce ratio for the machine, F2/F1, is A1/A2.Thus, in this case (and in the related hy-draulic braking system and hydraulic jack)the force exerted by the user is less than theforce applied; the force ratio is greater than1. If the distance moved by the effort is s1and that moved by the load is s2 then, sincethe same volume is transmitted through thesystem, s1A1 = s2A2; i.e. the distance ratiois A2/A1. In practical terms, the device isnot very efficient since frictional effects arelarge.

hydrostatics /hÿ-drŏ-stat-iks/ The studyof fluids (liquids and gases) in equilibrium.

hyperbola /hÿ-per-bŏ-lă/ (pl. hyperbolasor hyperbolae) A CONIC with an eccentric-ity greater than 1. The hyperbola has twobranches and two axes of symmetry. Anaxis through the foci cuts the hyperbola attwo vertices. The line segment joining thesevertices is the transverse axis of the hyper-bola. The conjugate axis is a line at rightangles to the transverse axis through thecenter of the hyperbola. A chord through afocus perpendicular to the transverse axisis a latus rectum.

In Cartesian coordinates the equation:x2/a2 – y2/b2 = 1

represents a hyperbola with its center at theorigin and the transverse axis along the x-axis. 2a is the length of the transverse axis.2b is the length of the conjugate axis. Thisis the distance between the vertices of a dif-ferent hyperbola (the conjugate hyperbola)with the same asymptotes as the given one.The foci of the hyperbola are at the points(ae,0) and (–ae,0), where e is the eccentric-ity. The asymptotes have the equations:

x/a – y/b = 0

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hyperbola

O

V1V2

F2 F1

Y1

Y2

asymptoteasymptote

x

y

Hyperbola: F1 and F2 aere foci. V1 and V2 are the vertices with coordinates (a,0) and (–a,0)respectively. Y1 and Y2 have coordinates (b,0) and (–b,0).

x/a + y/b = 0The equation of the conjugate hyper-

bola isx2/a2 – y2/b2 = –1

The length of the latus rectum is 2b2/ae.A hyperbola for which a and b are equal isa rectangular hyperbola:

x2 – y2 = a2

If a rectangular hyperbola is rotated sothat the x- and y-axes are asymptotes, thenits equation is

xy = kwhere k is a constant.

hyperbolic functions /hÿ-per-bol-ik/ Aset of functions that have properties similarin some ways to the trigonometric func-tions, called the hyperbolic sine, hyperboliccosine, etc. They are related to the hyper-bola in the way that the trigonometricfunctions (circular functions) are related tothe circle.

The hyperbolic sine (sinh) of an angle ais defined as:

sinha = ½(ea – e–a)The hyperbolic cosine (cosh) of an

angle a is defined as:cosha = ½(ea + e–a)

The hyperbolic tangent (tanh) of anangle a is defined as:

tanha = sinha/cosha =(ea – e–a)/(ea + e–a)

Hyperbolic secant (sech), hyperboliccosecant (cosech), and hyperbolic cotan-gent (coth) are defined as the reciprocals ofcosh, sinh, and tanh respectively. Some ofthe fundamental relationships between hy-perbolic functions are:

sinh(–a) = –sinhacosh(–a) = +coshacosh2a – sinh2a = 1sech2a + tanh2a = 1

coth2a – cosech2a = 1

hyperbolic functions

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3 –

2 –

111 –

0

-1 –

-2 –

-3 –

– – – – – – –

y = cosh x

y = sinh x

y = tanh x

x

y = sinh x

y = tanh x

-3 -2 -1 1 2 3

The graphs of the hyperbolic functions cosh x, sinh x, and tanh x.

hyperbolic paraboloid A CONICOID thatis described by the equation x2/a2 – y2/b2 =2z/c, where a, b, and c are constants andthe coordinate system is such that theorigin is a SADDLE POINT. Cross-sectionsthrough the z-axis cut this surface inparabolas, with the origin being the vertexof all these parabolas. Cross-sections of thesurface formed by planes parallel to the xy-planes are hyperbolas. The cross-sectionwith the xy-plane is a pair of straight lines.Cross-sections with planes which are par-allel to the xz- and yz-planes are parabolas.The xz- and yz-planes are symmetryplanes.

hyperboloid /hÿ-per-bŏ-loid/ A surfacegenerated by rotating a hyperbola aboutone of its axes of symmetry. Rotationabout the conjugate axis gives a hyper-boloid of one sheet. Rotation about thetransverse axis gives a hyperboloid of twosheets

hypertext /hÿ-per-text/ A method of cod-ing and displaying text on a computerscreen in such a way that key words orphrases in the document can act as directlinks to other documents or to other partsof the document. It is extensively used onthe World Wide Web. An extension of hy-pertext, known as hypermedia, allowslinkage to sounds, images, and video clips.

hypocycloid /hÿ-pŏ-sÿ-kloid/ A cuspedcurve that is the locus of a point on the cir-cumference of a circle that rolls around the

inside of a larger fixed circle. See also cusp;epicycloid.

hypotenuse /hÿ-pot-ĕ-news/ The sideopposite the right angle in a right-angledtriangle. The ratios of the hypotenuselength to the lengths of the other sides areused in trigonometry to define the sine andcosine functions of angles.

hypothesis /hÿ-poth-ĕ-sis/ (pl. hypoth-eses) A statement, theory, or formula thathas yet to be proved but is assumed to betrue for the purposes of the argument.

hypothesis test (significance test) A rulefor deciding whether an assumption (hy-pothesis) about the distribution of a ran-dom variable should be accepted orrejected, using a sample from the distribu-tion. The assumption is called the null hy-pothesis, written H0, and it is tested againstsome alternative hypothesis, H1. For ex-ample, when a coin is tossed H0 can beP(heads) = ½ and H1 that P(heads) > ½. Astatistic is computed from the sample data.If it falls in the critical region, where thevalue of the statistic is significantly differ-ent from that expected under H0, H0 is re-jected in favor of H1. Otherwise H0 isaccepted. A type I error occurs if H0 is re-jected when it should be accepted. A type IIerror occurs if it is accepted when it shouldbe rejected. The significance level of thetest, α, is the maximum probability withwhich a type I error can be risked. For ex-ample, α = 1% means H0 is wrongly re-jected in one case out of 100.

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hypothesis test

icosahedron /ÿ-kos-ă- hee-drŏn/ (pl.icosahedrons or icosahedra) A POLYHE-DRON that has twenty triangular faces. Aregular icosahedron has twenty congruentfaces, each one an equilateral triangle.

identity, law of See laws of thought.

identity element An element of a set that,combined with another element, leaves itunchanged. See group.

identity matrix See unit matrix.

identity set A set consisting of the sameelements as another. For example, the setof natural numbers greater than 2 and theset of integers greater than 2 are identitysets.

if and only if (iff) See biconditional.

if…then… See implication.

image The result of a geometrical trans-formation or a mapping. For example, ingeometry, when a set of points are trans-formed into another set by reflection in aline, the reflected figure is called the image.Similarly, the result of a rotation or a pro-jection is called the image. The algebraicequivalent of this occurs when a functionf(x) acts on a set A of values of x to pro-duce an image set B. See also domain;range; transformation.

imaginary axis The axis on the complexplane that purely imaginary numbers lieon. This axis is usually drawn as the y-axis.

imaginary number A multiple of i, thesquare root of minus one. The use of imag-inary numbers is needed to solve equations

such as x2 + 2 = 0, for which the solutionsare x = +i√2 and x = –i√2. See complexnumber.

imaginary part Symbol Imz. The part iyof a complex number z that can be writtenas z = x + iy, where x and y are both realnumbers.

impact A collision between two bodies. Ifthe initial masses and velocities of the bod-ies are known then the velocities followingthe impact can be calculated in terms of theinitial velocities and the COEFFICIENT OF

RESTITUTION by using the principle of theconservation of momentum. The instanceof a direct (head-on) impact is easiest toanalyze theoretically but the motions ofbodies which are not initially moving alongthe same line can also be analyzed by amodification of the method used for bodiesthat are initially moving along the sameline.

Imperial units The system of measure-ment based on the yard and the pound. Thef.p.s. system was a scientific system basedon Imperial units.

implication 1. (material implication; con-ditional) Symbol: → or ⊃ In logic, the rela-tionship if…then… between twopropositions or statements. Strictly, impli-cation reflects its ordinary language inter-

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I

P Q P → Q

T T TT F FF T TF F T

Implication

pretation (if…then…) much less than con-junction, disjunction, and negation dotheirs. Formally, P → Q is equivalent to ‘ei-ther not P or Q’ (∼ P ∨ Q), hence P → Q isfalse only when P is true and Q is false.Thus, logically speaking, if ‘pigs can fly’ issubstituted for P and ‘grass is green’ for Qthen ‘if pigs can fly then grass is green’ istrue. The truth-table definition of implica-tion is given in the illustration.2. In algebra, the symbol ⇒ is used be-tween two equations when the first impliesthe second. For example:

x = y ⇒ x2 = y2

See also condition; truth table.

implicit Denoting a function that con-tains two or more variables that are notindependent of each other. An implicitfunction of x and y is one of the form f(x,y)= 0, for example,

x2 + y2 – 4 = 0Sometimes an explicit function, that is,

one expressed in terms of an independentvariable, can be derived from an implicitfunction. For example,

y + x2 – 1 = 0can be written as

y = 1 – x2

where y is an explicit function of x.

improper fraction See fraction.

improper integral An integral in whicheither the interval of integration is infiniteor the value of the function f(x) which isbeing integrated becomes infinite at somepoint in the interval of integration.

An improper integral of the type∫∞a f(x)dx is said to exist if the value of theintegral of the function f(x) with respect tox in which the interval of integration runsfrom a to b tends to a finite limit l as b→∞,i.e. ∫∞a f(x)dx exists if lim∫baf(x)dx = l. An ex-ample of an improper integral of this typeis given by ∫∞1 (1/x2)dx = 1.

It is also possible to define an improperintegral if the interval of integration is from–∞ to a. This means that if the integral∫∞–∞ f(x)dx is written as the sum ∫ a

–∞f(x)dxand ∫∞a f(x)dx and if these two integralsexist and have values l1 and l2 respectively

then the improper integral ∫∞–∞ f(x)dx existsand has the value l1 + l2.

If the function f(x) which is being inte-grated becomes infinite at some point overthe interval of integration it is possible toexamine whether the improper integral ex-ists. This is the case whether the point atwhich the function becomes infinite is oneof the limits of integration or is within theinterval of integration. The case when thepoint is one of the limits is easier to ana-lyze. If the point is the lower limit a thenone can consider the integral with thelower limit a+δ, where δ is a small number,perform the integration and take the limitas δ→0. If such a limit exists then the im-proper integral exists and has a value givenby this limit. The case when the functionbecomes infinite at the upper limit b can beinvestigated in a similar way.

If the function becomes infinite at somepoint c which is between a and b the inte-gral is split into two integrals, with c beingthe upper limit of one integral and thelower limit of the other integral. If boththese improper integrals exist then theoriginal improper integral exists, with itsvalue being equal to the sum of the valuesof the two integrals into which it was split.

impulse (impulsive force) A force actingfor a very short time, as in a collision. If theforce (F) is constant the impulse is Fδt, δtbeing the time period. If the force is vari-able the impulse is the integral of this overthe short time period. An impulse is equalto the change of momentum that it pro-duces.

impulsive force See impulse.

incenter /in-sen-ter/ See incircle.

inch Symbol: in or ″ A unit of length equalto one twelfth of a foot. It is equivalent to0.025 4 m.

incircle /in-ser-kăl/ (inscribed circle) A cir-cle drawn inside a figure, touching all thesides. The center of the circle is the incen-ter of the figure. Compare circumcircle.

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incircle

inclined plane A type of MACHINE. Effec-tively a plane at an angle, it can be used toraise a weight vertically by movement upan incline. Both distance ratio and forceratio depend on the angle of inclination (θ)and equal 1/sinθ. The efficiency can befairly high if friction is low. The screw andthe wedge are both examples of inclinedplanes.

inclusion See subset.

inclusive disjunction (inclusive or) Seedisjunction.

inconsistent Describing a set of equationsfor which the solution set is empty. Geo-metrically, a set of equations is inconsistentif there is no point common to all the linesor curves represented by the equations. Inthe case of two linear equations in twovariables x and y the two equations can berepresented by straight lines L1 and L2. Thetwo equations are inconsistent if L1 and L2are parallel. In the case of three linearequations in three variables x, y, and z thethree equations can be represented byplanes P1, P2, and P3. There are three waysin which the equations can be inconsistent.The first way is that the three planes are allparallel and distinct. The second way isthat two of the planes are parallel and dis-tinct. The third way is that one plane isparallel to the line of intersection of theother two planes.

increasing function A real function f ofx for which f(x1) ≤ f(x2) for any x1 and x2in an interval I for which x1 < x2. If f(x1) <f(x2) when x1 < x2 then f is said to be astrictly increasing function.

increment A small difference in a vari-able. For example, x might change by anincrement ∆x from the value x1 to the valuex2; ∆x = x2 – x1. In calculus, infinitelysmall increments are used. See also differ-entiation; integration.

indefinite integral The general integra-tion of a function f(x), of a single variable,x, without specifying the interval of x towhich it applies. For example, if f(x) = x2,the indefinite integral

∫f(x)dx = ∫x2dx = (x3/3) + Cwhere C is an unknown constant (the con-stant of integration) that depends on the in-terval. Compare definite integral. See alsointegration.

independence See probability.

independent See dependent.

independent variable See variable.

indeterminate equation An equationthat has an infinite number of solutions.For example,

x + 2y = 3is indeterminate because an infinite num-ber of values of x and y will satisfy it. Anindeterminate equation in which the vari-ables can take only integer values is calleda Diophantine equation and it has an infi-nite but denumerable set of solutions. Dio-phantine equations are named for theGreek mathematician Diophantus ofAlexandria (fl. AD 250).

indeterminate form An expression thatcan have no quantitative meaning; for ex-ample 0/0.

index (pl. indexes or indices) A numberthat indicates a characteristic or functionin a mathematical expression. For exam-ple, in y4, the exponent, 4, is also known as

inclined plane

108

Incircle: an incircle touches all the sides of itssurrounding figure.

the index. Similarly in 3√27 and log10x, thenumbers 3 and 10 respectively are calledindices (or indexes).

indirect proof (reductio ad absurdum) Alogical argument in which a proposition orstatement is proved by showing that itsnegation or denial leads to a CONTRADIC-TION. Compare direct proof.

induction /in-duk-shŏn/ 1. (mathematicalinduction) A method of proving mathe-matical theorems, used particularly for se-ries sums. For instance, it is possible toshow that the series 1 + 2 + 3 + 4 + … hasa sum to n terms of n(n + 1)/2. First weshow that if it is true for n terms it mustalso be true for (n + 1) terms. According tothe formula

Sn = n(n + 1)/2 if the formula is correct, the sum to (n + 1)terms is obtained by adding (n + 1) to this

Sn+1 = n(n + 1)/2 + (n + 1)Sn+1 = (n + 1)(n + 2)/2

This agrees with the result obtained byreplacing n in the general formula by (n +1), i.e.:

Sn+1 = (n + 1)(n + 1 + 1)/2Sn+1 = (n + 1)(n + 2)/2

Thus, the formula is true for (n + 1)terms if it is true for n terms. Therefore, ifit is true for the sum to one term (n = 1), itmust be true for the sum to two terms (n +1). Similarly, if true for two terms, it mustbe true for three terms, and so on throughall values of n. It is easy to show that it istrue for one term:

Sn = 1(1 + 1)/2Sn = 1

which is the first term in the series. Hencethe theorem is true for all integer values ofn.2. In logic, a form of reasoning from indi-vidual cases to general ones, or from ob-served instances to unobserved ones.Inductive arguments can be of the form: F1is A, F2 is A … Fn is A, therefore all Fs areA (‘this swan has wings, that swan haswings … therefore all swans have wings’);or: all Fs observed so far are A, thereforeall Fs are A (‘all swans observed so far arewhite, therefore all swans are white’). Un-like deduction, asserting the premisses

while denying the conclusion in an induc-tion does not lead to a CONTRADICTION.The conclusion is not guaranteed to be trueif the premisses are. Compare deduction.

inelastic collision /in-i-las-tik/ A collisionfor which the restitution coefficient is lessthan one. In effect, the relative velocityafter the collision is less than that before;the kinetic energy of the bodies is not con-served in the collision, even though the sys-tem may be closed. Some of the kineticenergy is converted into internal energy.See also restitution, coefficient of.

inequality A relationship between twoexpressions that are not equal, often writ-ten in the form of an equation but with thesymbols > or < meaning ‘is greater than’and ‘is less than’. For example, if x < 4 thenx2 < 16. If y2 > 25, then y > 5 or y < –5. Ifthe end values are included, the symbols ≥(is greater than or equal to) and ≤ (is lessthan or equal to) are used. When one quan-tity is very much smaller or greater thananother, it is shown by << or >>. For ex-ample, if x is a large number x >> 1/x or 1/x<< x. See also equality.

inequation /in-i-kway-zhŏnz/ Anotherword for inequality.

inertia /i-ner-shă/ An inherent property ofmatter implied by Newton’s first law ofmotion: inertia is the tendency of a body toresist change in its motion. See also inertialmass; Newton’s laws of motion.

inertial mass /i-ner-shăl/ The mass of anobject as measured by the property of iner-tia. It is equal to the ratio force/accelera-tion when the object is accelerated by aconstant force. In a uniform gravitationalfield, it appears to be equal to GRAVITA-TIONAL MASS – all objects have the samegravitational acceleration at the sameplace.

inertial system A frame of reference inwhich an observer sees an object that is freeof all external forces to be moving at con-stant velocity. The observer is called an in-ertial observer. Any FRAME OF REFERENCE

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inertial system

that moves with constant velocity andwithout rotation relative to an inertialframe is also an inertial frame. NEWTON’SLAWS OF MOTION are valid in any inertialframe (but not in an accelerated frame),and the laws are therefore independent ofthe velocity of an inertial observer.

inf See infimum.

inference /in-fĕ-rĕns/ 1. The process ofreaching a conclusion from a set of pre-misses in a logical argument. An inferencemay be deductive or inductive. See also de-duction; induction.2. See sampling.

infimum (inf) The greatest lower BOUND

of a set.

infinite number The smallest infinitenumber is '0 (aleph zero). This is the num-ber of members in the set of integers. Awhole hierarchy of increasingly large infi-nite numbers can be defined on this basis.'1, the next largest, is the number of sub-sets of the set of integers. See also aleph;continuum; countable.

infinite sequence See sequence.

infinite series See series.

infinite set A set in which the number ofelements is infinite. For example, the set of‘positive integers’, z = 1, 2, 3, 4, …, is in-finite but the set of ‘positive integers lessthan 20’ is a finite set. Another example ofan infinite set is the number of circles in aparticular plane. Compare finite set.

infinitesimal /in-fi-nă-tess-ă-măl/ Infi-nitely small, but not equal to zero. Infini-tesimal changes or differences are made useof in CALCULUS (infinitesimal calculus).

infinity Symbol: ∞ The value of a quantitythat increases without limit. For example,if y = 1/x, then y becomes infinitely large,or approaches infinity, as x approaches 0.An infinitely large negative quantity is de-noted by –∞ and an infinitely large positive

quantity by +∞. If x is positive, y = –1/xtends to –∞ as x tends to 0.

inflection See point of inflection.

information theory The branch of prob-ability theory that deals with uncertainty,accuracy, and information content in thetransmission of messages. It can be appliedto any system of communication, includingelectrical signals and human speech. Ran-dom signals (noise) are often added to amessage during the transmission process,altering the signal received from that sent.Information theory is used to work out theprobability that a particular signal receivedis the same as the signal sent. Redundancy,for example simply repeating a message, isneeded to overcome the limitations of thesystem. Redundancy can also take the formof a more complex checking process. Intransmitting a sequence of numbers, theirsum might also be transmitted so that thereceiver will know that there is an errorwhen the sum does not correspond to therest of the message. The sum itself gives noextra information since, if the other num-bers are correctly received, the sum caneasily be calculated. The statistics of choos-ing a message out of all possible messages(letters in the alphabet or binary digits forexample) determines the amount of infor-mation contained in it. Information is mea-sured in bits (binary digits). If one out oftwo possible signals are sent then the infor-mation content is one bit. A choice of oneout of four possible signals contains moreinformation, although the signal itselfmight be the same.

inner product Consider a vector space Vover a scalar field F. An inner product on Vis a mapping of ordered pairs of vectors inV into F; i.e. with every pair of vectors xand y there is associated a scalar, which iswritten ⟨x,y⟩ and called the inner productof x and y, such that for all vectors x, y, zand scalars α(i) ⟨x+y,z⟩ = ⟨x,z⟩ + ⟨y,z⟩(ii) ⟨x,y⟩ = α⟨x,y⟩(iii) ⟨x,y⟩ = ⟨y,x ⟩, where ⟨a,b ⟩ is the com-plex conjugate of ⟨a,b⟩(iv) ⟨x,x⟩ ≥ 0, ⟨x,x⟩ = 0 if and only if x = 0.

inf

110

An inner product on V defines a norm onV given by ||x|| = √⟨x,x⟩. See norm.

input 1. The signal or other form of in-formation that is applied (fed in) to an elec-trical device, machine, etc. The input to acomputer is the data and programmed in-structions that a user communicates to themachine. An input device accepts com-puter input in some appropriate form andconverts the information into a code ofelectrical pulses. The pulses are then trans-mitted to the central processor of the com-puter. 2. The process or means by which input isapplied.3. To feed information into an electricaldevice or machine.

See also input/output; output.

input/output (I/O) The equipment andoperations used to communicate with acomputer, and the information passed inor out during the communication. Input/output devices include those used only forINPUT or for OUTPUT of information andthose, such as visual display units, used forboth input and output.

inscribed Describing a geometric figurethat is drawn inside another geometricalfigure. Compare circumscribed.

inscribed circle See incircle.

instantaneous value /in-stăn-tay-nee-ŭs/The value of a varying quantity (e.g. veloc-ity, acceleration, force, etc.) at a particularinstant in time.

integer /in-tĕ-jer/ Symbol: z Any of the setof whole numbers, …, –2, –1, 0, 1, 2, …used for counting. The integers includezero and the negative whole numbers.

integer part Symbol [x]. An integer nwhich is related to a real number x by n ≤

x < n + 1. The integer n is said to be the in-teger part of x. For example, [13/3] = 4 and[e] = 2. Care needs to be taken in the case of negative numbers. For example,[–13/3] = –5. The difference between posi-tive numbers can be significant if one has acomputer program that converts a realnumber into an integer by truncation. Inthe case of positive numbers and zero trun-cation and taking the integer part of thereal number would give the same integerbut this is not the case for negative num-bers.

integer variable See variable.

integral /in-tĕ-grăl, in-teg-răl/ The resultof integrating a function. See integration.

integral equation An equation in whichan unknown function appears under an in-tegral sign. Many problems in physical sci-ence and engineering can be expressed asintegral equations. It is sometimes easier tosolve problems which can be expressed asordinary or partial differential equationsby expressing them in terms of integralequations. In particular, solving initialvalue problems and boundary-value prob-lems of differential equations is frequentlyperformed by converting the differentialequation into an integral equation. Just asthere is no general technique for perform-ing integration so there is no general tech-nique for solving all integral equations.

integral transform A transform in whicha function f(x) of a variable x is trans-formed into a function g(y) of a variable yby the integral equation:

g(y) = ∫K(x, y)f(x)dx, where K(x,y) is a part of the integral equa-tion known as its kernel. For example,integral transforms can map a function oftime into another function in frequencywhen wave phenomena are being analyzed.See Laplace transform.

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integral transform

– – – – – – – –

-3 -2 -1 0 1 2 3 4

Integers: a number line showing positive and negative integers.

integrand /in-tĕ-grand/ A function that isto be integrated. For example, in the inte-gral of f(x).dx, f(x) is the integrand. Seealso integration.

integrating factor A multiplier used tosimplify and solve DIFFERENTIAL EQUA-TIONS, usually given the symbol ξ. For ex-ample, xdy – ydx = 2x3dx may bemultiplied by ξ(x) = 1/x2 to give the stan-dard form:

d(y/x) = (x.dy – y.dx)/x2 = 2xdxwhich has the solution y/x = x2 + C, whereC is a constant.

integration The continuous summing ofchange in a function, f(x), over an intervalof the variable x. It is the inverse process ofDIFFERENTIATION in calculus, and its resultis known as the integral of f(x) with respectto x. An integral:

x1∫x2

vdt

can be regarded as the area between thecurve and the x-axis, between the values x1and x2. It can be considered as the sum ofa number of column areas of width ∆x andheights given by f(x). As ∆x approacheszero, the number of columns increases infi-nitely and the sum of the column areas ap-proaches the area under the curve. The

integral of velocity is distance. For exam-ple, a car traveling with a velocity V in atime interval t1 to t2 goes a distance s givenby

t1∫t2

vdt

Integrals of this type, between definite lim-its, are known as definite integrals. An in-definite integral is one without limits. Theresult of an indefinite integral contains aconstant – the constant of integration. Forexample,

∫xdx = x2/2 + Cwhere C is the constant of integration. Atable of integrals is given in the Appendix.

integration by parts A method of inte-grating a function of a variable by express-ing it in terms of two parts, both of whichare differentiable functions of the samevariable. A function f(x) is written as theproduct of u(x) and the derivative dv/dx.The formula for the differential of a prod-uct is:

d(u.v)/dx = u.dv/dx + v.du/dxIntegrating both sides over x and rearrang-ing the equation gives

∫u.(dv/dx)dx = uv – ∫v(du/dx)dxwhich can be used to evaluate the integralof a product. For example, to integratexsinx, dv/dx is taken to be sinx, so v =–cosx + C (C is a constant of integration).

integrand

112

y

f(x )

x2

f(x )dxx1

0x1 x2

dxdxdx

x

The integration of a function y = f(x) as the area between the curve and the x-axis.

u is taken to be x, so du/dx = 1. The inte-gral is then given by

∫xsinxdx =x(–cosx + C) – ∫(–cosx + C) =

–xcosx + sinx + kwhere k is a constant of integration. Usu-ally a trigonometric or exponential func-tion is chosen for dv/dx.

integration by substitution A methodof integrating a function of one variable byexpressing it as a simpler or more easily in-tegrated function of another variable. Forexample, to integrate √(1 – x2) with respectto x, we can make x = sinu, so that √(1 – x2)= √(1 – sin2u) = √(cos2u) = cosu, and dx =(dx/du).du = cosudu. Therefore:

a

b

(1 – x2)dx = c

d

cos2u.du

= [u/2 – 12sinucosu]dc

Note that for a definite integral the limitsmust also be changed from values of x tocorresponding values of u.

integro-differential equation /in-teg-roh-dif-ĕ-ren-shăl/ A differential equationthat also has an integral as part of it. Suchequations occur in the quantitative descrip-tion of transport processes. General meth-ods for the solution of integro-differentialequations do not exist and approximationmethods have to be used.

interaction Any mutual action betweenparticles, systems, etc. Examples of interac-tions include the mutual forces of attrac-tion between masses (gravitationalinteraction) and the attractive or repulsiveforces between electric charges (electro-magnetic interaction).

intercept A part of a line or plane cut offby another line or plane.

interest The amount of money paid eachyear at a stated rate on a borrowed capital,or the amount received each year at astated rate on loaned or invested capital.The interest rate is usually expressed as apercentage per annum. See compound in-terest; simple interest.

interface A shared boundary. It is thearea(s) or place(s) at which two devices ortwo systems meet and interact. For exam-ple, there is a simple interface between anelectric plug and socket. A far more com-plicated interface of electronic circuits pro-vides the connection between the centralprocessor of a computer and each of its pe-ripheral units. Human-machine interfacerefers to the interaction between peopleand machines, including computers. Forgood, i.e. efficient, interaction, devicessuch as visual display units and easily un-derstandable programming languages havebeen introduced.

interior angle An angle formed on the in-side of a plane figure by two of its straightsides. For example, there are three interiorangles in a triangle, which add up to 180°.Compare exterior angle.

intermediate value theorem A theoremconcerning continuous functions whichstates that if a real function f is continuouson an interval I bounded by a and b forwhich f(a) ≠ f(b) then if n is some real num-ber between f(a) and f(b) there must besome number c in the interval, i.e. a < c < bfor which f(c) = n.

The intermediate value theorem can beused to find the solutions of an equation.For example, if f(a) > 0 and f(b) < 0 thenf(x) = 0 between a and b.

internal store See central processor;store.

interpolation /in-terp-ŏ-lay-shŏn/ Theprocess of estimating the value of a func-tion from known values on either side of it.For example, if the speed of an engine, con-trolled by a lever, increases from 40 to 50revolutions per second when the lever ispulled down by four centimeters, one caninterpolate from this information and as-sume that moving it two centimeters willgive 45 revolutions per second. This is thesimplest method of interpolation, calledlinear interpolation. If known values ofone variable y are plotted against the othervariable x, an estimate of an unknownvalue of y can be made by drawing a

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interpolation

straight line between the two nearestknown values.

The mathematical formula for linear in-terpolation is:

y3 = y1 + (x3 – x1)(y2 – y1)/(x2 – x1)y3 is the unknown value of y (at x3) and y2and y1 (at x2 and x1) are the nearest knownvalues, between which the interpolation ismade. If the graph of y against x is asmooth curve, and the interval between y1and y2 is small, linear interpolation cangive a good approximation to the truevalue, but if (y2 – y1) is large, it is less likelythat y will fit sufficiently well to a straightline between y1 and y2. A possible sourceof error occurs when y is known at regularintervals, but oscillates with a periodshorter than this interval.

Compare extrapolation.

interquartile range /in-ter-kwor-tÿl/ Ameasure of dispersion given by (P75 – P25)where P75 is the upper QUARTILE and P25the lower quartile. The semi-interquartilerange is ½(P75 – P25).

intersection 1. The point at which two ormore lines cross each other, or a set ofpoints that two or more geometrical fig-ures have in common.2. In set theory, the set formed by the ele-ments common to two or more sets. Forexample, if set A is black four-legged ani-mals and set B is sheep then the intersec-tion of A and B, written A∩B, is blacksheep. This can be represented on a VENN

DIAGRAM by the intersection of two circles,one representing A and the other B.

interval A set of numbers, or points in acoordinate system, defined as all the valuesbetween two end points. See also closed in-terval; open interval.

into A mapping from one set to another issaid to be into if the range of the mappingis a proper subset of the second set; i.e. ifthere are members of the second set whichare not the image of any element of the firstset under the mapping. Compare onto.

intransitive /in-tran-să-tiv/ Describing arelation that is not transitive; i.e. when xRyand yRz then it is not true that xRz. An ex-ample of an intransitive relation is beingthe square of. If x = y2 and y = z2 it does notfollow that x = z2.

intuitionism /in-too-ish-ŏ-niz-ăm/ An ap-proach to the foundations of mathematicsthat was developed in the late nineteenthcentury and the early decades of the twen-tieth century. Intuitionism tried to build upmathematics based on intuitive conceptsrather than from a rigid series of axioms.Using this approach the intuitionists wereable to build up various parts of algebraand geometry and to formulate calculus,albeit in a complicated way. As with otherattempts such as FORMALISM to providefoundations for mathematics, intuitionismwas not accepted universally as providing afoundation for mathematics.

The main difficulties for all attempts toprovide foundations for mathematics, in-cluding formalism and intuitionism involvedealing with infinite sets and infiniteprocesses.

interquartile range

114

E

A B

A ∩ B

The shaded area in the Venn diagram is the intersection of set A and set B.

invariant /in-vair-ee-ănt/ Describing aproperty of an equation, function, or geo-metrical figure that is unaltered after theapplication of any member of some givenfamily of transformations. For example,the area of a polygon is invariant under thegroup of rotations.

inverse element An element of a set that,combined with another element, gives theidentity element. See group.

inverse function The reverse mapping ofthe set B into the set A when the FUNCTION

that maps A into B has already been de-fined.

inverse hyperbolic functions The in-verse functions of hyperbolic sine, hyper-bolic cosine, hyperbolic tangent, etc.,defined in an analogous way to the inversetrigonometric functions. For instance, theinverse hyperbolic sine of a variable x,written arc sinhx or sinh–1x, is the angle (ornumber) of which x is the hyperbolic sine.Similarly, the other inverse hyperbolicfunctions are:inverse hyperbolic cosine of x (written arccoshx or cosh–1x)inverse hyperbolic tangent of x (written arctanhx or tanh–1x)inverse hyperbolic cotangent of x (writtenarc cothx or coth–1x)inverse hyperbolic secant of x (written arcsechx or sech–1x)inverse hyperbolic cosecant of x (writtenarc cosechx or cosech–1x).

inverse mapping A mapping, denotedf–1, from one set B to another set A, wheref is a one-to-one and onto mapping from Ato B which satisfies the definition in the fol-lowing sentence. The inverse mapping f–1

from B to A exists if, for an element b of B,there is a unique element a of A given byf–1(b) = a which satisfies f(a) = b. Like themapping f, the inverse mapping f–1 is a one-to-one and onto mapping.

inverse matrix The unique n × n matrix,denoted A–1 corresponding to the n × n ma-trix A that satisfies AA–1 = A–1A = I, whereI is the UNIT MATRIX of order n. If a matrixis not a square matrix it cannot have an in-verse matrix. It is not necessarily the casethat a square matrix has an inverse. If amatrix does have an inverse then the ma-trix is invertible or non-singular.

There are several techniques for findinginverse matrices. In general, the inverse ofA is given by (1/detA) adj A, where adjA IS

the ADJOINT of A, provided that detA ≠ 0. IfdetA = 0 then the matrix is said to be SIN-GULAR and an inverse does not exist. In thecase of a 2 × 2 matrix A of theform

The inverse A–1 exists if ad–bc ≠ 0 and isgiven by

In the case of large matrices it is notconvenient to find the inverse of a matrix

A−1 = d −b

−c a1

ad − bc

A =a bc d

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inverse matrix

x + 3y = 52x + 4y = 6

<=> 1 32 4

xxy

56

1 32 4

x 1 00 1

=

=

56

xy

-2 1 -½

23

-2 1 -½

23

-1 2

=> = x =

The solution of simultaneous equations by finding the inverse of a matrix. The equations are writ-ten as the equivalent matrix equation, both sides of which are then multiplied by the inverse ofthe coefficient matrix.

by using the general formula involving thedeterminant. It is more convenient to use atechnique similar to GAUSSIAN ELIMINATION,which is used to solve sets of linear equa-tions.

inverse of a complex number A com-plex number, denoted by 1/z or z–1, whichis the inverse of the complex number z = x+ i y, i.e. it is the number z–1 for which zz–1

= 1. It is given by z = x/(x2 + y2) – i y/(x2 +y2). If z is written in terms of polar coordi-nates z = r (cos θ + i sin θ) = r exp (iθ) thenz–1 is given by (1/r) exp (–iθ) = (1/r) (cos θ – i sin θ), which exists if r ≠ 0.

inverse ratio A reciprocal ratio. For ex-ample, the inverse ratio of x to y is the ratioof 1/x to 1/y.

inverse square law A physical law inwhich an effect varies inversely as thesquare of the distance from the source pro-ducing the effect. An example is Newton’slaw of universal gravitation.

inverse trigonometric functions The in-verse functions of sine, cosine, tangent, etc.For example, the inverse sine of a variableis called the arc sine of x; it is written arcsinx or sin–1x and is the angle (or number)of which the sine is x. Similarly, the otherinverse trigonometric functions are:inverse cosine of x (arc cosine, written arccosx or cos–1x)inverse tangent of x (arc tangent, writtenarc tanx or tan–1x)inverse cotangent of x (arc cotangent, writ-ten arc cotx or cot–1x)inverse cosecant of x (arc cosecant, writtenarc cosecx or cosec–1x)inverse secant of x (arc secant, written arcsecx or sec–1x).

inverter gate See logic gate.

involute /in-vŏ-loot/ The involute of acurve is a second curve that would be ob-tained by unwinding a taut string wrappedaround the first curve. The involute is thecurve traced out by the end of the string.

I/O See input/output.

irrational number /i-rash-ŏ-năl/ A num-ber that cannot be expressed as a ratio oftwo integers. The irrational numbers areprecisely those infinite decimals that arenot repeating. Irrational numbers are oftwo types:(i) Algebraic irrational numbers are rootsof polynomial equations with rationalnumbers as coefficients. For example, √3 =1.732 050 8… is a root of the equation x2

= 3. This equation does not have a rationalsolution since such a solution could bewritten x = m/n with m2 = 3n2, but this isimpossible since 3 divides the left-handside an even number of times and the right-hand side an odd number of times.(ii) Transcendental numbers are irrationalnumbers that are not algebraic, e.g. e andπ.

irrational term A term in which at leastone of the indices is an irrational number.For example, xy√2 and 2xπ are irrationalterms.

irrotational vector A vector V for whichthe curl of V vanishes, i.e. ∇ × V = 0. Ex-amples of irrotational vectors include thegravitational force described by Newton’slaw of gravity and the electrostatic forcegoverned by Coulomb’s law. If u is the dis-placement of a plane wave in an elasticmedium (or a spherical wave a long wayfrom the source of the wave) then the con-dition that u is an irrotational vectormeans that the wave is a longitudinal wave.When this result is combined with theconcept of a SOLENOIDAL VECTOR andHELMHOLTZ’S THEOREM it is useful in ana-lyzing waves in seismology.

If a vector is an irrotational vector thenit can be written as –1 times the gradient ofa scalar function. In physical applicationsthe scalar function is usually called thescalar potential.

isolated point A point that satisfies theequation of a curve but is not on the mainarc of the curve. For example, the equationy2(x2 – 4) = x4 has a solution at x = 0 and y= 0, but there is no real solution at any

inverse of a complex number

116

point near the origin, so the origin is an iso-lated point. See also double point; multiplepoint.

isolated system See closed system.

isometric paper /ÿ-sŏ-met-rik/ Paper onwhich there is a grid of equilaterial trian-gles printed. This type of paper is useful fordrawing shapes such as cubes and cuboids,with each face being a parallelogram in thedrawing. Using this type of paper it is pos-sible to draw all the dimensions of the fig-ures correctly and to measure them byusing the regular grid pattern.

isometry /ÿ-som-ĕ-tree/ A transformationin which the distances between the pointsremain constant.

isomorphic /ÿ-sŏ-mor-fik/ See homomor-phism.

isomorphism /ÿ-sŏ-mor-fiz-ăm/ See ho-momorphism.

isosceles /ÿ-soss-ĕ-leez/ Having two equalsides. See triangle.

issue price See nominal value.

iterated integral /it-ĕ-ray-tid/ (multiple

integral) A succession of integrations per-formed on the same function. For example,a DOUBLE INTEGRAL or a TRIPLE INTEGRAL.

iteration /it-ĕ-ray-shŏn/ A method of solv-ing a problem by successive approxima-tions, each using the result of the precedingapproximation as a starting point to obtaina more accurate estimate. For example, thesquare root of 3 can be calculated by writ-ing the equation x2 = 3 in the form 2x2 = x2

+ 3, or x = ½(x + 3/x). To obtain a solutionfor x by iteration, we might start with afirst estimate, x1 = 1.5. Substituting this inthe equation gives the second estimate, x2= ½(1.5 + 2) = 1.750 00. Continuing in thisway, we obtain:

x3 = ½(1.75 + 3/1.75) = 1.732 14x4 = ½(1.732 14 + 3/1.732 14) =

1.732 05and so on, to any required accuracy. Thedifficulty in solving equations by iterationis in finding a formula for iteration (algo-rithm) that gives convergent results. In thiscase, for example, the algorithm xn+1 =3/xn does not give convergent results.There are several standard techniques,such as NEWTON’S METHOD, for obtainingconvergent algorithms. Iterative calcula-tions, although often tedious for manualcomputation, are widely used in comput-ers.

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iteration

j An alternative to i for the square root of –1. The use of j rather than i in complexnumbers is particularly common amongelectrical engineers.

job A unit of work submitted to a com-puter. It usually includes several programs.The information necessary to run a job isinput in the form of a short program writ-ten in the job-control language (JCL) of thecomputer. The JCL is interpreted by theOPERATING SYSTEM and is used to identifythe job and describe its requirements to theoperating system. See also batch process-ing.

joule /jool/ Symbol: J The SI unit of energyand work, equal to the work done wherethe point of application of a force of onenewton moves one meter in the direction ofaction of the force. 1 J = 1 N m. The jouleis the unit of all forms of energy. The unitis named for the British physicist JamesPrescott Joule (1818–89).

Julia set /joo-lee-ă/ A set of points in thecomplex plane defined by iteration of a

complex number. One takes the expressionz2 + c, where z and c are complex numbers,and calculates it for a given value of z andtakes the result as a new starting value of z.This process can be repeated indefinitelyand three possibilities occur, depending onthe initial value for z. One is that the valuetends to zero with successive iterations.Another is that the value diverges to infin-ity. There is, however, a set of initial valuesof z for which successive iterations give val-ues that stay in the set. This set of values isa Julia set and it can be represented bypoints in the complex plane. The Julia set isthe boundary between values that have anattractor at zero and values that have anATTRACTOR at infinity. The actual form ofthe Julia set depends on the value of theconstant complex number c. Thus, if c = 0,the iteration is z → z2. In this case the Juliaset is a circle with radius 1. There is an in-finite number of Julia sets showing a widerange of complex patterns. The set isnamed for the French mathematician Gas-ton Julia (1893–1978). See also Mandel-brot set.

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kelvin Symbol: K The SI base unit of ther-modynamic temperature. It is defined asthe fraction 1/273.16 of the thermody-namic temperature of the triple point ofwater. Zero kelvin (0 K) is absolute zero.One kelvin is the same as one degree on theCelsius scale of temperature. The unit isnamed for the British theoretical and ex-perimental physicist Baron William Thom-son Kelvin (1824–1907).

Kendall’s method A method of measur-ing the degree of association between twodifferent ways of ranking n objects, usingtwo variables (x and y), which give data(x1,y1),…,(xn,yn). The objects are rankedusing first the xs and then the ys. For eachof the 2n(n – 1)/2 pairs of objects a score isassigned. If the RANK of the jth object isgreater (or less) than that of the kth, re-gardless of whether the xs or ys are used,the score is plus one. If the rank of the jthis less than that of the kth using one vari-able but greater using the other, the score isminus one. Kendall’s coefficient of rankcorrelation τ = (sum of scores)/½n(n – 1).The closer τ is to one, the greater the degreeof association between the rankings. Themethod is named for the British statisticianMaurice Kendall (1907–83). See alsoSpearman’s method.

Kepler’s laws /kep-lerz/ Laws of plane-tary motion deduced in about 1610 by theGerman astronomer Johannes Kepler(1571–1630) using astronomical observa-tions made by the Danish astronomerTycho Brahe (1546–1601):(1) Each planet moves in an elliptical orbitwith the Sun at one focus of the ellipse.(2) The line between the planet and the Sunsweeps out equal areas in equal times.

(3) The square of the period of each planetis proportional to the cube of the semi-major axis of the ellipse.

Application of the third law to the orbitof the Moon about the Earth gave supportto Newton’s theory of gravitation.

keyboard A computer input device that ahuman user can operate to type in data inthe form of alphanumeric characters. It hasa standard QWERTY key layout with someadditional characters and function keys.See alphanumeric; input device.

kilo- Symbol: k A prefix denoting 103. Forexample, 1 kilometer (km) = 103 meters(m).

kilogram /kil-ŏ-gram/ (kilogramme) Sym-bol: kg The SI base unit of mass, equal tothe mass of the international prototype ofthe kilogram, which is a piece of plat-inum–iridium kept at Sèvres in France.

kilogramme /kil-ŏ-gram/ An alternativespelling of kilogram.

kilometer /kil-ŏ-mee-ter, kă-lom-ĕ-ter/(km) A unit of distance equal to 1000 me-tres, approximating to 0.62 mile.

kilowatt-hour /kil-ŏ-wot/ Symbol: kwh Aunit of energy, usually electrical, equal tothe energy transferred by one kilowatt ofpower in one hour. It is the same as theBoard of Trade unit and has a value of 3.6× 106 joules.

kinematics /kin-ĕ-mat-iks/ The study ofthe motion of objects without considera-tion of its cause. See also mechanics.

kinetic energy /ki-net-ik/ Symbol: T The

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K

work that an object can do because of itsmotion. For an object of mass m movingwith velocity v, the kinetic energy is mv2/2.This gives the work the object would do incoming to rest. The rotational kinetic en-ergy of an object of moment of inertia Iand angular velocity ω is given by Iω2/2.See also energy.

kinetic friction See friction.

kite A plane figure with four sides, andtwo pairs of adjacent sides equal. Two ofthe angles in a kite are opposite and equal.Its diagonals cross perpendicularly, one ofthem (the shorter one) being bisected bythe other. The area of a kite is equal to halfthe product of its diagonal lengths. In thespecial case in which the two diagonalshave equal lengths, the kite is a rhombus.

Klein bottle /klÿn/ A curved surface withthe unique topological property that it hasonly one surface, no edges, and no insideand outside. It can be thought of as formedby taking a length of flexible stretchy tub-ing, cutting a hole in the side throughwhich one end can fit exactly, passing anend of the tube through this hole, and thenjoining it to the other end from the inside.Starting at any point on the surface a con-tinuous line can be drawn along it to anyother point without crossing an edge. Thebottle is named for the German mathe-matician (Christian) Felix Klein (1849–1925). See also topology.

knot 1. A curve formed by looping and in-terlacing a string and then joining the ends.The mathematical theory of knots is abranch of TOPOLOGY.2. A unit of velocity equal to one nauticalmile per hour. It is equal to 0.414 m s–1.

Koch curve /kokh/ See fractal.

Königsberg bridge problem /koh-nigz-berg/ A classical problem in topology. Theriver in the Prussian city of Königsberg di-vided into two branches and was crossedby seven bridges in a certain arrangement.The problem was to show that it is impos-sible to walk in a continuous path acrossall the bridges and cross each one onlyonce. The problem was solved by Euler inthe eighteenth century, by replacing thearrangement by an equivalent one of linesand vertices. He showed that a networklike this (called a graph) can be traversed ina single path if and only if there are fewerthan three vertices at which an odd numberof line segments meet. In this case there arefour.

Kronecker delta /kroh-nek-er/ Symbolδij. A quantity defined by δij = 1, if i = j, andδij = 0 if i ≠ j. It is named for the Germanmathematician Leopold Kronecker (1823–91).

Kuratowski–Zorn lemma See Zornlemma.

kinetic friction

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d

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islandl

a b

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l

c

d

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Z

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map graph

Königsberg bridge problem

lambda calculus /lam-dă/ A branch ofmathematical logic that is used to expressthe idea of COMPUTABILITY. Any mathemat-ical process that can be performed by usingan ALGORITHM can be achieved using thelambda calculus. The ideas of the lambdacalculus are very closely related to thefoundations of the theory of how comput-ers operate, particularly the concept of aTURING MACHINE.

lamina /lam-ă-nă/ (pl. laminae) An ideal-ization of a thin object that is taken to haveuniform density and has the dimensions ofarea but zero thickness. A sheet of metal orpaper or a card can be represented by alamina.

Lami’s theorem /lam-ee, lah-mee/ A rela-tion between three forces A, B, and C thatare in equilibrium and the angles α, β, andγ between the forces shown in the figure.The theorem states that A/sinα = B/sinβ =C/sinγ. Lami’s theorem is derived by apply-ing the SINE RULE to the triangle of forcesassociated with A, B, and C and using theresult that sin(180°– θ) – sin θ. It is used tosolve problems involving three forces.

Langlands program /lang-lănds/ An at-tempt to unify different, apparently uncon-nected, branches of mathematics that wasinitiated by the American mathematicianRobert Langlands in the 1960s. This pro-gram started off as a series of conjecturesrelating different branches of mathematics.A major boost to the Langlands programwas supplied in 1995 when FERMAT’S LAST

THEOREM was proved by showing that aconjecture relating two different branchesof mathematics is true. If the other conjec-tures in the Langlands program are true itwould mean that problems in one branchof mathematics could be related to prob-lems in another branch of mathematics,with the hope that problems that are verydifficult to solve in one branch of mathe-matics might be easier to solve in anotherbranch. In addition, the program would, ifrealized, give a great deal of unity to math-ematics.

Laplace’s equation /lah-plahs/ See par-tial differential equation. The equation isnamed for the French mathematicianPierre Simon, Marquis de Laplace (1749–1827).

Laplace transform An integral trans-form f(x) of a function F(t) defined by:

f(x) = ∫ ∞0 exp(–xt)F(t)dt

It is possible for this integral not to existbut it is possible to specify under what con-ditions it does exist.

The Laplace transform of a functionF(t) is sometimes denoted by LF(t). Forexample, if F(t) = 1, LF(t) = 1/x. If F(t) =sin at, LF(t) = a/(x2 + a2). If F(t) =exp(–at), LF(t) = 1/(x + a). Tables ofLaplace transforms are available and theycan be used in the solution of ordinary andpartial differential equations.

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α

β

Lami’s theorem

Laplacian /lă-plass-ee-ăn/ (Laplace opera-tor) Symbol ∇2. The operator defined inthree dimensions by

∇2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2. The Laplacian occurs in many PARTIAL DIF-FERENTIAL EQUATIONS, including equationsof physical interest such as the LAPLACE

EQUATION and the POISSON EQUATION.

laptop /lap-top/ A type of small light-weight portable computer with a flip-upscreen and powered by rechargeable bat-teries.

laser printer A type of computer printerin which the image is formed by scanning acharged plate with a laser. Powdered inkadheres to the charged areas and is trans-ferred to paper as in a photocopier. Laserprinters can produce text almost as good astypesetting machines.

lateral /lat-ĕ-răl/ Denoting the side of asolid geometrical figure, as opposed to thebase. For instance, a lateral edge of a pyra-mid is one of the edges from the vertex(apex). A lateral face of a pyramid or prismis a face that is not a base. The lateral sur-face (or area) of a cylinder or cone is thecurved surface (or area), excluding theplane base.

latin square An n × n square array of ndifferent symbols with the property thateach symbol appears once and only once ineach row and each column. Such anarrangement is possible for every n. For ex-ample, for n = 4 and the letters A, B, C, D:

A B C DC D A BD C B AB A D C

Such arrays are used in statistics to analyzeexperiments with three factors influencingthe outcome; for example the experi-menter, the method, and the materialunder test. The significance of each factorin the experiment may be tested by using alatin square. For example, denoting thefour methods by A, B, C, D we may use thelatin square

materialA B C D

experimenter C D A BD C B AB A D C

The latin-square array is used since, instudying the effects of one factor, the influ-ences of the other factors occur to the sameextent.

latitude The distance of a point on theEarth’s surface from the equator, measured

Laplacian

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N

S

O

P

all points on thiscircle havelatitude

equator

The latitude θ of a point P on the Earth’s surface.

as the angle in degrees between a planethrough the equator (the equatorial plane)and the line from the point to the center ofthe Earth. A point on the equator has a lat-itude of 0° and the North Pole has a lati-tude of 90° N. See also longitude.

lattice A partially ordered set such thateach pair of elements a and b has both: 1. A greatest lower bound c; i.e. an elementc such that c ≤ a and c ≤ b and if c′ ≤ a andc′ ≤ b then c′ ≤ c.2. A least upper bound d; i.e. an element dsuch that d ≥ a and d ≥ b and if d′ ≥ a andd′ ≥ b then d′ ≥ d.

The elements c and d are called the meetand join respectively of a and b and are de-noted by c = a∩b and d = a∪b. An exampleof a lattice is the set of all subsets of a givenset, where A ≤ B means that each elementof A is also an element of B. In this exam-ple, A∩B is the intersection of the sets Aand B and A∪B is their union.

latus rectum /lay-tŭs rek-tŭm/ (pl. laterarecta) See ellipse; hyperbola; parabola.

law of flotation See flotation; law of.

law of large numbers A theorem inprobability stating that if an event E hasprobability p, and if N(E) represents thenumber of occurrences of the event in n tri-als, then N(E)/n is very close to p if n is a

large number and N(E)/n converges to p asn tends to infinity.

law of moments See moment.

law of the mean (mean value theorem)The rule in differential calculus that, if f(x)is continuous in the interval a ≤ x ≤ b, andthe derivative f′(x) exists everywhere in thisinterval, then there is at least one value ofx (x0) between a and b for which:

[f(b) – f(a)]/(b – a) = f′(x0)Geometrically this means that if a straightline is drawn between two points, (a,f(a))and (b,f(b)), on a continuous curve, thenthere is at least one point between thesewhere the tangent to the curve is parallel tothe line. This law is derived from ROLLE’STHEOREM.

laws of conservation See conservationlaw.

laws of friction See friction.

laws of thought Three laws of logic thatare traditionally considered – as are otherlogic rules – to exemplify something funda-mental about the way we think; that is, it isnot arbitrary that we say certain forms ofreasoning are correct. On the contrary, itwould be impossible to think otherwise. 1. The law of contradiction (law of non-contradiction). It is not the case that some-

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tangent at Cparallel to AB

A

BC ••

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y = f(x )

xa x o b

0

y

The law of the mean for a function f(x) that is continuous between x = a and x = b.

thing can both be true and not be true:symbolically

∼(p ∧ ∼p)2. The law of excluded middle. Somethingmust either be true or not be true: symbol-ically

p ∨ ∼p3. The law of identity. If something is true,then it is true: symbolically

p → p

LCD Lowest common denominator. Seecommon denominator.

LCM Lowest common multiple. See com-mon multiple.

leading diagonal See square matrix.

least common denominator See com-mon denominator.

least common multiple See commonmultiple.

least squares method A method of fit-ting a REGRESSION LINE to a set of data. Ifthe data are points (x1,y1),…(xnyn), thecorresponding points (x1,y1′),…(xn,yn′) arefound using the linear equation y = ax + b.The least squares line minimizes

[(y1 – y1′)2 + (y2 – y2′)2 + … + (yn –yn′)2]

It is found by solving the normal equations∑y = an + b∑x

∑xy = a∑x + b∑x2

for a and b. The technique is extended forregression quadratics, cubics, etc.

least upper bound See bound.

left-handed system See right-handed sys-tem.

Legendre polynomial /lă-zhahn-drĕ/ Aseries of functions that arises as solutionsto Laplace’s equation in spherical polar co-ordinates. They form an infinite series. Thepolynomial is named for the French math-ematician Adrien-Marie Legendre (1752–1833). See also partial differential equa-tion.

Leibniz theorem /lÿb-nits/ A formula forfinding the nth derivative of a product oftwo functions. The nth derivative with re-spect to x of a function f(x) = u(x)v(x),written Dn(uv) = dn(uv)/dxn, is equal to theseries:

uDnv + nC1DuDn–1v + nC2D2uDn–2v +… + nCn–1Dn–1uDv + vDnu

where nCr = n!/[(n – r)!r!].The formula holds for all positive integervalues of n.For n = 1, D(uv) = uDv + vDuFor n = 2, D2(uv) = uD2v + 2DuDv + vD2uFor n = 3, D3(uv) = uD3v + 3DuD2v +3D2uDv + D3uNote the similarity between the differentialcoefficients and the binomial expansioncoefficients. The theorem is named for theGerman mathematician, philosopher, his-torian, and physicist Gottfried WilhelmLeibniz (1646–1716).

lemma /lem-ă/ (pl. lemmas or lemmata) ATHEOREM proved for use in the proof of an-other theorem.

length The distance along a line, plane fig-ure, or solid. In a rectangle, the greater ofthe two dimensions is usually called thelength and the smaller the breadth.

lever A class of MACHINE; a rigid objectable to turn around some point (pivot orfulcrum). The force ratio and the distanceratio depend on the relative positions of thefulcrum, the point where the user exerts theeffort, and the point where the lever appliesforce to the load. There are three types (or-ders) of lever.

First order, in which the fulcrum is be-tween the load and the effort. An exampleis a crowbar.

Second order, in which the load is be-tween the effort and the fulcrum. An ex-ample is a wheelbarrow.

Third order, in which the effort is be-tween the load and the fulcrum. An exam-ple is a pair of sugar tongs.Levers can have high efficiency; the mainenergy losses are by friction at the pivot,and by bending of the lever itself.

LCD

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library programs Collections of com-puter programs that have been purchased,contributed by users, or supplied by thecomputer manufacturers for the use of thecomputing community.

light pen A computer input device thatenables a user to write or draw on thescreen of a VISUAL DISPLAY UNIT (VDU).

light-year Symbol: ly A unit of distanceused in astronomy, defined as the distancethat light travels through space in one year.It is approximately equal to 9.460 5 × 1015

meters.

limit In general, the value approached bya function as the independent variable ap-proaches some value. The idea of a limit isthe basis of the branch of mathematicsknown as analysis. There are several exam-ples of the use of limits. 1. The limit of a function is the value it ap-proaches as the independent variable tendsto some value or to infinity. For instance,the function x/(x + 3) for positive values ofx is less than 1. As x increases it ap-proaches 1 – the value approached as x be-comes infinite. This is written

Lim x/(x + 3) = 1x → ∞

stated as ‘the limit of x/(x + 3) as x ap-proaches (or tends to) infinity is 1’. 1 is thelimiting value of the function.2. The limit of a CONVERGENT SEQUENCE isthe limit of the nth term as n approachesinfinity.3. The limit of a CONVERGENT SERIES is thelimit of the sum of n terms as n approachesinfinity.4. A DERIVATIVE of a function f(x) is thelimit of [f(x + δx) – f(x)]/δx as δx ap-proaches zero.5. A definite INTEGRAL is the limit of a finitesum of terms yδx as δx approaches zero.

limit cycle A closed curve in phase spaceto which a system evolves. A limit cycle isa type of ATTRACTOR characteristic of oscil-lating systems.

limiting friction See friction.

line A join between two points in space oron a surface. A line has length but nobreadth, that is, it has only one dimension.A straight line is the shortest distance be-tween two points on a flat surface.

linear Relating to a straight line. Twovariables x and y have a linear relationshipif it can be represented graphically by astraight line, i.e. by an equation of the formy = mx + c (where m is the slope of the linewhen plotted and c is a constant). Theequation is known as a LINEAR EQUATION.

linear dependence See dependent.

linear equation An EQUATION in whichthe highest power of an unknown variableis one. The general form of a linear equa-tion is

mx + c = 0where m and c are constants. On a Carte-sian coordinate graph

y = mx + cis a straight line that has a gradient m andcrosses the y-axis at y = c. The equation

x + 4y2 = 4is linear in x but not in y. See also equation.

linear extrapolation See extrapolation.

linear independence See dependent.

linear interpolation See interpolation.

linearly ordered A linearly ordered set isa partially ordered set that satisfies the tri-chotomy principle: for any two elements xand y exactly one of x > y, x < y, is true. Forexample, the set of positive integers withtheir natural order is a linearly ordered set.

linear momentum See momentum.

linear momentum, conservation of Seeconstant linear momentum; law of.

linear programming The process offinding maximum or minimum values of alinear function under limiting conditionsor constraints. For example, the function x– 3y might be minimized subject to the con-straints that x + y ≤ 10, x ≤ y – 2, 8 ≥ x ≥ 0

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and y ≥ 0. The constraints can be shown asthe area on a Cartesian coordinate graphbounded by the lines x + y = 10; x = y – 2,x = 8, and y = 0. The minimum value for x– 3y is chosen from points within this area.A series of parallel lines x – 3y = k aredrawn for different values of k. The line k= –9 just reaches the constraint area at thepoint (10,0). Lower values are outside it,and so x = 10, y = 0 gives the minimumvalue of x – 3y within the constraints. Lin-ear programming is used to find the bestpossible combination of two or more vari-able quantities that determine the value ofanother quantity. In most applications, forexample, finding the best combination ofquantities of each product from a factoryto give the maximum profit, there aremany variables and constraints. Linearfunctions with large numbers of variables

and constraints are maximized or mini-mized by computer techniques that aresimilar in principle to this graphical tech-nique for two variables.

linear transformation 1. In one dimen-sion, the general linear transformation isgiven by:

x′ = (ax + b)/(dx + c)where a, b, c, and d are constants. In twodimensions, the general linear transforma-tion is given by:

x′ = (a1x + b1y + c1)/(d1x + e1y + f2)and

y′ = (a2x + b2y + c2)/(d2x + e2y + f2)General linear transformations in morethan two dimensions are defined similarly.2. In an n-dimensional vector space, a lin-ear transformation has the form y = Ax,where x and y are column vectors and A is

linear transformation

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−3

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x = y − 2

x = 8

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y

x

Linear programing: possible values lie in the shaded area. The minimum value is at (8,0).

a matrix. A linear transformation takes ax+ by into ax′ + by′ for all a and b if it takesx and y into x′ and y′.

line integral (contour integral; curvilinearintegral) The integration of a functionalong a particular path, C, which may be asegment of straight line, a portion of spacecurve, or connected segments of severalcurves. The function is integrated with re-spect to the position vector r = ix + jy + kz,which denotes the position of each pointP(x,y,z) on a curve C.

For example, the direction and magni-tude of a force vector F acting on a particlemay depend on the particle’s position in agravitational field or a magnetic field. Thework done by the force in moving the par-ticle over a distance dr is F.dr. The totalwork done in moving the particle along aparticular path from point P1 to point P2 isthe line integral shown in the diagram.

line printer An output device of a com-puter system that prints characters (letters,numbers, punctuation marks, etc.) onpaper a complete line at a time and cantherefore operate very rapidly.

Lissajous figures /lee-sa-zhoo/ Patternsobtained by combining two simple har-monic motions in different directions. Forexample, an object moving in a plane sothat two components of the motion at rightangles are simple harmonic motions, tracesout a Lissajous figure. If the componentshave the same frequency and amplitude

and are in phase the motion is a straightline. If they are out of phase by 90°, it is acircle. Other phase differences produce el-lipses. If the frequencies of the componentsdiffer, more complex patterns are formed.Lissajous figures can be demonstrated withan oscilloscope by deflecting the spot withone oscillating signal along one axis andwith another signal along the other axis.The patterns are named for the Frenchphysicist Jules Antoine Lissajous (1822–80).

liter /lee-ter/ Symbol: l A unit of volumedefined as 10–3 meter3. The name is notrecommended for precise measurements.Formerly, the liter was defined as the vol-ume of one kilogram of pure water at 4°Cand standard pressure. On this definition,1 l = 1000.028 cm3.

literal equation An equation in which thenumbers are, like the unknowns, repre-sented by letters. For example, ax + b = c isa literal equation in x. To solve it, it is nec-essary to find x in terms of a, b, and c, i.e.x = (c – b)/a. Finding the solution of a lit-eral equation can be regarded as changingthe subject of a formula.

load The force generated by a machine.See machine.

local maximum (relative maximum) Avalue of a function f(x) that is greater thanfor the adjacent values of x, but is not the

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force at anypoint P inspace

P2(x2, y2, z2)

z

C

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r0

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y

Fx

F

PP1(x1, y1, z1)

FFFyyy

FFF3

The line integral of a force vector F along a path C from a point P1 to a point P2.

P1

P2

F,dr =x1

x2

Fxdx +y1

y2

Fydy +z1

z2

Fzdz

greatest of all values of x. See maximumpoint.

local meridian See longitude.

local minimum (relative minimum) Avalue of a function f(x) that is less than forthe adjacent values of x but is not the low-est of all values of x. See minimum point.

location See store.

locus of points A set of points, often de-fined by an equation relating coordinates.For example, in rectangular Cartesian co-ordinates, the locus of points on a linethrough the origin at 45° to the x-axis andthe y-axis is defined by the equation x = y;the line is said to be the locus of the equa-tion. A circle is the locus of all points thatlie a fixed distance from a given point.

logarithm /lôg-ă-rith-m/ A number ex-pressed as the exponent of another num-ber. Any number x can be written in theform x = ay. y is then the logarithm to thebase a of x. For example, the logarithm tothe base ten of 100 (log10100) is two, since100 = 102. Logarithms that have a base often are known as common logarithms (orBriggsian logarithms), named for the Eng-lish mathematician Henry Briggs (1561–1630). They are used to carry out multipli-cation and division calculations becausenumbers can be multiplied by adding theirlogarithms. In general p × q can be writtenas ac × ad = a(c+d), p = ac and q = ad. Bothlogarithms and antilogarithms (the inversefunction) are available in the form ofprinted tables, one used for calculations.For example, 4.91 × 5.12 would be calcu-lated as follows: log104.91 is 0.6911 (fromtables) and log105.12 is 0.7093 (from ta-bles). Therefore, 4.91 × 5.12 is given by an-tilog (0.6911 + 0.7093), being 25.14 (fromantilog tables). Similarly, division can becarried out by subtraction of logarithmsand the nth root of a number (x) is the an-tilogarithm of (logx)/n.

For numbers between 0 and 1 the com-mon logarithm is negative. For example,log100.01 = –2. The common logarithm ofany positive real number can be written in

the form n + log10x, where x is between 1and 10 and n is an integer. For example,log1015 = log10(10×1.5) = log1010 +log101.5 = 1 + 0.1761log10150 = log10(100×1.5) = 2.1761log100.15 = log10 (0.1×1.5) = –1 + 0.1761.This is written in the notation 1

_.1761.

The integer part of the logarithm iscalled the characteristic and the decimalfraction is the mantissa. Natural loga-rithms (Napierian logarithms) use the basee = 2.718 28…, logex is often written aslnx.

logarithmic function /lôg-ă-rith-mik/The function logax, where a is a constant.It is defined for positive values of x.

logarithmic scale 1. A line in which thedistance, x, from a reference point is pro-portional to the logarithm of a number.For example, one unit of length along theline might represent 10, two units 100,three units 1000, and so on. In this case thedistance x along the logarithmic scale isgiven by the equation x = log10a. Logarith-mic scales form the basis of the slide rulesince two numbers may be multiplied byadding lengths on a logarithmic scalelog(a × b) = log a + log b.

The graph of the curve y = xn, whenplotted on graph paper with logarithmiccoordinate scales on both axes (log-loggraph paper), is a straight line since logy =nlogx. This method can be used to estab-lish the equation of a non-linear curve.Known values of x and y are plotted onlog-log graph paper and the gradient n ofthe resulting line is measured, enabling theequation to be found. See also log-lineargraph.2. Any scale of measurement that varieslogarithmically with the quantity mea-sured. For instance, pH in chemistry is ameasure of acidity or alkalinity – i.e. of hy-drogen-ion concentration. It is defined aslog10(1/[H+]). An increase in pH from 5 to6 represents a decrease in [H+] from 10–5 to10–6, i.e. a factor of 10. An example of alogarithmic scale in physics is the decibelscale used for noise level.

local meridian

128

logarithmic series The infinite power se-ries that is the expansion of the functionloge(1+x), namely:

x – x2/2 + x3/3 – x4/4 + …This series is convergent for all values of xgreater than –1 and less than or equal to 1.

log graph See logarithmic scale.

logic The study of the methods and prin-ciples used in distinguishing correct orvalid arguments and reasoning from incor-rect or invalid arguments and reasoning.The main concern in logic is not whether aconclusion is in fact accurate, but whetherthe process by which it is derived from a setof initial assumptions (premisses) is cor-rect. Thus, for example, the following formof argument is valid:

all A is Ball B is C

therefore all A is C,and thus the conclusion

all fish have wingscan be derived, correctly, from the pre-misses

all fish are mammalsand

all mammals have wingseven though the premisses and the conclu-sion are untrue. Similarly, true premissesand true conclusions are no guarantee of a valid argument. Therefore, the true con-clusion

all cats are mammalsdoes not follow, logically, from the truepremisses:

all cats are warm-bloodedand

all mammals are warm-bloodedbecause it is an example of the invalid ar-gument form

all A is Ball C is B

therefore, all A is C.

The incorrectness of the argument showsup clearly when, after making reasonablesubstitutions for A, B, and C, we get truepremisses but a false conclusion:

all dogs are mammalsall cats are mammals

therefore all dogs are cats.Such an argument is called a fallacy.

Logic puts forward and examines rulesthat will insure that – given true premisses– a true conclusion can automatically bereached. It is not concerned with examin-ing or assessing the truth of the premisses;it is concerned with the form and structureof arguments, not their content. See deduc-tion; induction; symbolic logic; truth-value; validity.

logic circuit An electronic switching cir-cuit that performs a logical operation, suchas ‘and’ and ‘implies’ on its input signals.There are two possible levels for the inputand output signals, high and low, some-times indicated by the binary digits 1 and0, which can be combined like the values‘true’ and ‘false’ in a truth table. For exam-ple, a circuit with two inputs and one out-put might have a high output only whenthe inputs are different. The output there-fore is the logical function ‘either…or…’ ofthe two inputs (the exclusive disjunction).See truth table.

logic gate An electronic circuit that car-ries out logical operations. Examples ofsuch operations are ‘and’, ‘either – or’,

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logic gate

| | | |

| | | | | | | | | |

–1 0 1 2

0.1 0.2 0.5 1 2 5 10 20 50 100

Logarithmic scale

input 1 input 2 output

high high lowhigh low highlow high highlow low low

Table for a logic circuit

‘not’, ‘neither – nor’, etc. Logic gates oper-ate on high or low input and output volt-ages. Binary logic circuits, those thatswitch between two voltage levels (highand low), are widely used in digital com-puters. The inverter gate or NOT gate sim-ply changes a high input to a low outputand vise versa. In its simplest form, theAND gate has two inputs and one output.The output is high if and only if both in-puts are high. The NAND gate (not and) issimilar, but has the opposite effect; that is,a low output if and only if both inputs arehigh. The OR gate has a high output if oneor more of the inputs are high. The exclu-sive OR gate has a high input only if one ofthe inputs, but not more than one, is high.The NOR gate has a high output only if allthe inputs are low. Logic gates are con-structed using transistors, but in a circuitdiagram they are often shown by symbolsthat denote only their logical functions.These functions are, in effect, those rela-tionships that can hold between proposi-tions in symbolic logic, with combinationsthat can be represented in a TRUTH TABLE.See also conjunction; disjunction; nega-tion.

log-linear graph (semilogarithmic graph)A graph on which one axis has a logarith-mic scale and the other has a linear scale.On a log-linear graph, an exponentialfunction (one of the form y = keax where kand a are constants) is a straight line. Val-

ues of x are plotted on the linear scale andvalues of y on the LOGARITHMIC SCALE.

log–log graph A graph on which bothaxes have LOGARITHMIC SCALES.

long division A method used to divideone number by another number or one al-gebraic expression by another algebraic ex-pression. In both cases it is customary towork from the left. In the case of numbers,the first figure of the number to be dividedis divided by the dividing number. Thisgives an integer (which may be zero) and aremainder. The remainder is added to thenext figure of the number to be divided andthen divided by the dividing number. Thisprocess is repeated until all the figures inthe number to be divided have been dividedby the dividing number, with the answerbeing an integer and a remainder. The cal-culation of numbers using long divisionhas largely been replaced by the use of elec-tronic calculators. Nevertheless, the samesort of procedure is used to divide onealgebraic expression by another algebraicexpression.

longitude The east-west position of apoint on the Earth’s surface measured asthe angle in degrees from a standard merid-ian (taken as the Greenwich meridian). Ameridian is a great circle that passesthrough the North and South poles. Thelocal meridian of a point is a great circle

log-linear graph

130

100 –

10 –

1 –

–––

0.1 1 2 3x

y

In this log-linear graph, the function y = 4.9 e1 5x is shown as a straight line with a gradient of 1.5.

passing through that point and the twopoles. See also latitude.

longitudinal wave A wave motion inwhich the vibrations in the medium are inthe same direction as the direction of en-ergy transfer. Sound waves, transmitted byalternate compression and rarefaction ofthe medium, are an example of longitudi-nal waves. Compare transverse wave.

long multiplication A technique for themultiplication of two numbers or algebraicexpressions. It is customary to write thetwo numbers to be multiplied above eachother. The top number is multiplied by thefigures of the bottom number separately,with a separate line for each product in theappropriate place underneath the initialnumbers, with a final line at the bottomgiving the final product. The calculation ofnumbers by long multiplication has largelybeen replaced by the use of electronic cal-culators. Long multiplication of algebraicexpressions can be performed by a similarprocedure.

loop A sequence of instructions in a com-puter program that is performed either aspecified number of times or is performedrepeatedly until some condition is satisfied.See also branch.

Lorentz–Fitzgerald contraction /lo-rents fits-je-răld/ A reduction in the lengthof a body moving with a speed v relative toan observer, as compared with the lengthof an identical object at rest relative to theobserver. The object is supposed to con-tract by a factor √(1 – v2/c2), c being thespeed of light in free space. The contractionwas postulated to account for the negativeresult of the Michelson–Morley experi-ment using the ideas of classical physics.The idea behind it was that the electro-magnetic forces holding atoms togetherwere modified by motion through theether. The idea was made superfluous(along with the concept of the ether) by thetheory of relativity, which supplied an al-ternative explanation of the Michelson–Morley experiment. The contraction wasnamed for the Dutch theoretical physicistHendrik Antoon Lorentz (1853–1928) andthe Irish physicist George Francis Fitzger-ald (1851–1901), who arrived at the solu-tion independently.

Lorentz force /lo-rents/ The force F ex-erted on an electric charge q that is movingin a magnetic field of field strength B witha velocity v. This force is given by: F = q v× b. The force is perpendicular to both thedirection of the magnetic field and the ve-locity. This means that the Lorentz force

131

Lorentz force

all points onthis circle havelongitude φ

φ

φ

N

zerolongitude

S

P’

P

equator

The longitude φ of a point P on the Earth’s surface.

deflects the electric charge but does notchange its speed. There are many physicalapplications of the Lorentz force.

Lorentz transformation A set of equa-tions that correlate space and time coordi-nates in two frames of reference. See alsorelativity, theory of.

lower bound See bound.

lowest common denominator See com-mon denominator.

lowest common multiple See commonmultiple.

low-level language See program.

lumen /loo-mĕn/ Symbol: lm (pl. lumensor lumina) The SI unit of luminous flux,equal to the luminous flux emitted by apoint source of one candela in a solid angleof one steradian. 1 lm = 1 cd sr.

lune /loon/ A portion of the area of asphere bounded by two great semicirclesthat have common end points.

lux /luks/ Symbol: lx (pl. lux) The SI unitof illumination, equal to the illuminationproduced by a luminous flux of one lumenfalling on a surface of one square meter.1 lx = 1 lm m–2.

Lorentz transformation

132

machine A device for transmitting forceor energy between one place and another.The user applies a force (the effort) to themachine; the machine applies a force (theload) to something. These two forces neednot be the same; in fact the purpose of themachine is often to overcome a large loadwith a small effort. For any machine this

relationship is measured by the force ratio(or mechanical advantage) – the force ap-plied by the machine (load, F2) divided bythe force applied by the user (effort, F1).

The work done by the machine cannotexceed the work done to the machine.Therefore, for a 100% efficient machine:if F2 > F1 then s2 < s1and if F2 <F1 then s2 > s1.Here s2 and s1 are the distances moved byF2 and F1 in a given time.

The relationship between s1 and s2 in agiven case is measured by the distance ratio(or velocity ratio) – the distance moved bythe effort (s1) divided by the distancemoved by the load (s2).

Neither distance ratio nor force ratiohas a unit; neither has a standard symbol.See also hydraulic press; inclined plane;lever; pulley; screw; wheel and axle.

machine code See program.

machine language See program.

Maclaurin series /mă-klor-in/ See Taylorseries.

magic square A square array of numberswhose columns, rows and diagonals add tothe same total. For the magic square:

6 7 21 5 98 3 4

the total is 15.

magnetic disk See disk.

magnetic tape A long strip of flexibleplastic with a magnetic coating on whichinformation can be stored. Its use in therecording and reproduction of sound iswell-known. It is also widely used in com-

133

M

x y

ELE

Lα

L

E

A1

E

A2

L

E L

radius Rradius r

(a)

(b)

(c)

(d)

(e)

Machine: (a) lever; (b) inclined plane; (c) pul-ley; (d) hydraulic press; (e) wheel and axle.

puting to store information. The data isstored on the tape in the form of smallclosely packed magnetic spots arranged inrows across the tape. The spots are magne-tized in one of two directions so that thedata is in binary form. The magnetizationpattern of a row of spots represents a letter,digit (0–9), or some other character.fromthe central processor. It is widely used as abacking store.

magnificent seven problems A set ofseven mathematical problems put forwardby a committee of mathematicians in 2000as the most outstanding problems at thattime. The POINCARÉ CONJECTURE was one ofthe seven problems. It is hoped that at-tempts to solve these problems will stimu-late progress in mathematics in thetwenty-first century just as attempts tosolve HILBERT’S PROBLEMS stimulated pro-gress in mathematics in the twentieth cen-tury. A reward of one million dollars isbeing offered for the solution of each ofthese problems. In order to claim this re-ward the solution has to be published in amathematics journal and be scrutinized fortwo years. At the time of writing, no re-wards have yet been given.

magnitude 1. The absolute value of anumber (without regard to sign).2. The non-directional part of a VECTOR,corresponding to the length of the line rep-resenting it.

mainframe See computer.

main store See store; central processor.

major arc See arc.

major axis See ellipse.

major sector See sector.

major segment The segment of a CIRCLE

divided by a chord that contains the largerpart of the circle.

Mandelbrot set /man-dĕl-brot/ A set ofpoints in the complex plane generated byconsidering the iterations of the form z →

z2 + c, where z and c are complex numbers.This process can produce an infinite num-ber of patterns depending on the value cho-sen for the constant complex number c (seeJulia set). These patterns are of two types:they are either connected, so that there is asingle area within a boundary, or they aredisconnected and broken into distinctparts. The Mandelbrot set is the set of allJulia sets that are connected. It can be rep-resented by points on the complex planeand has a characteristic shape.

It is not necessary to generate Julia setsfor the Mandelbrot set to be produced. Itcan be shown that a given value of c is inthe Mandelbrot set if the starting value z =0 is bounded for the iteration. Thus, if thesequence

c, c2 + c, (c2 + c)2 + c, …remains bounded, then the point c is in theMandelbrot set.

The set has been the subject of much in-terest. The figure is a FRACTAL and can beexamined on the computer screen in high‘magnification’ (i.e. by calculating the setover a short range of values of c). It showsan amazingly complex self-similar struc-ture characterized by the presence ofsmaller and smaller copies of the set at in-creasing levels of detail. The set is namedfor the American mathematician BenoitMandelbrot (1924– ).

manifold A space that is locally Euclideanbut is not necessarily Euclidean globally.Examples of manifolds that are not Euclid-ean globally include the circle and the sur-face of a sphere. It is convenient to discussmany aspects of geometry and topology interms of manifolds.

mantissa /man-tiss-ă/ See logarithm.

many-valued function A function forwhich one value of x gives more than onevalue of y. If one value of x gives n valuesof y then the function is said to be an n-valued function of x. If n = 2 then the func-tion is called a two-valued function of x.For example, y2 = x is a two-valued func-tion of x since every real value of x givestwo values of y.

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134

map See function.

mapping See function.

market price See nominal value; yield.

Markov chain /mar-koff/ A sequence ofdiscrete random events or variables thathave probabilities depending on previousevents in the chain. The sequence is namedfor the Russian mathematician Andrey An-dreyevich Markov (1856–1922).

mass Symbol: m A measure of the quan-tity of matter in an object. The SI unit ofmass is the kilogram. Mass is determined intwo ways: the inertial mass of a body de-termines its tendency to resist change inmotion; the GRAVITATIONAL MASS deter-mines its gravitational attraction for othermasses. See also inertial mass; weight.

mass, center of See center of mass.

mass-energy equation The equation E =mc2, where E is the total energy (rest mass

energy + kinetic energy + potential energy)of a mass m, c being the speed of light infree space. The equation is a consequenceof Einstein’s special theory of relativity. Itis a quantitative expression of the idea thatmass is a form of energy and energy alsohas mass. Conversion of rest-mass energyinto kinetic energy is the source of power inradioactive substances and the basis of nu-clear-power generation.

material implication See implication.

mathematical induction See induction.

mathematical logic See symbolic logic.

matrix /may-triks, mat-riks/ (pl. matricesor matrixes) A set of quantities arranged inrows and columns to form a rectangulararray. The common notation is to enclosethese in parentheses. Matrices do not havea numerical value, like DETERMINANTS.They are used to represent relations be-tween the quantities. For example, a planevector can be represented by a single col-

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matrix

Mandelbrot set: the shape of the set with a detail showing the fine structure (top left).

umn matrix with two numbers, a 2 × 1 ma-trix, in which the upper number representsits component parallel to the x-axis and thelower number represents the componentparallel to the y-axis. Matrices can also beused to represent, and solve, simultaneousequations. In general, an m × n matrix –one with m rows and n columns – is writ-ten with the first row:

a11a12 … a1nThe second row is:

a21a22 … a2nand so on, the mth row being:

am1am2 … amnThe individual quantities a11, a21, etc., arecalled elements of the matrix. The numberof rows and columns, m × n, is the order ordimensions of the matrix. Two matricesare equal only if they are of the same orderand if all their corresponding elements areequal. Matrices, like numbers, can beadded, subtracted, multiplied, and treatedalgebraically according to certain rules.However, the commutative, associative,and distributive laws of ordinary arith-metic do not apply. Matrix addition con-sists of adding corresponding elementstogether to obtain another matrix of thesame order. Only matrices of the sameorder can be added. Similarly, the result ofsubtracting two matrices is the matrixformed by the differences between corre-sponding elements.

Matrix multiplication also has certainrules. In multiplication of an m × n matrix

by a number or constant k, the result is an-other m × n matrix. If the element in the ithrow and jth column is aij then the corre-sponding element in the product is kaij.This operation is distributive over matrixaddition and subtraction, that is, for twomatrices A and B,

k(A + B) = kA + kBAlso, kA = Ak, as for multiplication ofnumbers. In the multiplication of two ma-trices, the matrices A and B can only bemultiplied together to form the product ABif the number of columns in A is the sameas the number of rows in B. In this casethey are called conformable matrices. If Ais an m × p matrix with elements aij and Bis a p × n matrix with elements bij, thentheir product AB = C is an m × n matrixwith elements cij, such that cij is the sum ofthe products

aijbij + ai2b2j + ai3b3j + … + aipbpjMatrix multiplication is not commutative,that is, AB ≠ BA.

See also square matrix.

matrix of coefficients An m × n matrix,with general entry aij associated with thecoefficients in the set of m linear equationsfor n unknown variables x1, x2, ... xn:

a11x1 + a12x2 + ... a1nxn = b1,a21x1 + a22x2 + ... a2nxn = b2,

am1x1 + am2x2 + ... amnxn = bm,Expressing a set of linear equations interms of the matrix of coefficients can be astep towards finding the set of solutions of

matrix of coefficients

136

1 2 34 5 6

2 6 1 04 8 1 2

3 8 1 38 13 18

1 2 34 5 6

3 6 91 2 1 5 1 8

1 2 34 5 6

6 7 8 91 0 1 1

(6 + 16 + 30) (7 + 18 + 33)

(24 + 40 + 60) (28 + 45 + 66)

Matrix addition.

Multiplication of a matrix by a number.

Matrix multiplication

+ =

3 x =

X =

Matrix: addition and multiplication of matrices.

the equations. The matrix formed by in-cluding the constants b1, b2, etc., is theaugmented matrix.

maximum likelihood A method of esti-mating the most likely value of a parame-ter. If a series of observations x1,x2,…,xnare made, the likelihood function, L(x), isthe joint probability of observing these val-ues. The likelihood function is maximizedwhen [dlogL(x)]/dp = 0. In many cases, anintuitive estimate, such as the mean, is alsothe maximum likelihood estimate.

maximum point A point on the graph ofa function at which it has the highest valuewithin an interval. If the function is asmooth continuous curve, the maximum isa TURNING POINT, that is, the slope of thetangent to the curve changes continuouslyfrom positive to negative by passingthrough zero. If there is a higher value ofthe function outside the immediate neigh-borhood of the maximum, it is a local max-imum (or relative maximum). If it is higherthan all other values of the function it is anabsolute maximum. See also stationarypoint.

mean A representative or expected valuefor a set of numbers. The arithmetic meanor average (called mean) of x1, x2, …, xn isgiven by:

(x1 + x2 + x3 + … + xn)/nIf x1, x2, …, xk occur with frequencies f1,f2, …, fk then the arithmetic mean is

(f1x1 + f2x2 + … + fkxk)/(f1 + f2 + … + fk)

When data is classified, as for example in afrequency table, x1 is replaced by the classmark.

The weighted mean W =(w1x1 + w2x2 + … + wnxn)/

(w1 + w2 + … + wn)where weight wi is associated with xi.

The harmonic mean H =n/[(1/x1) + (1/x2) + … + (1/xn)]

The geometric mean G =(x1.x2 … xn)1/n

The mean of a random variable is its ex-pected value.

mean center See centroid.

mean deviation A measure of the disper-sion of a set of numbers. It is equal to theaverage, for a set of numbers, of the differ-ences between each number and the set’smean value. If x is a random variable withmean value µ, then the mean deviation isthe average, or expected value, of |x – µ|,written

i∑|xi – µ|/n

mean value theorem See law of themean.

measurements of central tendency Thegeneral name given for the types of averageused in statistics, i.e. the MEAN, the MEDIAN,and the MODE.

measure theory A branch of mathemat-ics concerned with constructing the theoryof integration in a mathematically rigorousway, starting from set theory.

137

measure theory

|||

|||

| |

||||

CA’

B

C’

A

B’P

Median: the three medians intersect at P, the centroid of the triangle.

mechanical advantage See force ratio;machine.

mechanics The study of forces and theireffect on objects. If the forces on an objector in a system cause no change of momen-tum the object or system is in equilibrium.The study of such cases is statics. If theforces acting do change momentum thestudy is of dynamics. The ideas of dynam-ics relate the forces to the momentumchanges produced. Kinematics is the studyof motion without consideration of itscause.

median /mee-dee-ăn/ 1. The middle num-ber of a set of numbers arranged in order.When there is an even number of numbersthe median is the average of the middletwo. For example, the median of1,3,5,11,11 is 5, and of 1,3,5,11,11,14 is(5 + 11)/2 = 8. The median of a large pop-ulation is the 50th percentile (P50). Com-pare mean. See also percentile; quartile.2. In geometry, a straight line joining thevertex of a triangle to the mid-point of theopposite side. The medians of a triangle in-tersect at a single point, which is the cen-troid of the triangle.

mediator A line that bisects another lineat right angles.

mega- Symbol: M A prefix denoting 106.For example, 1 megahertz (MHz) = 106

hertz (Hz).

member See element.

memory See store.

mensuration /men-shŭ-ray-shŏn/ Thestudy of measurements, especially of thedimensions of geometric figures in order tocalculate their areas and volumes.

Mercator’s projection /mer-kay-terz/ Amethod of mapping points from the sur-face of a sphere onto a plane surface. Mer-cator’s projection is used to make maps ofthe World. The lines of longitude on thesphere become straight vertical lines on theplane. The lines of latitude on the sphere

become straight horizontal lines. Areas fur-ther from the equator are more stretchedout in the horizontal direction. For a par-ticular point on the surface of the sphere atangle θ latitude and angle φ longitude, thecorresponding Cartesian coordinates onthe map are:

x = kθy = k log tan(φ/2)

Mercator’s projection is an example of aconformal mapping, in which the anglesbetween lines are preserved (except at thepoles). This method of mapping is namedfor the Dutch cartographer and geographerGerardus Mercator (1512–94).See also projection.

meridian /mĕ-rid-ee-ăn/ See longitude.

Mersenne prime /mair-sen/ a primenumber that can be written as 2p–1, wherep is a prime number. Mersenne primes arediscovered using computers, with the num-ber being discovered increasing as thepower of computers increases. For exam-ple, the Mersenne prime 219937–1 was dis-covered in this way. Mersenne primes arerelated to even PERFECT NUMBERS since theonly even perfect numbers are numbers ofthe form 2(p–1)(2p–1). The prime is namedfor the French mathematician MarinMersenne (1588–1648).

meter /mee-ter/ Symbol: m The SI baseunit of length, defined as the distance trav-eled by light in a vacuum in 1/299 792 458of a second.

metric /met-rik/ An expression for thedistance between two points in some geo-metrical space that is not necessarily Eu-clidean. The expressions for the metric innon-Euclidean geometrics are generaliza-tions of the distance between two points inEuclidean geometry which is given by ap-plying PYTHAGORAS’ THEOREM to Cartesiancoordinates. The concept of the metric isvery important in the geometries that de-scribe flat space–time in special relativitytheory and curved space–time in generalrelativity theory.

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138

metric prefix Any of various numericalprefixes used in the metric system.

metric space Any set of points, such as aplane or a volume in geometrical space, inwhich a pair of points a and b with a dis-tance d(a,b) between them, satisfy the con-ditions that d(a,b) ≥ 0 and d(a,b) = 0 if andonly if a and b are the same point. Anotherproperty of a metric space is that d(a,b) +d(b,c) ≥ d(a,c). The set of all the functionsof x that are continuous in the interval x =a to x = b is also a metric space. If f(x) andg(x) are in the space,

a

b

[f(x) – g(x)]dx

is defined for all values of x between a andb, and the integral is zero if and only if f(x)= g(x) for all values of x between a and b.

metric system A system of units based onthe meter and the kilogram and using mul-tiples and submultiples of 10. SI units,c.g.s. units, and m.k.s. units are all scien-tific metric systems of units.

metric ton See tonne.

metrology /mĕ-trol-ŏ-jee/ The study ofunits of measurements and the methods ofmaking precise measurements. Every phys-ical quantity that can be quantified is ex-pressed by a relationship of the type q = nu,where q is the physical quantity, n is anumber, and u is a unit of measurement.One of the prime concerns of metrologistsis to select and define units for all physicalquantities.

m.g.f. See moment generating function.

Michelson–Morley experiment /mÿ-kĕl-sŏn mor-lee/ A famous experiment con-ducted in 1887 in an attempt to detect theether, the medium that was supposed to benecessary for the transmission of electro-magnetic waves in free space. In the exper-iment, two light beams were combined toproduce interference patterns after travel-ing for short equal distances perpendicularto each other. The apparatus was thenturned through 90° and the two interfer-ence patterns were compared to see if therehad been a shift of the fringes. If light has avelocity relative to the ether and there is anether ‘wind’ as the Earth moves throughspace, then the times of travel of the twobeams would change, resulting in a fringeshift. No such shift was detected, not evenwhen the experiment was repeated sixmonths later when the ether wind wouldhave reversed direction. The experiment isnamed for its instigators, the Americanphysicist Albert Abraham Michelson(1852–1931) and the American chemistand physicist Edward William Morley(1838–1923). See also relativity; theory of.

micro- Symbol: µ A prefix denoting 10–6.For example, 1 micrometer (µm) = 10–6

meter (m).

microcomputer /mÿ-kroh-kŏm-pyoo-ter/See computer.

micron /mÿ-kron/ (micrometer) Symbol:µm A unit of length equal to 10–6 meter.

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micron

METRIC PREFIXESPrefix Symbol Multiple Prefix Symbol Multiple atto- a × 10–18 deca- da × 10 femto f × 10–15 hecto- h × 102

pico- p × 10–12 kilo- k × 103

nano- n × 10–9 mega- M × 106

micro- µ × 10–6 giga- G × 109

milli- m × 10–3 tera- T × 1012

centi- c × 10–2 peta- P × 1015

deci- d × 10–1 exa- E × 1018

microprocessor /mÿ-kroh-pross-ess-er/See central processor.

mid-point theorem A result in geometrythat states that the line that joins the mid-points of two sides of a triangle has half thelength of the third side of the triangle andis parallel to the third side. This result canbe proved very easily using vectors in thefollowing way.Denote AB by b and AC by c. The directedline BC is given by BC = BA + AC = –b + c= c–b. One also has AM = (1/2) AB = (1/2)b and AN = (1/2)AC = (1/2) c. Thus, MN =MA + AN = –(1/2) b + (1/2) c = (1/2) c–(1/2) b = (1/2) (c – b). The last result canbe written as MN = (1/2)BC. This meansthat MN has half the magnitude of BC andis parallel to it.

mil 1. A unit of length equal to one thou-sandth of an inch. It is commonly called a‘thou’ and is equivalent to 2.54 × 10–5 m.2. A unit of area, usually called a circularmil, equal to the area of a circle having adiameter of 1 mil.

mile A unit of length equal to 1760 yards.It is equivalent to 1.6093 km.

milli- Symbol: m A prefix denoting 10–3.For example, 1 millimeter (mm) = 10–3

meter (m).

million A number equal to 1 000 000(106).

minicomputer /min-ee-kŏm-pyoo-ter/ Seecomputer.

minimum point A point on the graph ofa function at which it has the lowest valuewithin an interval. If the function is asmooth continuous curve, the minimum isa turning point, that is, the slope of the tan-gent to the curve changes continuouslyfrom negative to positive by passingthrough zero. If there is a lower value ofthe function outside the immediate neigh-borhood of the minimum, it is a local min-imum (or relative minimum). If it is lowerthan all other values of the function it is anabsolute minimum. See also stationarypoint; turning point.

minor arc See arc.

minor axis See ellipse.

minor sector See sector.

minor segment The segment of a CIRCLE

divided by a chord that contains thesmaller part of the circle.

minuend /min-yoo-end/ The term fromwhich another term is subtracted in a dif-ference. In 5 – 4 = 1, 5 is the minuend (4 isthe subtrahend).

minute (of arc) A unit of plane angleequal to one sixtieth of a degree.

mirror line A line that is the axis of sym-metry for reflection symmetry. If part of anobject is in contact with the mirror linethen the image of the object is also in con-tact with the mirror line. If part of an ob-ject is a certain distance from the mirrorline then the image of that part of the ob-ject is the same distance from the mirrorline. If the mirror line passes through anobject then it also passes through the imageof the object.

mixed number See fraction.

m.k.s. system A system of units based onthe meter, the kilogram, and the second. Itformed the basis for SI units.

mmHg (millimeter of mercury) A formerunit of pressure defined as the pressure that

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A

CB

M N

Midpoint theorem.

will support a column of mercury one mil-limeter high under specified conditions. Itis equal to 133.322 4 Pa. It is also calledthe torr.

Möbius strip /moh-bee-ŭs/ (Möbius band)A continuous flat loop with one twist. It isformed by taking a flat rectangular strip,twisting it in the middle so that each endturns through 180° with respect to theother, and then joining the ends together.Because of the twist, a continuous line canbe traced along the surface between anytwo points, without crossing an edge. Theunique topological property of the Möbiusstrip is that it has one surface and one edge.If a Möbius strip is cut along a line parallelto the edge it is transformed into a doublytwisted band that has two edges and twosides. It is named for the German mathe-matician August Ferdinand Möbius(1790–1868). See also topology.

modal class /moh-d’l/ The CLASS that oc-curs with the greatest frequency, for exam-ple in a frequency table. See also mode.

mode The number that occurs most fre-quently in a set of numbers. For example,the mode (modal value) of 5, 6, 2, 3, 2, 1,2 is 2. If a continuous random variable hasprobability density function f(x), the modeis the value of x for which f(x) is a maxi-mum. If such a variable has a frequencycurve that is approximately symmetricaland has only one mode, then

(mean – mode) = 3(mean – median).

modem /moh-dem/ A device for sendingcomputer data over long distances usingtelephone lines. It is short formodulator/demodulator.

modulation /moj-ŭ-lay-shŏn/ A processin which the characteristics of one wave arealtered by some other wave. Modulation isused extensively in radio transmission. Thewave which is altered is called the carrierwave and the wave responsible for thechange is called the modulating wave. In-formation is transmitted in this way.

There are several ways in which modu-lation can be performed. In amplitudemodulation the amplitude of the carrierwave rises and falls as the amplitude of themodulating wave rises and falls.

If the carrier wave is regarded as a sinewave then amplitude modulation alters theamplitude of the sine wave but does notchange the angular aspect of the carrierwave, either as regards phase angle or fre-quency. Modulation which involveschange of the angle is known as angle mod-ulation. There are two types of angle mod-ulation: frequency modulation and phasemodulation. In frequency modulation thefrequency of the carrier wave rises and fallsas the amplitude of the modulating waverises and falls. In phase modulation the rel-ative phase of the carrier wave changes byan amount proportional to the amplitudeof the modulating wave. In angle modula-tion the amplitude of the carrier wave isnot altered.

There exist several other types of mod-ulation, including some which make use ofpulses (pulse modulation).

modulus /moj-ŭ-lŭs/ (pl. moduli) The ab-solute value of a quantity, not consideringits sign or direction. For example, the mod-ulus of minus five, written |–5|, is 5. Themodulus of a vector quantity correspondsto the length or magnitude of the vector.The modulus of a COMPLEX NUMBER x + iy

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Möbius strip, which has one surface and one edge.

is √(x2 + y2). If the number is written in theform r(cosθ + isinθ), the modulus is r. Seealso argument.

mole Symbol: mol The SI base unit ofamount of substance, defined as theamount of substance that contains as manyelementary entities as there are atoms in0.012 kilogram of carbon-12. The elemen-tary entities may be atoms, molecules, ions,electrons, photons, etc., and they must bespecified. One mole contains 6.022 52 ×1023 entities. One mole of an element withrelative atomic mass A has a mass of Agrams (this was formerly called one gram-atom). One mole of a compound with rel-ative molecular mass M has a mass of Mgrams (this was formerly called one gram-molecule).

moment (of a force) The turning effectproduced by a force about a point. If thepoint lies on the line of action of the forcethe moment of the force is zero. Otherwiseit is the product of the force and the per-pendicular distance from the point to theline of action of the force. If a number offorces are acting on a body, the resultantmoment is the algebraic sum of all the in-dividual moments. For a body in equilib-rium, the sum of the clockwise moments isequal to the sum of the anticlockwise mo-ments (this law is sometimes called the lawof moments). See also couple; torque.

moment generating function (m.g.f.) Afunction that is used to calculate the statis-tical properties of a random variable, x. Itis defined in terms of a second variable, t,such that the m.g.f., M(t), is the expecta-tion value of etx, E(etx). For a discrete ran-dom variable

M(t) = ∑etxpand for a continuous random variable

M(t) = ∫etxf(x)dxTwo distributions are the same if theirm.g.f.s are the same. The mean and vari-ance of a distribution can be found by dif-ferentiating the m.g.f. The mean E(x) =M′(0) and the variance, Var(x) = M″(0) –(M′(0))2.

moment of area For a given surface, themoment of area is the moment of mass thatthe surface would have if it had unit massper unit area.

moment of inertia Symbol: I The rota-tional analog of mass. The moment of in-ertia of an object rotating about an axis isgiven by

I = mr2

where m is the mass of an element a dis-tance r from the axis. See also radius of gy-ration; theorem of parallel axes.

moment of mass The moment of mass ofa point mass about a point, line, or plane isthe product of the mass and the distancefrom the point or of the mass and the per-pendicular distance from the line or plane.For a system of point masses, the momentof mass is the sum of the mass-distanceproducts for the individual masses. For anobject the integral must be used over thevolume of the object.

momentum, conservation of /mŏ-men-tŭm/ See constant linear momentum; lawof.

momentum, linear Symbol: p The prod-uct of an object’s mass and its velocity: p =mv. The object’s momentum cannotchange unless a net outside force acts. Thisrelates to Newton’s laws and to the defini-tion of force. It also relates to the principleof constant momentum. See also angularmomentum.

monotonic /mon-ŏ-tonn-ik/ Alwayschanging in the same direction. A monoto-nic increasing function of a variable x in-creases or stays constant as x increases, butnever decreases. A monotonic decreasingfunction of x decreases or stays constant asx increases, but never increases. Each termin a monotonic series is either greater thanor equal to the one before it (monotonic in-creasing) or less than or equal to the onebefore it (monotonic decreasing). Comparealternating series.

mouse A computer INPUT DEVICE that isheld under the palm of the hand and

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moved on a flat surface to control themovements of a cursor (pointer) on thecomputer screen. Instructions can be sentto the computer by pressing (‘clicking on’)one or more buttons on the mouse.

multiple A number or expression that hasa given number or expression as a factor.For example, 26 is a multiple of 13.

multiple integral See iterated integral.

multiple point A point on the curve of afunction at which several arcs intersect, orwhich forms an isolated point, and where asimple derivative of the function does notexist. If the equation of the curve is writtenin the form:

(a1x + b1y) + (a2x2 + b2xy + c2y2)+ (a3x3 + …) + … = 0

in which the multiple point is at the originof a Cartesian-coordinate system, the val-ues of the coefficients of x and y indicatethe type of multiple point. If a1 and b1 arezero, that is, if all the first degree terms arezero, then the origin is a singular point. Ifthe terms a2, b2, and c2 are also zero it is adouble point. If, in addition, the terms a3,b3, etc., of the third degree terms are zero,it is a triple point, and so on. See also dou-ble point; isolated point.

multiplicand /mul-ti-plă-kand/ A numberor term that is multiplied by another (themultiplier) in a multiplication.

multiplication Symbol: × The operationof finding the product of two or morequantities. In arithmetic, multiplication ofone number, a, by another, b, consists ofadding a to itself b times. This kind of mul-tiplication is commutative, that is, a × b =b × a. The identity element for arithmeticmultiplication is 1, i.e. multiplication by 1produces no change. In a series of multipli-cations, the order in which they are carriedout does not change the result. For exam-

ple, 2 × (4 × 5) = (2 × 4) × 5. This is the as-sociative law for arithmetic multiplication.

Multiplication of vector quantities andmatrices do not follow the same rules.

multiplication of complex numbersThe rules of multiplication which emergefrom the definition of a complex number.If z1 = a + ib and z2 = c + id, then the prod-uct z1z2 is given by: z1z2 = (ac – bd) + i(ad+ bc).

The product of two complex numberscan be written neatly in terms of their polarforms:z1 = r1 exp (iθ1) = r1 (cosθ1 + i sinθ1) andz2 = r2 exp (iθ2) = r2 (cosθ2 + i sinθ2).This givesz1z2 = r1r2 exp [i(θ1 + θ2)] = r1r2 [cos (θ1 +θ2) + i sin (θ1 + θ2)]. In addition to beingvery convenient for many purposes, the ex-pression for the product of two complexnumbers in terms of polar forms has a sim-ple geometrical interpretation that themodulus of the product is the product ofthe moduli of the two complex numbersbeing multiplied together and the argu-ment of the product is the sum of the argu-ments of the two complex numbers.

multiplication of fractions See frac-tions.

multiplication of matrices See matrix.

multiplication of vectors See scalarproduct; vector product.

multiplier See multiplicand.

mutually exclusive events Two eventsthat cannot occur together. If the twoevents are denoted by M1 and M2 respec-tively then the two events being mutuallyexclusive is denoted by M1 ∩ M2 = ∅ (thenull event).

myria- Symbol: my A prefix used inFrance to denote 104.

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myria-

NAND gate See logic gate.

nano- Symbol: n A prefix denoting 10–9.For example, 1 nanometer (nm) = 10–9

meter (m).

Napierian logarithm /nă-peer-ee-ăn/ Seelogarithm.

Napier’s formulae /nay-pee-erz/ A set ofequations used in spherical trigonometryto calculate the sides and angles in a spher-ical triangle. In a spherical triangle withsides a, b, and c, and angles opposite theseof α, β, and γ respectively:

sin½(a – b)/sin½(a + b) =tan½(α – β)/tan½γ

cos½(a – b)/cos½(a + b) =tan½(α + β)/tan½γ

sin½(α – β)/sin½(α + β) =tan½(a – b)/cot½c

cos½(α – β)/cos½(α + β) =tan½(a + b)/cot½c

The formulae are named for the Scot-tish mathematician John Napier (1550–1617). See also spherical triangle.

nappe /nap/ One of the two parts of a con-ical surface that lie either side of the vertex.See cone.

natural frequency The frequency atwhich an object or system will vibratefreely. A free vibration occurs when there isno external periodic force and little resis-tance. The amplitude of free vibrationsmust not be too great. For instance, a pen-dulum swinging with small swings underits own weight moves at its natural fre-quency. Normally, an object’s natural fre-quency is its fundamental frequency.

natural logarithm See logarithm.

natural numbers Symbol: N The set ofnumbers 1,2,3, … used for counting sep-arate objects.

nautical mile A unit of length equal to6080 feet (about 1.852 km), equivalent to1 minute of arc along a great circle on theEarth’s surface. A speed of 1 nautical mileper hour is a knot.

necessary condition See condition.

negation Symbol: ∼ or ¬ In logic, the op-eration of putting not or it is not the casethat in front of a proposition or statement,thus reversing its truth value. The negationof a proposition p is false if p is true andvice versa. The truth-table definition fornegation is shown in the illustration. Seealso truth table.

negative Denoting a number of quantitythat is less than zero. Negative numbers arealso used to denote quantities that arebelow some specified reference point. Forexample, in the Celsius temperature scale atemperature of –24°C is 24° below thefreezing point of water. Compare positive.

negative binomial distribution See Pas-cal’s distribution.

neighborhood See topology.

nested intervals A sequence of intervals

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N

P ~P F T T F

Negation

such that each interval contains the previ-ous one. The nested interval theorem statesthat for any sequence of bounded andclosed nested intervals there is at least onepoint that belongs to all the intervals. If thelengths of the intervals tend to zero as onegoes through the sequence then there is ex-actly one such point.

nesting The embedding of a computersubroutine or a loop of instructions withinanother subroutine or loop, which in turnmay lie within yet another, and so on.

net 1. Denoting a weight of goods exclud-ing the weight of the containers or packing.2. Denoting a profit calculated after de-ducting all overhead costs, expenses, andtaxes. Compare gross.3. A surface that can be folded to form asolid.4. A network.

network A graph consisting of VERTICES

that are joined by arcs (directed edges),with each arc having an arrow on it to in-dicate its direction, and each arc is associ-ated with a non-negative number called itsweight. There are many physical applica-tions of networks including electrical cir-cuits and representations of streets intowns, with the physical significance of theweight depending on the physical problem.For example, in some applications there isphysical flow (transport) of something be-tween the vertices, such as electrical cur-rent in an electrical circuit or cars movingalong a street, with the weight being the ca-pacity of the arc. Another type of exampleis the one in which a network representssome process, with the vertices represent-ing the steps in the process and the weightof an arc joining two vertices represents thetime between the two steps.

neutral equilibrium Equilibrium suchthat if the system is disturbed a little, thereis no tendency for it to move further nor toreturn. See stability.

newton Symbol: N The SI unit of force,equal to the force needed to accelerate onekilogram by one meter per second. 1 N = 1

kg m s–2. The unit is named for the Englishphysicist and mathematician Sir IsaacNewton (1642–1727).

Newtonian mechanics /new-toh-nee-ăn/Mechanics based on Newton’s laws of mo-tion; i.e. relativistic or quantum effects arenot taken into account.

Newton’s law of universal gravitationThe force of gravitational attraction be-tween two point masses (m1 and m2) isproportional to each mass and inverselyproportional to the square of the distance(r) between them. The law is often given inthe form

F = Gm1m2/r2

where G is a constant of proportionalitycalled the gravitational constant. The lawcan also be applied to bodies; for example,spherical objects can be assumed to havetheir mass acting at their center. See alsorelativity; theory of.

Newton’s laws of motion Three laws ofmechanics formulated by Sir Isaac Newtonin 1687. They can be stated as:1. An object continues in a state of rest orconstant velocity unless acted on by an ex-ternal force.2. The resultant force acting on an object is proportional to the rate of change of mo-mentum of the object, the change of mo-mentum being in the same direction as theforce.3. If one object exerts a force on anotherthen there is an equal and opposite force(REACTION) on the first object exerted bythe second.

The first law was discovered by Galileo,and is both a description of inertia and adefinition of zero force. The second lawprovides a definition of force based on theinertial property of mass. The third law isequivalent to the law of conservation oflinear momentum.

Newton’s method A technique for ob-taining successive approximations (itera-tions) to the solution of an equation, eachmore accurate than the preceding one. Theequation in a variable x is written in the

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form f(x) = 0, and the general formula oralgorithm:

xn+1 = xn – f(xn)/f′(xn)is applied, where xn is the nth approxima-tion. Newton’s method can be thought ofas repeated estimates of the position on agraph of f(x) against x at which the curvecrosses the x-axis, by extrapolation of thetangent to the curve. The slope of the tan-gent at (x1,f(x1)) is df/dx at x = x1, that is

f′(x1) = f(x1)/(x2 – x1)x2 = x1 – f(x1)/f′(x1) is therefore the pointwhere the tangent crosses the x-axis, and isa closer approximation to x at f(x) = 0 thanx1 is. Similarly,

x3 = x2 – f(x2)/f′(x2)is a better approximation still. For exam-ple, if f(x) = x2 – 3 = 0, then f′(x) = 2x andwe obtain the algorithm

xn+1 = xn – (x2n – 3)/2xn = ½(xn + 3/xn)

See also iteration.

node A point of minimum vibration in astationary wave pattern, as near the closedend of a resonating pipe. Compare antin-ode. See also stationary wave.

Noether’s theorem /noh-terz/ A funda-mental result in physics which relates sym-metry to conservation laws. It states thatfor each continuous symmetry underwhich a physical system is invariant there isa conserved quantity. For example, invari-ance of a system under rotational symme-try is associated with the conservation ofangular momentum. It does not follow thatif a certain symmetry and conservation lawassociated with Noether’s theorem existsin a system described by classical physicsthen the symmetry and conservation lawnecessarily have to exist in the correspond-ing system described by QUANTUM ME-CHANICS.

nominal value (per value) The valuegiven to a stock or share by the governmentor corporation that offers it for sale. Stockshave a nominal value of $100. Shares,however, may have any nominal value. Forexample, a corporation wishing to raise$100 000 by an issue of shares may issue100 000 $1 shares or 200 000 50¢ shares,or any other combination. The issue price,

i.e. the price paid by the first buyers of theshares, may not be the same as the nominalvalue, although it is likely to be close to it.A share with a nominal value of 50¢ maybe offered at an issue price of 55¢; it is thensaid to be offered at a premium of 5¢. If of-fered at an issue price of 45¢ it is said to beoffered at a discount of 5¢. Once estab-lished as a marketable share on a stock ex-change, the nominal value has littleimportance and it is the market price atwhich it is bought and sold. However, thedividend is always expressed as a percent-age of the nominal value.

nomogram /nom-ŏ-gram/ A graph thatconsists of three parallel lines, each one ascale for one of three related variables. Astraight line drawn between two points,representing known values of two of thevariables, crosses the third line at the cor-responding value of the third variable. Forexample, the lines might show the temper-ature, volume, and pressure of a knownmass of gas. If the volume and pressure areknown, the temperature can be read off thenomogram.

nonagon /non-ă-gon/ A plane figure withnine straight sides. A regular nonagon hasnine equal sides and nine equal angles.

non-Cartesian coordinates Coordinateswhich are not Cartesian coordinates.POLAR COORDINATES and SPHERICAL POLAR

COORDINATES are examples of non-Carte-sian coordinates.

In many physical problems which arespecified in terms of a partial differentialequation it is frequently the case that theequation can only be solved exactly using aparticular set of non-Cartesian coordi-nates, with the coordinate system beingused depending on the geometry or sym-metry of the problem. For example, spher-ical polar coordinates are convenient forproblems with spherical symmetry.

non-contradiction, law of See laws ofthought.

non-Euclidean geometry Any system ofgeometry in which the parallel postulate of

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Euclid does not hold. This postulate can bestated in the form that, if a point lies out-side a straight line, only one line parallel tothe straight line can be drawn through thepoint. In the early nineteenth century it wasshown that it is possible to have a wholeself-consistent formal system of geometrywithout using the parallel postulate at all.There are two types of non-Euclideangeometry. In one (called elliptic geometry)there are no parallel lines through thepoint. An example of this is a system de-scribing the properties of lines, figures, an-gles, etc., on the surface of a sphere inwhich all lines are parts of great circles (i.e.circles that have the same center as the cen-ter of the sphere). Since all great circles in-tersect, no parallel can be drawn throughthe point. Note also that the angles of a tri-angle on such a sphere do not add up to180°. The other type of non-Euclideangeometry is called a hyperbolic geometry –here an infinite number of parallels can bedrawn through the point.

Note that a type of geometry is not in it-self based on ‘experiment’ – i.e. of mea-surements of distance, angles, etc. It is apurely abstract system based on certain as-sumptions (such as Euclid’s axioms).Mathematicians study such systems fortheir own sake – without necessarily look-ing for practical applications. The practicalapplications come in when a particularmathematical system gives an accurate de-scription of physical properties – i.e. theproperties of the ‘real world’. In practicaluses (in architecture, surveying, engineer-ing, etc.) it is assumed that Euclideangeometry applies. However, it is found thatthis is only an approximation, and that thespace-time continuum of relativity theoryis non-Euclidean in its properties.

non-isomorphism See isomorphism.

non-linear oscillations Oscillations inwhich the force is not proportional to thedisplacement from the equilibrium posi-tion; i.e. the oscillations are not simple har-monic motion. The general motion of apendulum is an example of a non-linear os-cillation. The motion of a pendulum is acase of a non-linear oscillation that can be

solved exactly but, in general, non-linearoscillations require approximate methods.Associated with this, it is possible forchaotic motion to occur for non-linear os-cillators.

non-linear waves Waves that are associ-ated with NON-LINEAR OSCILLATIONS. Theform of a non-linear wave is more compli-cated than a sine wave and only approxi-mates to a sine wave in the case of smalloscillations. It has been suggested that verylarge ‘freak waves’ in oceans are non-linearwaves. Other physical examples exist. ASOLITON is a particular type of non-linearwave.

non-uniform motion Motion in whichthe velocity is not constant. This can meanthat there is acceleration (including thecase of negative acceleration, i.e. decelera-tion) and/or the direction of the motion isnot constant. For example, a body movingin a circle is an example of non-uniformmotion since the direction of the body is al-ways changing, which means that there isalways an acceleration, even if the speed ofthe body is constant. If the speed of a bodymoving in a circle is not uniform then thereis said to be non-uniform circular motion.The analysis of non-uniform circular mo-tion requires a slight generalization of theanalysis of uniform circular motion. In thecase of non-uniform circular motion thereis a component of the acceleration directedalong the tangent to the circle as well as acomponent directed towards the center ofthe circle.

NOR gate See logic gate.

norm A generalization of the concept ofmagnitude to any vector space. The normof a vector x is usually written ||x||. It is areal number associated with the vector andis positive or zero (for the zero vector). If ais a real number, then

||ax|| = a||x||and

||x + y|| ≤ ||x|| + ||y||

normal Denoting a line or plane that isperpendicular to another line or plane. A

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normal

line or plane is said to be normal to a curveif it is perpendicular to the tangent to thecurve at the point at which the line and thecurve meet. A radius of a circle, for exam-ple, is normal to the circumference. A planepassing through the center of a sphere isnormal to the surface at all points at whichthey meet.

normal chord A straight line joining twopoints on a curve that is also a normal tothe curve at one or both of the two points.The diameter of a circle can be regarded asa normal chord. It is also possible to con-struct normal chords of a hyperbola thatconnect the two branches of the hyperbola.

normal distribution (Gaussian distribu-tion) The type of statistical distributionfollowed by, for example, the same mea-surement taken several times, where thevariation of a quantity (x) about its meanvalue (µ) is entirely random. A normal dis-tribution has the probability density func-tion

f(x) = exp[–(x – µ)2/2σ2]/σ√2πwhere σ is known as the standard devia-tion. The distribution is written N(µ,σ2).The graph of f(x) is bell-shaped and sym-metrical about x = µ. The standard normaldistribution has µ = 0 and σ2 = 1. x can bestandardized by letting z = (x – µ)/σ. Thevalues zα, for which the area under thecurve from – ∞ to zα is α, are tabulated; i.e.z is such that P(z ≤ zα) = α. Hence

P(a<x≤b) =P(a – µ)/σ < z ≤ (b – µ)/σ

can be found. The alternative term ‘Gauss-ian distribution’ is named for the Germanmathematician Karl Friedrich Gauss(1777–1855).

normal form See canonical form.

normalize To multiply a quantity (e.g. avector or matrix) by a suitable constant sothat its norm is equal to one.

normal subgroup A subgroup H of agroup G is normal if and only if for any el-ement h in H, h–1 gh is in H for all elementsg of G.

NOT gate See logic gate.

NP-problem A type of problem wherethe size of the problem is characterized bysome number n and the number of stepswhich an algorithm would need to solvethis problem is N, and the dependence of Non n is such that N increases with n morerapidly than any polynomial of n. Suchproblems are called NP-problems becausethe time it takes to solve them increasesmore rapidly than any polynomial of n.Examples of NP-problems include theTRAVELING SALESMAN PROBLEM and the factorization of large integers. NP-prob-lems are more difficult to solve than P-PROBLEMS. It may be the case that someNP-problems, such as the factorization oflarge integers, could be solved much morerapidly using a quantum computer.

n-th root of unity A complex number zthat satisfies the relation zn = 1. There aren such n-th roots of unity. They are givenby z = exp (i2πl/n), where l takes all the in-teger values from 0 to n–1. The complexnumbers that are the n-th root of unity canbe represented in an ARGAND DIAGRAM forthe complex plane as vertices of a regularpolygon with n sides. These points lie on acircle with its center at the origin of thecomplex plane and a radius of one unit. Ifn is an even integer then the roots must in-clude both 1 and –1. If n is an odd integerthe roots that are not real numbers have tobe pairs of complex numbers that are com-plex conjugates of each other. The n-th

normal chord

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T

P

N

Normal: NP is the normal to the curve

roots of unity can also be written as: z =cos(2πl/n) + i sin(2πl/n), with l taking all in-teger values from 0 to n–1.

NTP Normal temperature and pressure.See STP.

null hypothesis See hypothesis test.

null matrix (zero matrix) A MATRIX inwhich all the elements are equal to zero.

null set See empty set.

number line A straight horizontal line onwhich each point represents a real number.Integers are points marked at unit distanceapart.

numbers Symbols used for counting andmeasuring. The numbers now in generaluse are based on the Hindu-Arabic system,which was introduced to Europe in the14th and 15th centuries. The Roman nu-merals used before this made simple arith-metic very difficult, and most calculationsneeded an abacus. Hindu-Arabic numerals(0, 1, 2, … 9) enabled calculations to beperformed with far greater efficiency be-cause they are grouped systematically inunits, tens, hundreds, and so on. See alsointegers; irrational numbers; natural num-bers; rational numbers; real numbers;whole numbers.

numeral /new-mĕ-răl/ A symbol that de-

notes a number. Examples include 0, 1, 2,3, 4, 5, 6, 7, 8, 9 of Arabic numerals and I,V, X, L, C, D, M of ROMAN NUMERALS.

numerator /new-mĕ-ray-ter/ The top partof a fraction. For example, in the fraction¾, 3 is the numerator and 4 is the denomi-nator. The numerator is the dividend.

numerical analysis /new-me-ră-kăl/ Thestudy of methods of calculation that in-volve approximations, for example, itera-tive methods. See also iteration.

numerical integration A procedure forcalculating approximate values of inte-grals. Sometimes a function is known onlyas a set of values for corresponding valuesof a variable and not as a general formulathat can be integrated. Also, many func-tions cannot be integrated in terms ofknown standard integrals. In these cases,numerical integration methods, such as theTRAPEZIUM RULE and SIMPSON’S RULE, can beused to calculate the area under a graphcorresponding to the integral. The area isdivided into vertical columns of equalwidth, the width of each column represent-ing an interval between two values of x forwhich f(x) is known. Usually a calculationis first carried out with a few columns;these are further subdivided until the de-sired accuracy is attained, i.e. when furthersubdivision makes no significant differenceto the result.

149

numerical integration

–1 0 1 2 3 4 5 6 7

Number line: this shows an open interval of the real numbers between –1 and +2 and a closed interval from 4 to 6 (including 4 and 6)

object The set of points that undergoes ageometrical transformation or mapping.See also projection.

oblate /ob-layt, ŏ-blayt/ Denoting a spher-oid that has a polar diameter that is smallerthan the equatorial diameter. The Earth,for example, is not a perfect sphere but isan oblate spheroid. Compare prolate. Seealso ellipsoid.

oblique /ŏ-bleek/ Forming an angle that isnot a right angle.

oblique coordinates See Cartesian coor-dinates.

oblique solid A solid geometrical figurethat is ‘slanted’; for example, a cone, cylin-der, pyramid, or prism with an axis that isnot at right angles to its base. Compareright solid.

oblique spherical triangle See sphericaltriangle.

oblique triangle A triangle that does notcontain a right angle.

oblong /ob-long/ An imprecise term for aRECTANGLE.

obtuse /ŏb-tewss/ Denoting an angle thatis greater than 90° but less than 180°.Compare acute; reflex.

OCR (optical character recognition) Asystem used to input information to a com-puter. The information, usually in the formof letters and numbers, is printed, typed, orsometimes hand-written. The charactersused can be read and identified optically byan OCR reader. This machine interprets

each character and translates it into a seriesof electrical pulses.

octagon /ok-tă-gon/ A plane figure witheight straight sides. A regular octagon haseight equal sides and eight equal angles.

octahedron /ok-tă-hee-drŏn/ (pl. octahe-drons or octahedra) A POLYHEDRON thathas eight faces. A regular octahedron haseight faces, each one an equilateral trian-gle.

octal /ok-tăl/ Denoting or based on thenumber eight. An octal number system haseight different digits instead of the ten inthe decimal system. Eight is written as 10,nine as 11, and so on. Compare binary;decimal; duodecimal; hexadecimal.

octant /ok-tănt/ 1. One of eight regionsinto which space is divided by the threeaxes of a three-dimensional Cartesian co-ordinate system. The first octant is the onein which x, y, and z are all positive. Thesecond, third, and fourth octants are num-

150

O

Octahedron: a regular octahedron

bered anticlockwise around the positive z-axis. The fifth octant is underneath thefirst, the sixth under the second, etc.2. A unit of plane angle equal to 45 degrees(π/4 radians).

odd Not divisible by two. The set of oddnumbers is 1, 3, 5, 7, …. Compare even.

odd function A function f(x) of a variablex for which f(–x) = –f(x). For example, sinxand x3 are odd functions of x. Compareeven function.

odds When bets are placed on some eventthe odds are the probability of it happen-ing.

oersted Symbol Oe A unit of magneticfield strength in the c.g.s. system. It is equalto 103/4π amperes per meter (103/4πA m–1). The unit is named for the Danishphysicist Hans Christian Oersted (1777–1851).

ohm /ohm/ Symbol: Ω The SI unit of elec-trical resistance, equal to a resistancethrough which a current of one ampereflows when there is an electric potentialdifference of one volt across it. 1 Ω = 1V A–1. The unit is named for the Germanphysicist Georg Simon Ohm (1787–1854).

one-to-one correspondence A functionor mapping between two sets of things ornumbers, such that each element in the firstset maps into only one element in the sec-

ond, and vice versa. See also function; ho-momorphism.

one-to-one mapping A mapping f froma set M to a set N such that if m1, and m2are different elements of M then their im-ages f(m1) and f(m2) are different elementsof N. This definition means that if f is aone-to-one mapping then the result f(m1) =f(m2) means that m1 = m2.

onto A mapping from one set S to anotherset T is onto if every member of T is theimage of some member of S under the map-ping. Compare into.

open curve A curve in which the ends donot meet, for example, a parabola or a hy-perbola. Compare closed curve.

open interval A set consisting of thenumbers between two given numbers (endpoints), not including the end points, forexample, all the real numbers greater than1 and less than 4.5 constitute an open in-terval. The open interval between two realnumbers a and b is written (a,b). Here, theround brackets indicate that the points aand b are not included in the INTERVAL. Ona number line, the end points of an open in-terval are circled. Compare closed interval.

open sentence In formal logic, a sentencethat contains one or more free variables.

open set A set defined by limits that arenot included in the set itself. The set of all

151

open set

A

B

C

O

Octant: ABCO is an octant of the sphere.

rational numbers greater than 0 and lessthan ten, written x:0<x<10; x∈R, and theset of all the points inside a circle, but notincluding the circle itself, are examples ofopen sets. Compare closed set.

operating system (OS) The collection ofprograms used in the control of a computersystem. It is generally supplied by the com-puter manufacturer. An operating systemhas to decide at any instant which of themany demands on the attention of the cen-tral processor to satisfy next. These de-mands include input from and output tovarious devices, the execution of a numberof programs, and accounting and timing.Large computers, in which many jobs canbe run simultaneously, will have a highlycomplex operating system.

operation Any process that combines to-gether members of a set. Combining twomembers to produce a third is a binary op-eration. Chief of these are addition, sub-traction, multiplication, and division, themain operations in arithmetic. See opera-tor.

operator 1. A mathematical function,such as addition, subtraction, multiplica-tion, or taking a square root or a loga-rithm, etc. See function.2. The symbol denoting a mathematicaloperation or function, for example: +, –, ×,√, log10.

opposite Denoting the side facing a givenangle in a triangle, i.e. the side not formingone arm of the angle. In trigonometry, theratios of the length of the opposite side tothe other side lengths in a right-angled tri-angle are used to define the sine and tan-gent functions of the angle.

optical character recognition See OCR.

or See disjunction.

orbit The curved path or trajectory alongwhich a moving object travels under the in-fluence of a gravitational field. An objectwith a negligible mass moving under the in-fluence of a planet or other body has an

orbit that is a conic section; i.e. a parabola,ellipse, or hyperbola.

order 1. (of a matrix) The number of rowsand columns in a matrix. See matrix.2. (of a derivative) The number of times avariable is differentiated. For example,dy/dx is a first-order derivative; d2y/dx2 issecond-order; etc.3. (of a DIFFERENTIAL EQUATION) The orderof the highest derivative in an equation.For example,

d3y/dx3 + 4xd2y/dx2 = 0is a third-order differential equation.

d2y/dx2 – 3x(dy/dx)3 = 0is a second-order differential equation.Compare degree.

ordered pair Two numbers indicatingvalues of two variables in a particularorder. For example, the x- and y-coordi-nates of points in a two-dimensional Carte-sian coordinate system form a set ofordered pairs (x,y).

ordered set A set of entities in a particu-lar order. See sequence.

ordered triple Three numbers indicatingvalues of three variables in a particularorder. The x-, y-, and z-coordinates of apoint in a three-dimensional coordinatesystem form an ordered triple (x,y,z).

order of rotational symmetry the valueof n in an n-fold axis of symmetry, i.e. thevalue of n when rotation about this axis by(360/n) degrees gives a result indistinguish-able from the original position of the objectwhich is being rotated. For example, if abody has a threefold axis of symmetry thenthe order of rotational symmetry is three.

ordinal numbers Whole numbers thatdenote order, as distinct from number orquantity. That is, first, second, third, andso on. Compare cardinal numbers.

ordinary differential equation Anequation that contains total derivatives butno partial derivatives. See differentialequation.

operating system

152

ordinate The vertical coordinate (y-coor-dinate) in a two-dimensional rectangularCartesian coordinate system. See Cartesiancoordinates.

OR gate See logic gate.

origin The fixed reference point in a co-ordinate system, at which the values of allthe COORDINATES are zero and at which theaxes meet.

orthocenter /or-thoh-sen-ter/ A point in atriangle that is the point of intersection oflines from each vertex perpendicular to theopposite sides. The triangle formed byjoining the feet of these vertices is the pedaltriangle.

orthogonal circles /or-thog-ŏ-năl/ Twocircles that intersect at right angles to eachother. Circles which intersect in this wayare said to cut orthogonally. If A and B arethe centers of two circles that cut orthogo-nally at the points P and Q then the tangentat P to the circle with center A is perpen-dicular to the radius AP. In a similar waythe tangent at P to the circle with center Bis perpendicular to the radius BP. Analo-gous results hold for the tangents at Q.Since the circles are orthogonal the tan-gents at P are perpendicular, as are the tan-gents at Q. This means that each of thetangents at both P and Q goes through thecenter of the other circle. This means thatAP2 + BP2 = AQ2 + BQ2 = AB2, i.e. the

square of the distance between two orthog-onal circles is equal to the sum of thesquares of their two radii.

orthogonal group See orthogonal ma-trix.

orthogonal matrix A matrix for whichits transpose is also its inverse. Thus, if A isa matrix and its transpose is denoted by

~A

then A is an orthogonal matrix if AÃ = I,where I is the unit matrix. The set of all n ×n orthogonal matrices forms a group calledthe orthogonal group. The set of all n × northogonal matrices with determinantsthat have the value 1 is a subgroup of theorthogonal group called the special orthog-onal group. The orthogonal group is de-noted O(n) and the special orthogonalgroup is denoted SO(n). The groups O(n)and SO(n) have several important physicalapplications, including the description ofrotations.

orthogonal projection /or-thog-ŏ-năl/ Ageometrical transformation that producesan image on a line or plane by perpendicu-lar lines crossing the plane. If a line oflength l is projected orthogonally from aplane at angle θ to the image plane, itsimage length is lcosθ. The image of a circleis an ellipse. See also projection.

orthogonal vectors Vectors which areperpendicular. This means that if a and bare non-zero vectors then they are orthog-

153

orthogonal vectors

O

A

BC

X

Y

Z

Orthocenter of a triangle ABC: the pedal triangle is XYZ.

onal vectors if and only if their SCALAR

PRODUCT a.b = 0. If there is a set of mutu-ally orthogonal unit vectors then there issaid to be an orthonormal set of vectors. Inthree-dimensional space the UNIT VECTORS

i, j, and k along the x, y, and z axes respec-tively form a set of orthonormal vectors.

orthonormal /or-thŏ-nor-măl/ See or-thogonal vectors.

OS See operating system.

Osborn’s rule /oz-born/ A rule that canbe used to convert relations betweentrigonometric identities and the analogousrelations between hyperbolic functions. Itstates that a cos2 term becomes a cosh2

term without change of sign but that a sin2

term (or more generally a product of twosine terms) becomes a –sinh2 term, or,more generally, the sign of the term ischanged. For example, cos2x + sin2x = 1becomes cosh2x – sinh2x = 1 and cos2x –sin2x= cos2x becomes cosh2x + sinh2x =cosh2x. Osborn’s rule has to be used care-fully since there are some trigonometricfunctions, such as tan2x, that can be ex-pressed in terms of sines and cosines in sev-eral ways. Although Osborn’s rule can beused to find an identity for hyperbolicfunctions, finding such an identity in thisway does not constitute a proof for theidentity.

oscillating series /os-ă-lay-ting/ A specialtype of nonconvergent series for which thesum does not approach a limit but contin-ually fluctuates. Oscillating series caneither fluctuate between bounds, for ex-ample the series 1 – 1 + 1 – 1 + …, or it canbe unbounded, for example 1 – 2 + 3 – 4 +5 – … .

oscillation /os-ă-lay-shŏn/ A regularly re-peated motion or change. See vibration.

ounce 1. A unit of mass equal to one six-teenth of a pound. It is equivalent to0.0283 49 kg.2. A unit of capacity, often called a fluidounce, equal to one sixteenth of a pint. It isequivalent to 2.057 3 × 10–5 m3. In the UK,

it is one twentieth of a UK pint, equivalentto 2.841 3 × 10–5 m3. 1 UK fluid ounce isequal to 0.960 8 US fluid ounce.

output 1. The signal or other form of in-formation obtained from an electrical de-vice, machine, etc. The output of acomputer is the information or results de-rived from the data and programmed in-structions fed into it. This information istransferred as a series of electrical pulsesfrom the central processor of the computerto a selected output device. Some of theseoutput units convert the pulses into a read-able or pictorial form; examples includethe printer, plotter, and visual display unit(which can also be used as an input device).Other output devices translate the pulsesinto a form that can be fed back into thecomputer at a later stage; the magnetic tapeunit is an example.2. The process or means by which output isobtained.3. To deliver as output.

See also input; input/output.

overdamping See damping.

overflow The situation arising in com-puting when a number, such as the result ofan arithmetical operation, has a greatermagnitude than can be represented in thespace allocated to it in a register or a loca-tion in store.

overlay A technique used in computingwhen the total storage requirements for alengthy program exceeds the space avail-able in the main store. The program is splitinto sections so that only the section or sec-tions required at any one time will be trans-ferred. These program segments (oroverlays) will all occupy the same area ofthe main store.

overtones The wave patterns which arepresent in sound and music in which thefrequency is greater than the fundamentalfrequency. It is possible to analyze theovertones of a wave pattern using FOURIER

SERIES. The overtones which are heardalong with the main note of a specific fre-quency f vary in different musical instru-

orthonormal

154

ments. This means that the waveform asso-ciated with each note varies in differentmusical instruments. The quality or timbreof a note, i.e. how much the waveform for

a note deviates from being a sine wave, istherefore dependent on the presence or ab-sence of overtones and hence depends onthe instrument.

155

overtones

156

palindrome A number that is the samesequence of integers forward and back-ward. An example of a palindrome is thenumber 35753. A conjecture concerningpalindromes is that starting from any num-ber, if the integers of that number are re-versed and the resulting number is added tothe original number, then if this process isrepeated long enough a palindrome will re-sult. However, this conjecture has not beenproved.

pandigital number /pan-dij-i-tăl/ A num-ber which contains all of the numbers 0, 1,2, 3, 4, 5, 6, 7, 8, 9 only once.

paper tape A long strip of paper, or some-times thin flexible plastic, on which infor-mation can be recorded as a pattern ofround holes punched in rows across thetape, once used extensively in data process-ing.

Pappus’ theorems Two theorems con-cerning the rotation of a curve or a planeshape about a line that lies in the sameplane. The first theorem states that the sur-face area generated by a curve revolvingabout a line that does not cross it, is equalto the length of the curve times the circum-ference of the circle traced out by its cen-troid. The second theorem states that thevolume of a solid of revolution generatedby a plane area that rotates about a line notcrossing it, is equal to the area times thecircumference of the circle traced out bythe centroid of the area. (Note that theplane area and the line both lie in the sameplane.) The theorems are named for theGreek mathematician Pappus of Alexan-dria (fl. AD 320). The second Pappus theo-rem is sometimes known as the Guldinustheorem as it was rediscovered by the Swiss

mathematician and astronomer PaulGuldin (1577–1643).

parabola /pă-rab-ŏ-lă/ A conic with an ec-centricity of 1. The curve is symmetricalabout an axis through the focus at right an-gles to the directrix. This axis intercepts theparabola at the vertex. A chord throughthe focus perpendicular to the axis is thelatus rectum of the parabola.

In Cartesian coordinates a parabola canbe represented by an equation:

y2 = 4axIn this form the vertex is at the origin andthe x-axis is the axis of symmetry. Thefocus is at the point (0,a) and the directrixis the line x = –a (parallel to the y-axis).The latus rectum is 4a.

If a point is taken on a parabola andtwo lines drawn from it – one parallel tothe axis and the other from the point to thefocus – then these lines make equal angleswith the tangent at that point. This isknown as the reflection property of theparabola, and is utilized in parabolic re-flectors and antennas. See paraboloid.

The parabola is the curve traced out bya projectile falling freely under gravity. Forexample, a tennis ball projected horizon-tally with a velocity v has, after time t, trav-eled a distance d = vt horizontally, and hasalso fallen vertically by h = gt2/2 because ofthe acceleration of free fall g. These twoequations are parametric equations of theparabola. Their standard form, corre-sponding to y2 = 4ax, is

x = at2

y = 2atwhere x represents h, the constant a is g/2,and y represents d. See illustration over-leaf. See also conic.

paraboloid /pă-rab-ŏ-loid/ A curved sur-

P

face in which the cross-sections in anyplane passing through a central axis is aparabola. A paraboloid of revolution isformed when a parabola is rotated aroundits axis of symmetry. Parabolic surfaces areused in telescope mirrors, searchlights, ra-diant heaters, and radio antennas on ac-count of the focusing property of theparabola.

Another type of paraboloid is the hy-perbolic paraboloid. This is a surface withthe equation:

x2/a2 – y2/b2 = 2czwhere c is a positive constant. Sections par-allel to the xy plane (z = 0) are hyperbolas.

Sections parallel to the other two planes (x= 0 or y = 0) are parabolas. See illustrationoverleaf. See also conicoid.

paradox /pa-ră-doks/ (antinomy) Aproposition or statement that leads to acontradiction if it is asserted and if it is de-nied.

A famous example of a paradox is Rus-sell’s paradox in set theory. A set is a col-lection of things. It is possible to think ofsets as belonging to two groups: sets thatcontain themselves and sets that do notcontain themselves. A set that contains it-self is itself a member of the set. For in-

157

paradox

axis of rotation

circlecircumferencec

centroidof curve

curvelength l

Pappus’ theorem: the curved surfacearea A = l x c.

axis of rotation

circle circumference cplane area A

centroid ofplane area

Pappus’ theorem: the volume enclosed bythe curved surface V = A x c.

Pappus’ theorems

stance, the set of all sets is itself a set, so itcontains itself. Other examples are the ‘setof all things that one can think about’ and‘the set of all abstract ideas’. On the otherhand the set of all things with four legsdoes not itself have four legs – it is an ex-ample of a set that does not contain itself.Other examples of sets that do not containthemselves (or are not members of them-selves) are ‘the set of all oranges’ and ‘theset of all things that are green’.

Consider now the set of all sets that donot contain themselves. The paradox arisesfrom the question, ‘Does the set of all setsthat do not contain themselves contain it-self or not?’ If one asserts that it does con-tain itself then it cannot be one of thethings that do not contain themselves. Thismeans that it does not belong to the set, so

it doesn’t contain itself. On the other hand,if it does not contain itself it must be one ofall the things that do not contain them-selves, so it must belong to the set. Theshort answer to the question is, ‘If it does,it doesn’t; if it doesn’t, it does!’ – hence theparadox. Paradoxes like this can be used ininvestigating fundamental questions in settheory.

parallax /pa-ră-laks/ The angle betweenthe direction of an object, for example astar or a planet, from a point on the surfaceof the Earth and the direction of the sameobject from the center of the Earth. Themeasurement of parallax is used to find thedistance of an object from the Earth. Theobject is viewed from two widely separatedpoints on the Earth and the distance be-

parallax

158

V F (0,a)

P (x,y)

D

E

F

G

a a

y

x

y

xF

P

S

T

Q

N

Parabola: F is the focus and DE is the directrix. FG is the latus rectum. If PQ is parallel to the x-axis, angles SPF and TPQ are equal (ST is a tangent at P).

y

x

y

xO O

Paraboloid of revolution

tween these two points and the direction ofthe object as seen from each is measured.The parallax at the observation points andthe distance of the object from the observercan be found by simple trigonometry.

parallel Extending in the same directionand remaining the same distance apart.Compare antiparallel.

parallel axes, theorem of See theorem ofparallel axes.

parallelepiped /pa-ră-lel-ă-pÿ-ped/ Asolid figure with six faces that are parallel-ograms. In a rectangular parallelepiped thefaces are rectangles. If the faces aresquares, the parallelepiped is a cube. If thelateral edges are not perpendicular to thebase, it is an oblique parallelepiped.

parallel forces When the forces on an ob-ject pass through one point, their resultantcan be found by using the parallelogram ofvectors. If the forces are parallel the resul-tant is found by addition, taking sign intoaccount. There may also be a turning effectin such cases, which can be found by theprinciple of moments.

parallelogram /pa-ră-lel-ŏ-gram/ A planefigure with four straight sides, and with op-posite sides parallel and of equal length.The opposite angles of a parallelogram arealso equal. Its area is the product of the

length of one side and the perpendiculardistance from that side to the side opposite.In the special case in which the angles areall right angles the parallelogram is a rec-tangle; when all the sides are equal it is arhombus.

parallelogram (law) of forces See par-allelogram of vectors.

parallelogram (law) of velocities Seeparallelogram of vectors.

parallelogram of vectors A method forfinding the resultant of two vectors actingat a point. The two vectors are shown astwo adjacent sides of a parallelogram: theresultant is the diagonal of the parallelo-gram through the starting point. The tech-nique can be used either with careful scaledrawing or with trigonometry. Thetrigonometrical relations give:

F = √(F12 + F2

2 + 2F1F2cosθ)α = sin–1[(F2/F)sinθ]

where θ is the angle between F1 and F2 andα is the angle between F and F1. See vector.

parallel postulate See Euclidean geome-try.

parameter /pă-ram-ĕ-ter/ A quantity that,when varied, affects the value of another.For example, if a variable z is a function ofvariables x and y, that is z = f(x,y), then xand y are the parameters that determine z.

159

parameter

h

b

area = hb

Parallelepiped Parallelogram

parametric equations Equations that, inan implicit function (such as f(x,y) = 0) ex-press x and y separately in terms of a quan-tity, which is an independent variable orparameter. For example, the equation of acircle can be written in the form

x2 + y2 = r2

or as the parametric equationsx = rcosθy = rsinθ

where r is the radius. See also parabola.

parametrization /pa-ră-met-ri-zay-shŏn/A method for associating a parameter twith a point P, which lies on a curve so thateach point on the curve is associated with avalue of t, with t lying in some interval ofthe real numbers. This is frequently doneby expressing the x and y coordinates of Pin terms of t. The resulting equations for xand y in terms of t are called the PARAMET-RIC EQUATIONS for the curve. It is possibleto find the expression for dy/dx at anypoint on the curve in terms of the parame-ter t since dy/dx = (dy/dt)/(dx/dt).

Perhaps the simplest parametrization ofa curve is given by a circle of unit radius.The points on the circle are parametrizedby an angle θ, with (in radians) 0 ≤ θ ≤ 2π.One has x = cosθ and y = sinθ. In the caseof an ellipse x2/a2 + y2/b2 = 1 the parame-trization is given by x = a cosθ and y = bsinθ (0 ≤ θ ≤ 2π). In the case of a parabolay2 = 4ax the curve is parametrized by t withx = at2 and y = 2at.

paraplanar /pa-ră-play-ner/ Describing aset of vectors that are all parallel to a planebut are not necessarily contained in thatplane.

parsec /par-sek/ Symbol: pc A unit of dis-tance used in astronomy. A star that is oneparsec away from the earth has a parallax(apparent shift), due to the Earth’s move-ment around the Sun, of one second of arc.One parsec is approximately 3.085 61 ×1016 meters.

partial derivative The rate of change of afunction of several variables as one of thevariables changes and the others are heldconstant. For example, if z = f(x,y) the par-

tial derivative ∂z/∂x is the rate of change ofz with respect to x, with y held constant. Itsvalue will depend on the constant value ofy chosen. In three-dimensional Cartesiancoordinates, ∂z/∂x is the gradient of a lineat a tangent to the curved surface f(x,y) andparallel to the x-axis. See also total deriva-tive.

partial differential The infinitesimalchange in a function of two or more vari-ables resulting from changing only one ofthe variables while keeping the others con-stant. The sum of all the partial differen-tials is the total differential. Seedifferential.

partial differential equation An equa-tion that contains partial derivatives of afunction with respect to a number of vari-ables. General methods of solution areavailable only for certain types of linearpartial differential equation. Many partialdifferential equations occur in physicalproblems. Laplace’s equation, for exam-ple, is

∂2φ/∂x2 + ∂2φ/∂y2 + ∂2φ/∂z2 = 0It occurs in the study of gravitational andelectromagnetic fields. The equation isnamed for the French mathematician, as-tronomer, and physicist Marquis PierreSimon de Laplace (1749–1827). See alsodifferential equation.

partial fractions A sum of fractions thatis equal to a particular fraction, for exam-ple, ½ + ¼ = ¾. Writing a ratio in terms ofpartial fractions can be useful in solvingequations or calculating integrals. For ex-ample

1/(x2 + 5x + 6)can be written as partial fractions by fac-torizing the denominator into (x + 2)(x +3), and writing the fraction in the form

[A/(x + 2)] + [B/(x + 3)] =1/(x2 + 5x + 6)

This means thatA(x + 3) + B(x + 2) = 1

i.e.Ax + 3A + Bx + 2B = 1

The values of A and B can be found bycomparing the coefficients of powers of x,i.e.

parametric equations

160

3A + 2B = 1 (constant term)A + B = 0 (coefficient of x)

Solving these simultaneous equations givesA = 1 and B = –1. So the fraction 1/(x2 + 5x+ 6) can be expressed as partial fractions1/(x + 2) – 1/(x + 3). This is a form that canbe integrated as a sum of two simpler inte-grals.

partially ordered A partially ordered setis a set with a relation x < y defined forsome elements x and y of the set satisfyingthe conditions: 1. if x < y then y < x is false and x and y arenot the same element;2. if x < y and y < z then x < z.

It need not be the case that x < y or y <x for any two elements x and y. An exam-ple of a partially ordered set is the set ofsubsets of a given set where we define A <B for sets A and B to mean that A is aproper subset of B.

partial product A product of the first nterms a1 a2 ... an, called the n-th partialproduct, where these n terms are the first nterms of the infinite product.

partial sum The sum of a finite number ofterms from the start of an infinite series. Ina convergent series, the partial sum of thefirst r terms, Sr, is an approximation to thesum to infinity. See series.

particle An abstract simplification of areal object – the mass is concentrated at theobject’s center of mass; its volume is zero.Thus rotational aspects can be ignored.

particular integral A name given to thePARTICULAR SOLUTION in the case of asecond-order linear differential equation ofthe form ad2y/dx2 + bdy/dx + cy = f(x),where a, b, and c are constants and f issome function of x. A way of finding a par-ticular integral which frequently works isto use a function which is similar to f(x).For example, if f(x) is a polynomial ofsome degree in x then the particular inte-gral to be tried should also be a polynomialof the same degree. Similarly, if f(x) is atrigonometric function such as sinlx orcosmx or an exponential function exp(nx)

then a similar function can be tried. In allthese examples, unknown coefficients canbe found by substitution into the originaldifferential equation.

particular solution A solution of a dif-ferential equation that is given by someparticular values of the arbitrary constantsthat appear in the general solution of thedifferential equation. A particular solutioncan be found if it satisfies both the differ-ential equation and the BOUNDARY CONDI-TIONS that apply.

partition A partition of a set S is a finitecollection of disjoint sets whose union is S.

pascal /pas-kăl/ Symbol: Pa The SI unit ofpressure, equal to a pressure of one newtonper square meter (1 Pa = 1 N m–2). The pas-cal is also the unit of stress. It is named forthe French mathematician, physicist andreligious philosopher Blaise Pascal (1623–62).

Pascal’s distribution (negative binomialdistribution) The distribution of the num-ber of independent Bernoulli trials per-formed up to and including the rth success.The probability that the number of trials,x, is equal to k is given by

P(x=k) = k–1Cr–1prqk–r

The mean and variance are r/p and rq/p2

respectively. See also geometric distribu-tion.

Pascal’s triangle A triangle array of num-bers in which each row starts and endswith 1 and that is built up by summing twoadjacent numbers in a row to obtain thenumber directly below them in the next

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Pascal’s triangle

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

Pascal’s triangle

row. Each row in Pascal’s triangle is set ofbinomial coefficients. In the expansion of(x + y)n, the coefficients of x and y aregiven by the (n + 1)th row of Pascal’s trian-gle.

pedal curve The pedal curve of a givencurve C with respect to a fixed point P isthe locus of the foot of the perpendicularfrom P to a variable tangent to the curve C.The point P is called the pedal point. Forexample, if C is a circle and P is a point onits circumference the pedal curve is a car-dioid.

pedal triangle See orthocenter.

pencil A family of geometric objects thatshare a common property. For example, apencil of circles consists of all the circles ina given plane that pass through two givenpoints, a pencil of lines consists of all thelines in a given plane passing through agiven point, and a pencil of parallel linesconsists of all the lines parallel to a givendirection.

pendulum A body that oscillates freelyunder the influence of gravity. A simplependulum consists of a small mass oscillat-ing to and fro at the end of a very lightstring. If the amplitude of oscillation issmall (less than about 10°), it moves withsimple harmonic motion; the period doesnot depend on amplitude. There is a con-tinuous interchange of potential and ki-netic energy through the motion; at theends of the swings the potential energy is amaximum and the kinetic energy zero. Atthe mid-point the kinetic energy is a maxi-mum and the potential energy is zero. Theperiod is given by

T = 2π√(l/g)Here l is the length of the pendulum (fromsupport to center of the mass) and g is theacceleration of free fall. If the amplitude ofoscillation is not small it does not movewith simple harmonic motion but theproblem of its motion can still be solvedexactly.

A compound pendulum is a rigid bodyswinging about a point. The period of acompound pendulum depends on the mo-

ment of inertia of the body. For small os-cillations it is given by the same relation-ship as that of the simple pendulum with lreplaced by [√(k2 + h2)]/h. Here, k is the RA-DIUS OF GYRATION about an axis throughthe center of mass and h is the distancefrom the pivot to the center of mass.

Penrose pattern /pen-rohz/ A two-dimen-sional pattern of a tiling that has five-foldrotational axes of symmetry and also long-range order, in spite of it being impossibleto have a two-dimensional crystal in whichfive-fold rotational axes of symmetry canoccur. It is possible to obtain this type ofpattern by combining two sets of rhom-buses, with one of the sets being ‘thin’rhombuses and the other set being ‘fat’rhombuses, in specific ways. A three-dimensional version of Penrose patternsoccurs in materials that have QUASICRYS-TALLINE SYMMETRY. The pattern is namedfor the British mathematician Sir RogerPenrose (1931– ).

pentagon /pen-tă-gon/ A plane figure withfive straight sides. In a regular pentagon,one with all five sides and angles equal, theangles are all 108°. A regular pentagon canbe superimposed on itself after rotationthrough 72° (2π/5 radians).

percentage A number expressed as a frac-tion of one hundred. For example, 5 per-cent (or 5%) is equal to 5/100. Anyfraction or decimal can be expressed as apercentage by multiplying it by 100. Forexample, 0.63 × 100 = 63% and ¼ × 100 =25%.

percentage error The ERROR or uncer-tainty in a measurement expressed as a per-centage. For example, if, in measuring alength of 20 meters, a tape can measure tothe nearest four centimeters, the measure-ment is written as 20± 0.04 meters and thepercentage error is (0.04/20) × 100 = 0.2%.

percentile /per-sen-tÿl, -tăl/ One of the setof points that divide a set of data arrangedin numerical order into 100 parts. The rth

percentile, Pr, is the value below and in-cluding which r% of the data lies and

pedal curve

162

above which (100 – r)% lies. Pr can befound from the cumulative frequencygraph. See also quartile; range.

perfect number A number that is equalto the sum of all its factors except itself. 28is a perfect number since its factors are 1,2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 = 28.See also Mersenne prime.

perimeter /pĕ-rim-ĕ-ter/ The distanceround the edge of a plane figure. For ex-ample, the perimeter of a rectangle is twicethe length plus twice the breadth. Theperimeter of a circle is its circumference(2πr).

period Symbol: T The time for one com-plete cycle of an oscillation, wave motion,or other regularly repeated process. It is thereciprocal of the frequency, and is relatedto pulsatance, or angular frequency, (ω) byT = 2π/ω.

periodic function A function that repeatsitself at regular intervals of the variable.For example sinx is a periodic function ofx because sinx = sin (x + 2π) for all valuesof x.

periodic motion Any kind of regularlyrepeated motion, such as the swinging of apendulum, the orbiting of a satellite, the vi-bration of a source of sound, or an electro-magnetic wave. If the motion can berepresented as a pure sine wave, it is a sim-ple harmonic motion. Harmonic motionsin general are given by the sum of two ormore pure sine waves.

period of investment The length of timefor which a fixed amount of capital re-mains invested. In times of historically lowinterest rates, an investor prepared to com-mit his or her money for a long period,such as five or ten years, will gain a higherrate of interest than can be expected for ashort-term investment. However, if interestrates are historically high this will not bethe case and long-term rates may be lowerthan short-term rates.

peripheral unit (peripheral) A deviceconnected to and controlled by the centralprocessor of a computer. Peripherals in-clude input devices, output devices, andbacking store. Some examples are visualdisplay units, printers, magnetic tape units,and disk units. See also input; output.

permutation /per-myŭ-tay-shŏn/ An or-dered subset of a given set of objects. Forthree objects, A, B, and C, there are sixpossible permutations: ABC, ACB, BAC,BCA, CAB, and CBA. The total number ofpermutations of n objects is n!

The total number of permutations of robjects taken from n objects is given byn!(n – r)!, assuming that each object can beselected only once. This is written nPr. Forexample, the possible permutations of twoobjects from the set of three objects A, B,and C would be AB, BA, AC, CA, BC, CB.Note that each object is selected only oncein this case – if the objects could occur anynumber of times, the above set of permuta-tions would include AA, BB, and CC. Thenumber of permutations of r objects se-lected from n objects when each can occurany number of times is nr.

Note also the difference between per-mutations and combinations: permuta-tions are different if the order of selectionis different, so AB and BA are different per-mutations but the same combination. Thenumber of combinations of r objects fromn objects is written nCr, and nPr = nCr × r!

permutation group The set of permuta-tions of a number of indistinguishable ob-jects. The permutation group is importantin theoretical physics. It is used to study the energy levels of electrons in atoms and molecules and is also used in nucleartheory.

perpendicular /per-pĕn-dik-yŭ-ler/ Atright angles. The perpendicular bisector ofa line crosses it half way along its lengthand forms a right angle. A vertical surfaceis perpendicular to a horizontal surface.

personal computer (PC) See microcom-puter.

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personal computer

perturbation /per-ter-bay-shŏn/ A way ofobtaining approximate solutions of equa-tions that represent the behavior of a sys-tem, by making a slight change in some ofthe basic parameters. It is an importanttechnique in quantum mechanics and celes-tial mechanics.

phase /fayz/ The stage in a cycle that aWAVE (or other periodic system) hasreached at a particular time (taken fromsome reference point). Two waves are inphase if their maxima and minima coin-cide.

For a simple wave represented by theequation

y = asin2π(ft – x/λ)The phase of the wave is the expression

2π(ft – x/λ)The phase difference between two

points distances x1 and x2 from the originis

2π(x1 – x2)/λA more general equation for a progres-

sive wave isy = asin2π[ft – (x/λ) – φ]

Here, φ is the phase constant – thephase when t and x are zero. Two wavesthat are out of phase have different stagesat the origin. The phase difference is φ1 –φ2. It is equal to 2πx/λ, where x is the dis-tance between corresponding points on thetwo waves. It is the phase angle betweenthe two waves; the angle between two ro-tating vectors (phasors) representing thewaves.

phase angle See phase.

phase constant See phase.

phase difference See phase.

phase portrait A picture which plots theevolution of possible paths in PHASE SPACE

that a point in that space can have, startingfrom a certain set of initial conditions. Inthe case of SIMPLE HARMONIC MOTION thephase portrait consists of a set of ellipseswhich all have the same center. In the caseof NON-LINEAR OSCILLATIONS more compli-cated patterns appear in the phase portrait.The phase portrait is a convenient way of

showing the complicated behavior that canoccur in dynamical systems, including thepresence of ATTRACTORS, chaotic behavior,and LIMIT CYCLES.

phase space A multi-dimensional spacethat can be used to define the state of a sys-tem. Phase space has coordinates (q1, q2,…p1, p2,…), where q1, q2, etc., are degrees offreedom of the system and p1, p2, are themomenta corresponding to these degreesof freedom. For example, a single particlehas three degrees of freedom (correspond-ing to the three coordinates defining its po-sition). It also has three components ofmomentum corresponding to these degreesof freedom. This means that the state of theparticle can be defined by six numbers (q1,q2, q3, p1, p2, p3) and it is thus defined by apoint in six-dimensional phase space. If thesystem changes with time (i.e. the particlechanges its position and momentum), thenthe point in phase space traces out a path(known as the trajectory). The system mayconsist of more than one particle. Thus, ifthere are N particles in the system then thestate of the system is specified by a point ina phase space of 6N dimensions. The ideaof phase space is useful in chaos theory. Seealso attractor.

phase speed The speed with which thephase in a traveling wave is propagated. Itis equal to λ/T, where T is the period. Com-pare group speed.

phasor /fay-zer/ See simple harmonic mo-tion.

pi /pÿ/ (π) The ratio of the circumferenceof any circle to its diameter. π is approxi-mately equal to 3.14159… and is a tran-scendental number (its exact value cannotbe written down, but it can be stated to anydegree of accuracy). There are several ex-pressions for π in terms of infinite series.

pico- /pÿ-koh/ Symbol: p A prefix denot-ing 10–12. For example, 1 picofarad (pF) =10–12 farad (F).

pictogram /pik-tŏ-gram/ (pictograph) Adiagram that represents statistical data in

perturbation

164

a pictorial form. For example, the pro-portions of pink, red, yellow, and whiteflowers that grow from a packet of mixedseeds can be shown by rows of the appro-priate relative numbers of colored flowershapes.

piecewise A function is piecewise contin-uous on S if it is defined on S and can beseparated into a finite number of piecessuch that the function is continuous on theinterior of each piece. Terms such as piece-wise differentiable and piecewise linear aresimilarly defined.

pie chart A diagram in which proportionsare illustrated as sectors of a circle, the rel-ative areas of the sectors representing thedifferent proportions. For example, if, outof 100 workers in a factory, 25 travel towork by car, 50 by bus, 10 by train, andthe rest walk, the bus passengers are repre-sented by half of the circle, the car passen-gers by a quarter, the train users by a 36°sector, and so on.

pint A unit of capacity. The US liquid pintis equal to one eighth of a US gallon and is equivalent to 4.731 8 × 10–4 m3. In theUK it is equal to one eighth of a UK gallonand is equivalent to 5.682 6 × 10–4 m3. TheUS dry pint is equal to one sixty-fourth ofa US bushel and is equivalent to 5.506 1 ×10–4 m3.

pixel /piks-ĕl/ See computer graphics.

place value the position of an integer in anumber. For example, in the number 375the 5 represents 5 units, the 7 represents 7tens, and the 3 represents 3 hundreds.

plan An illustration that shows the ap-pearance of a solid object as viewed fromabove (vertically downward). See also ele-vation.

planar /play-ner/ Describing somethingthat occupies a plane or is flat.

plane A flat surface, either real or imagi-nary, in which any two points are joined bya straight line lying entirely on the surface.Plane geometry involves the relationshipsbetween points, lines, and curves lying inthe same plane. In Cartesian coordinates,any point in a plane can be defined by twocoordinates, x and y. In three-dimensionalcoordinates, each value of z corresponds toa plane parallel to the plane in which the xand y axs lie. For any three points, thereexists only one plane containing all three.A particular plane can also be specified bya straight line and a point.

plane shape A shape that exists in aplane, i.e. a two-dimensional shape thatdoes not have a depth but has a width andheight. Examples of plane shapes includetriangles, squares, rectangles, kites, rhom-buses, and parallelograms. Plane shapesalso include curves such as circles and el-lipses.

Platonic solids /plă-tonn-ik/ A namegiven to the set of five regular polyhedra.See polyhedron.

plot To draw on a graph. A series of indi-vidual points plotted on a graph may showa general relationship between the vari-ables represented by the horizontal andvertical axs. For example, in a scientific ex-periment one quantity can be representedby x and another by y. The values of y atdifferent values of x are then plotted as aseries of points on a graph. If these fall ona line or curve, then the line or curve drawn

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plot

50%bus

25%car

15%walk

10%train

Pie chart showing how people travel to work.

through the points is said to be a plot of yagainst x.

plotter An output device of a computersystem that produces a permanent recordof the results of some program by drawinglines on paper. One pen, or maybe two ormore pens with different colored ink, aremoved over the paper according to instruc-tions sent from the computer or from abacking store. Plotters are used for draw-ing graphs, contour maps, etc.

plotting The process of marking pointson a system of coordinates, or of drawinga graph by marking points.

Poincaré conjecture /pwank-ka-ray/ Aconjecture in TOPOLOGY that can be statedin the form that there is a HOMEOMORPHISM

between any n-dimensional MANIFOLD

which is closed, i.e. it is like a loop and hasno end points, and is SIMPLY CONNECTED

and the sphere of that dimension. TheFrench mathematician Henri Poincaré(1854–1912) originally made in conjecturein three dimensions (n = 3). The more gen-eral version of the conjecture is sometimesknown as the generalized Poincaré conjec-ture.

The generalized Poincaré conjecture isreadily proved for n = 1 and n = 2. It hasbeen proved with more difficulty in thecase of n > 4 in the 1960s. The case of n =4 was proved, with considerable difficultyin the 1980s. Proving the case of n = 3, i.e. the original Poincaré conjecture, hasproved to be the hardest of all dimensionsand a proof of this conjecture is one of theMAGNIFICENT SEVEN PROBLEMS of mathe-matics. It has been claimed that the conjec-ture has been proved in three dimensionsbut, at the time of writing, this has notbeen confirmed.

point A location in space, on a surface, orin a coordinate system. A point has no di-mensions and is defined only by its posi-tion.

point group A set of symmetry opera-tions on some body in which one of thepoints of the body is left fixed in space. An

important application of point groups is tothe symmetry of isolated molecules inchemistry. Examples of the symmetry op-erations in point groups include rotationabout a fixed axis, reflection about a planeof symmetry, inversion through a center ofsymmetry, i.e. invariance in the appearanceof the body when a point in the body withcoordinates (x,y,z) is carried to a pointwith coordinates (–x,–y,–z), and rotationabout some axis followed by reflection in aplane perpendicular to that axis.

An important set of point groups is theset of CRYSTALLOGRAPHIC POINT GROUPS, i.e.the 32 point groups in three dimensionsthat are compatible with the translationalsymmetry of a crystal lattice (see spacegroup). In a crystallographic point grouponly two-fold, three-fold, four-fold, andsix-fold rotational symmetries are possible.

point of contact A single point at whichtwo curves, or two curved surfaces touch.There is only one point of contact betweenthe circumference of a circle and tangent tothe circle. Two spheres also can have onlyone point of contact.

point of inflection A point on a curvedline at which the tangent changes its direc-tion of rotation. Approaching from oneside of the point of inflection, the slope ofthe tangent to the curve increases; andmoving away from it on the other side, itdecreases. For example, the graph of y = x3

– 3x2 in rectangular Cartesian coordinates,has a point of inflection at the point x = 1,y = –2. The second derivative d2y/dx2 onthe graph of a function y = f(x) is zero andchanges its sign at a point of inflection.Thus, in the example above, d2y/dx2 = 6x –6, which is equal to zero at the point x = 1.

Poisson distribution A probability dis-tribution for a discrete random variable. Itis defined, for a variable (r) that can takevalues in the range 0, 1, 2, …, and has amean value µ, as

P(r) = e–µr/r!It is named for the French mathematicianand mathematical physicist Siméon-DenisPoisson (1781–1840).

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A binomial distribution with a smallfrequency of success p in a large number nof trials can be approximated by a Poissondistribution with mean np.

Poisson equation A generalization of theLAPLACE EQUATION in which there is somefunction f(x,y,z) on the right-hand side ofthe equation. This means that the Poissonequation has the form ∇2φ = (x,y,z), where∇2 is the LAPLACIAN. Here, for example, φcan be the gravitational potential in a re-gion in which there is matter, the electro-static potential in a region in which there iselectric charge, or the temperature in a re-gion in which there is a source of heat. Ingeneral, the function f(x,y,z) is called thesource density. In the examples given thesource density is respectively proportionalto the density of mass, the density of elec-tric charge, and the amount of heat gener-ated per unit time, per unit volume. ThePoisson equation is the main equation ofPOTENTIAL THEORY.

Poisson’s ratio The ratio of the contrac-tion per unit diameter of a rod that isstretched to the elongation per unit lengthof the rod. If the starting diameter of therod is d and the contraction is ∆d the con-

traction per unit diameter, denoted c, is∆d/d. If the starting length of the rod is land it is extended by ∆l then the elongationper unit length of the rod, denoted s, isgiven by s = ∆l/l. Poisson’s ratio is definedto be c/s. If the elasticity of a solid is inde-pendent of direction then the ratio of c/s is0.25. Values of the ratio differ for actualmaterials. For example, it is found empiri-cally that for steels the ratio is about 0.28.In the case of aluminum alloys it is about0.33. The value of Poisson’s ratio for a ma-terial is constant for that material up to itselastic limit. If the value of Poisson’s ratiois less than 0.50 for a material then stretch-ing that material increases the net volumeof the material. If Poisson’s ratio is exactly0.50 then the volume is constant.

polar coordinates A method of definingthe position of a point by its distance anddirection from a fixed reference point(pole). The direction is given as the anglebetween the line from the origin to thepoint, and a fixed line (axis). On a flat sur-face only one angle, θ, and the radius, r, areneeded to specify each point. For example,if the axis is horizontal, the point (r,θ) =(1,π/2) is the point one unit length awayfrom the origin in the perpendicular direc-

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polar coordinates

y = x3 – 3x2

x

–

–

–

–

–

–

–

–

–

– – – – –

1 2 3–1

y

4

3

2

1

0

–1

–2

–3

–4

Point of inflection: in this case the point of inflection is x = 1, y = –2. The derivative dy/dx = –3 at this point.

tion. Conventionally, angles are taken aspositive in the anticlockwise sense.

In a rectangular Cartesian coordinatesystem with the same origin and the x-axisat θ = 0, the x- and y-coordinates of thepoint (r,θ) are

x = rcosθy = rsinθ

Converselyr = √(x2 + y2)

andθ = tan–1(y/x)

In three dimensions, two forms of polarcoordinate systems can be used. See cylin-drical polar coordinates; spherical polarcoordinates. See also Cartesian coordi-nates.

polar vector A true vector, i.e. a vectorwhich does not change sign in going froma right-handed coordinate system to a left-handed coordinate system. Compare axialvector.

pole 1. One of the points on the Earth’ssurface through which its axis of rotationpasses, or the corresponding point on anyother sphere.2. See stereographic projection.3. See polar coordinates.

Polish notation A notation in computerscience in which parentheses are unneces-sary since all formulae can be written un-ambiguously without them. In Polish

notation operators precede their operands.Thus, a + b is written +ab. The notationwas invented by a Polish mathematician,Jan Lukasiewicz (1878–1956).

polygon /pol-ee-gon/ A plane figurebounded by a number of straight sides. Ina regular polygon, all the sides are equaland all the internal angles are equal. In aregular polygon of n sides the exteriorangle is 360°/n.

polyhedron /pol-ee-hee-drŏn/ A solid fig-ure bounded by plane polygonal faces. Thepoint at which three or more faces intersecton a polyhedron is called a vertex, and aline along which two faces intersect iscalled an edge. In a regular polyhedron(also known as a Platonic solid), all thefaces are congruent regular polygons.There are only five regular polyhedrons:the regular tetrahedron, which has fourequilateral triangular faces; the regularhexahedron, or cube, which has six equi-lateral square faces; the regular octa-hedron, which has eight equilateral tri-angular faces; the regular dodecahedron,which has twelve regular pentagonal faces;and the regular icosahedron, which hastwenty equilateral triangular faces. Theseare all convex polyhedrons. That is, all theangles between faces and edges are convexand the polyhedron can be laid down flaton any one of the faces. In a concave poly-hedron, there is at least one face in a plane

polar vector

168

•

radius vector OP

P(r,)

= 0O

r

Polar coordinates: coordinates of a point P in two dimensions.

that cuts through the polyhedron. Thepolyhedron cannot be laid down on thisface.

polynomial /pol-ee-noh-mee-ăl/ A sum ofmultiples of integer powers of a variable.The general equation for a polynomial inthe variable x is

a0xn + a1xn–1 + a2xn–2 +…where a0, a1, etc., are constants and n is thehighest power of x, called the degree of thepolynomial. If n = 1, it is linear expression,for example, f(x) = 2x + 3. If n = 2, it is qua-dratic, for example, x2 + 2x + 4. If n = 3, itis cubic, for example, x3 + 8x2 + 2x + 2. Ifn = 4, it is quartic. If n = 5, it is quintic

On a Cartesian coordinate graph onwhich (n + 1) individual points are plotted,there is at least one polynomial curve thatpasses through all the points. By choosingsuitable values of a0 and a1, the straightline

y = a0x + a1can be made to pass through any twopoints. Similarly a quadratic

y = a0x2 + a1x + a2can be made to pass through any threepoints.

A polynomial may have more than onevariable:

4x2 + 2xy + y2

is a polynomial of degree 2 (second-degreepolynomial) of two variables.

polytope /pol-ee-tohp/ The analog in n di-mensions of point, line, polygon, and poly-hedron in 0, 1, 2, and 3 dimensionsrespectively.

position vector The vector that repre-sents the displacement of a point from agiven reference origin. If a point P in polarcoordinates has coordinates (r,θ), then r isthe position vector of P – a vector of mag-nitude r making an angle θ with the axis.See vector.

positive Denoting a number or quantitythat is greater than zero. Numbers that areused in counting things and measuringsizes are all positive numbers. If a change ina quantity is positive, it increases, that is itmoves away from zero if it is already posi-

tive, and toward it if it is negative. Com-pare negative.

possibility space A two-dimensionalarray that sets out all the possible out-comes of two events. By arranging all thepossible outcomes systematically in thisway there is no chance of missing any ofthe possible outcomes. For example if thepossible outcomes of rolling two dice arelisted as a possibility space then it can beseen that there are 36 possible outcomes,i.e. the possibility space is a 6 by 6 array.This enables probabilities of various out-comes to be calculated. For example, tofind the probability of finding a specifictotal for two dice it is possible to circle theentries corresponding to that total and cal-culate the probability of that total as theratio of circled entries to the total numberof entries in the array. To find the proba-bility of the total for two dice being 4 thethree entries corresponding to a total of 4are circled, meaning that the probability ofa total score of 4 is 3/36 = 1/12.

postulate See axiom.

potential energy Symbol: V The work anobject can do because of its position orstate. There are many examples. The workan object at height can do in falling is itsgravitational potential energy. The ENERGY

‘stored’ in elastic or a spring under tensionor compression is elastic potential energy.Potential difference in electricity is a simi-lar concept, and so on. In practice the po-tential energy of a system is the energyinvolved in bringing it to its current statefrom some reference state; i.e. it is the sameas the work that the system could do inmoving from its current state back to a ref-erence state.

potential theory The branch of mathe-matical physics that analyzes fields, such asgravitational, magnetic and electric fields,in terms of the potentials of these fields.The main equations used in potential the-ory are the POISSON EQUATION and the re-lated LAPLACE EQUATION. The developmentof potential theory had a considerable in-fluence on the development of the theory of

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potential theory

differential equations and the CALCULUS OF

VARIATIONS. Potential theory is also relatedto VECTOR CALCULUS.

pound A unit of mass now defined as0.453 592 37 kg.

poundal Symbol: pdl The unit of force inthe f.p.s. system. It is equal to 0.138 255newton (0.138 255 N).

power 1. The number of times a quantityis to be multiplied by itself. For example, 24

= 2 × 2 × 2 × 2 = 16 is known as the fourthpower of two, or two to the power four.See also exponent; power series.2. Symbol: P The rate of energy transfer (orwork done) by or to a system. The SI unitof power is the watt – the energy transfer injoules per second.

power series A series in which the termscontain regularly increasing powers of avariable. For example,

Sn = 1 + 2x + 3x2 + 4x3 +… + nxn–1

is a power series in the variable x. In gen-eral, a power series has the form

a0 + a1x + a2x2 + … + anxn

where a0, a1, etc. are constants.

P-problem A type of problem in whichthe size of the problem is characterized bysome number n and the number of stepsthat an algorithm would need to solve thisproblem is N, where n and N are related byN ≤ cnp, where c and p are constants. Suchproblems are called P-problems becausethe time it takes to solve them increases asa polynomial of n. An example of a P-prob-lem is the problem of multiplying twonumbers together. P-problems are easier tosolve than NP-PROBLEMs.

precession /pri-sesh-ŏn/ If an object isspinning on an axis and a force is appliedat right angles to this axis, then the axis ofrotation can itself move around anotheraxis at an angle to it. The effect is seen intops and gyroscopes, which ‘wobble’slowly while they spin as a result of theforce of gravity. The Earth also precesses –the axis of rotation slowly describes a cone.

The precession of Mercury is a movementof the whole orbit of the planet around anaxis perpendicular to the orbital plane.Relativistic mechanics needs to be used todescribe this precession quantitatively.

precision The number of figures in anumber. For example 2.342 is stated to aprecision of four significant figures, orthree decimal places. The precision of anumber normally reflects the ACCURACY ofthe value it represents.

premiss In LOGIC, an initial proposition orstatement that is known or assumed to betrue and on which a logical argument isbased.

premium 1. The difference between theissue price of a stock or share and its nom-inal value when the issue price is in excessof the nominal value. Compare discount.2. The amount of money paid each year toan insurance company to purchase insur-ance cover for a specified risk.

pressure Symbol: p The pressure on a sur-face due to forces from another surface orfrom a fluid is the force acting at right an-gles to unit area of the surface:

pressure = force/area.The unit is the pascal (Pa), equal to thenewton per square meter.

Objects are often designed to maximizeor minimize pressure applied. To give max-imum pressure, a small contact area isneeded – as with drawing pins and knives.To give minimum pressure, a large contactarea is needed – as with snowshoes and thelarge tires of certain vehicles.

Where the pressure on a surface iscaused by the particles of a fluid (liquid orgas), it is not always easy to find the forceon unit area. The pressure at a given depthin a fluid is the product of the depth, theaverage fluid density, and g (the accelera-tion of free fall):

pressure in a fluid = depth × meandensity × g

As it is normally possible to measurethe mean density of a liquid only, this rela-tion is usually restricted to liquids.

pound

170

The pressure at a point at a certaindepth in a fluid: 1. is the same in all directions;2. applies force at 90° to any contact sur-face;3. does not depend on the shape of the con-tainer.

pressure of the atmosphere The pres-sure at a point near the Earth’s surface dueto the weight of air above that point. Itsvalue varies around about 100 kPa(100 000 newtons per square meter).

prime A prime number is a positive inte-ger which is not 1 and has no factors ex-cept 1 and itself. The set of prime numbersis 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ….There is an infinite number of prime num-bers but no general formula for them. Theprime factors of a number are the primenumbers that divide into it exactly. For ex-ample, the prime factors of 45 are 3 and 5since 45 = 3 × 3 × 5. Each whole numberhas a unique set of prime factors. See Er-atosthenes, sieve of.

prime number theorem A result con-cerning the proportion of numbers that areprime numbers that is valid for large num-bers. It states that if π(x) is the number ofprime numbers less than or equal to x thenas x → ∞, [π(x)lnx]/x → 1. This means thatas x becomes a large number it becomes agood approximation that π(x) is given bythe ratio x/lnx, i.e. the ratio of the numberto its natural logarithm. This result, whichwas first proposed in the late 19th century,has not been given a simple proof in termsof elementary mathematics.

primitive A particular type of n-th root ofunity in which all the n roots of unity arepowers of that n-th root. For example, inthe case of the fourth roots of unity i and –iare primitive fourth roots of unity but theother two fourth roots of unity, 1 and –1,are not primitive.

principal A sum of money that is bor-rowed, on which interest is charged. Seecompound interest; simple interest.

principal diagonal See square matrix.

principle of equivalence See relativity;theory of.

principle of moments The principle thatwhen an object or system is in equilibriumthe sum of the moments in any directionequals the sum of the moments in the op-posite direction. Because there is no resul-tant turning force, the moments of theforces can be measured relative to anypoint in the system or outside it.

principle of superposition The resultfor a system described by a linear equationthat if f and g are both solutions to theequation then f + g is also a solution. Thisprinciple applies to more than two solu-tions. An example of the principle of su-perposition is given in the theory of small(harmonic) oscillations about an equilib-rium point. In the case of a vibrating stringthe behavior of the string can be describedby adding together all the harmonic modesof oscillation, with there being no inter-ference between these modes. The princi-ple works in this example because theequation for small harmonic oscillations islinear. However, the principle of super-position does not apply to NON-LINEAR OS-CILLATIONS or, more generally, systemsgoverned by non-linear equations. The in-applicability of the principle of superposi-tion in such cases is one of the mainreasons why analyzing such systems ismuch more difficult than for systems de-scribed by linear equations.

The principle of superposition also ap-plies to states in quantum-mechanical sys-tems.

printer A computer output device thatproduces hard copy as a printout. Thereare various types, including daisy-wheel,dot-matrix, ink-jet, and laser.

printout The computer output, in theform of characters printed on a continuoussheet of paper, produced by a printer.

prism /priz-ăm/ A polyhedron with twoparallel opposite faces, called bases, that

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prism

are congruent polygons. All the otherfaces, called lateral faces, are parallelo-grams formed by the straight parallel linesbetween corresponding vertices of thebases. If the bases have a center, the linejoining the centers is the axis of the prism.If the axis is at right angles to the base, theprism is a right prism (in which case the lat-eral faces are rectangles); otherwise it is anoblique prism. A triangular prism has tri-angular bases and three lateral faces. Thisis the shape of many of the glass prismsused in optical instruments. A quadran-gular prism has a quadrilateral base andfour lateral faces. The cube is a special caseof this with square bases and square lateralfaces.

probability The likelihood of a givenevent occurring. If an experiment has npossible and equally likely outcomes, m ofwhich are event A, then the probability ofA is P(A) = m/n. For example, if A is aneven number coming up when a die isthrown, then P(A) = 3/6. When the proba-bilities of the different possible results arenot already known, and event A has oc-curred m times in n trials, P(A) is defined asthe limit of m/n as n becomes infinitelylarge.

In set theory, if S is a set of events(called the sample space) and A and B areevents in S (i.e. subsets of S), the probabil-ity function P can be represented in set no-tation. P(A) = 1 and P(0) = 0 mean that Ais 100% certain and the probability ofnone of the events in S occurring is zero.

Here, 0 ≤ P(A) ≤ 1 for all A in S. If A and Bare separate independent events, i.e., ifA∩B = 0, then P(A∪B) = P(A) + P(B). IfA∩B ≠ 0 then P(A∪B) = P(A) + P(B) –P(A∩B).

The conditional probability is the prob-ability that A occurs when it is known thatB has occurred. It is written as

P(A|B) = P(A∩B)/P(B)If A and B are independent events, P (A|B)= P(A) and P(A∩B) = P(A) P(B). If A and Bcannot occur simultaneously, i.e. are mutu-ally exclusive events, P(A∩B) = 0.

probability density function See ran-dom variable.

probability function See probability.

procedure See subroutine.

processor /pros-ess-er/ See central proces-sor.

product The result obtained by multipli-cation of numbers, vectors, matrices, etc.See also Cartesian product.

product formulae See addition formu-lae.

product notation Symbol Π. The symbol(a capital Greek letter pi) used to denotethe product of either a finite number ofterms in the sequence a1, a2, ..., an or an in-finite number of terms in the sequence a1,a2, ....

probability

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Prism: examples of prisms

right triangular prism oblique triangular prism right hexagonal prism

program A complete set of instructions toa computer, written in a programming lan-guage. (The word is also used as a verb,meaning to write such instructions.) Theseinstructions, together with the facts (usu-ally called data) on which the instructionsoperate, enable the computer to perform awide variety of tasks. For example, thereare instructions to do arithmetic, to movedata between the main store and the cen-tral processor of the computer, to performlogical operations, and to alter the flow ofcontrol in the program.

The instructions and data must be ex-pressed in such a way that the centralprocessor can recognize and interpret theinstructions and cause them to be carriedout on the right data. They must in fact bein binary form, i.e. in a code consisting ofthe binary digits 0 and 1 (bits). This binarycode is known as machine code (or ma-chine language). Each type of computerhas its own machine code.

It is difficult and time-consuming forpeople to write programs in machine code.Instead programs are usually written in asource language, and these source pro-grams are then translated into machinecode. Most source programs are written ina high-level language and are convertedinto machine code by a complicated pro-gram called a compiler. High-level lan-guages are closer to natural language andmathematical notation than to machinecode, with the instructions taking the formof statements. They are fairly easy to use.They are designed to solve particular sortsof problems and are therefore described as‘problem-orientated’.

It is also possible to write source pro-grams in a low-level language. These lan-guages resemble machine code moreclosely than natural language. They are de-signed for particular computers and arethus described as ‘machine-orientated’. As-sembly languages are low-level languages.A program written in an assembly lan-guage is converted into machine code bymeans of a special program known as anassembler. See also routine; software; sub-routine.

programming language See program.

progression See sequence.

progressive wave See wave.

projectile An object falling freely in agravitational field, having been projectedat a speed v and at an angle of elevation θto the horizontal. In the special case that θ= 90°, the motion is linear in the vertical di-rection. It may then be treated using theequations of motion. In all other cases thevertical and horizontal components of ve-locity must be treated separately. In the ab-sence of friction, the horizontal componentis constant and the vertical motion may betreated using the equations of motion. Thepath of the projectile is then an arc of aparabola. Some useful relations are givenbelow.Time to reach maximum height:

t = vsinθ/gMaximum height:

h = v2sin2θ/2gHorizontal range:

R = v2sin2θ/gThe last result means that the angle formaximum horizontal range is θ = 45°. Seealso orbit.

projection A geometrical transformationin which one line, shape, etc., is convertedinto another according to certain geometri-cal rules. A set of points (the object) is con-verted into another set (the image) by theprojection. See central projection; Merca-tor’s projection; orthogonal projection;stereographic projection.

projective geometry /prŏ-jek-tiv/ Thestudy of how the geometric properties of afigure are altered by projection. There is aone-to-one correspondence between pointsin a figure and points in its projectedimage, but often the ratios of lengths willbe changed. In central projection for exam-ple, a triangle maps into a triangle and aquadrilateral into a quadrilateral, but thesides and angles may change. See also pro-jection.

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projective geometry

prolate /proh-layt/ Denoting a spheroidthat has a polar diameter that is greaterthan the equatorial diameter. Compareoblate. See ellipsoid.

proof A logical argument showing that astatement, proposition, or mathematicalformula is true. A proof consists of a set ofbasic assumptions, called axioms or pre-misses, that are combined according to log-ical rules, to derive as a conclusion theformula that is being proved. A proof of aproposition or formula P is just a valid ar-gument from true premises to give P as aconclusion. See also direct proof; indirectproof.

proof by contradiction See reductio adabsurdum.

proper fraction See fraction.

proper subset A subset S of a set T is aproper subset if there are elements of T thatare not in S, i.e. S has fewer elements thanT (if T is a finite set).

proportion The relation between two setsof numbers when the ratio between theircorresponding members is constant. Forexample, the sets 2, 6, 7, 11 and 6, 18,21, 33 are in proportion (because 2:6 =6:18 = 7:21 = 11:33). See also propor-tional.

prolate

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central projection of a triangle ABC ontoa triangle A’B’C’

orthogonal projection of a triangle ABC ontoa triangle A’B’C’

center of projection

A C

B

A’C’

B ’imageplane

image figure

A

B

C

A’C’

B’

Projection from one plane onto another

proportional Symbol: ∝ Varying in aconstant ratio to another quantity. For ex-ample, if the length l of a metal bar in-creases by 1 millimeter for every 10°C risein its temperature T, then the length is pro-portional to temperature and the constantof proportionality k is 1/10 millimeter perdegree Celsius; l = l0 + kT, where l0 is theinitial length. If two quantities a and b aredirectly proportional then a/b = k, where kis a constant. If they are inversely propor-tional then their product is a constant; i.e.ab = k, or a = k/b.

proposition A sentence or formula in alogical argument. A proposition can have atruth value; that is, it can be either true or

false but not both. Any logical argumentconsists of a succession of propositionslinked by logical operations with a propo-sition as conclusion.

Propositions may be simple or com-pound. A compound proposition is onethat is made up of more than one proposi-tion. For example, a proposition P mightconsist of the constituent parts ‘if R, then Sor Q’; i.e. in this case P = R → (S ∨ Q). Asimple proposition is one that is not com-pound. See also logic; symbolic logic.

propositional calculus See symboliclogic.

propositional logic See symbolic logic.

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propositional logic

The image P’ of a point P in Mercator’sprojection.

N

P’k log (tan ø/2)

k

S

x

S

N

P(,ø)

Stereographic projection of a point P on thesurface of a sphere onto a plane perpen-dicular to the line joining the poles N and S.The image of P is P’.

Mercator’sprojection ofa point Ponto a pointP’ in an imageplane or map.

S

x

N

P

P’

Projection from a sphere onto a plane

protractor A drawing instrument usedfor marking out or measuring angles. Itusually consists of a semicircular piece oftransparent plastic sheet, marked with ra-dial lines at one-degree intervals.

pseudoscalar /soo-doh-skay-ler/ A quan-tity that is similar to a SCALAR quantity butchanges sign on going from a right-handedcoordinate system to a left-handed coordi-nate system or if the order of the vectors isinterchanged. An example of a pseu-doscalar is the quantity formed in theTRIPLE SCALAR PRODUCT. The dot product ofany AXIAL VECTOR and a POLAR VECTOR

(true vector) results in a pseudoscalar. Bycontrast, the dot product of two polar vec-tors or two axial vectors results in a truescalar, i.e. a scalar that does not change itssign on going from a right-handed coordi-nate system to a left-handed coordinatesystem.

pseudovector / soo-doh-vek-ter/ See axialvector.

Ptolemy’s theorem /tol-ĕ-meez/ Seecyclic polygon.

pulley A class of machine. In any pulleysystem power is transferred through thetension in a string wound over one or morewheels. The force ratio and distance ratio

depend on the relative arrangement ofstrings and wheels. The efficiency is notusually very high as work must be done toovercome friction in the strings and thewheel bearings and to lift any movingwheels. See machine.

pulsatance /pul-să-tăns/ See angular fre-quency.

pulse modulation See modulation.

punched card See card.

pure mathematics The study of mathe-matical theory and structures, without nec-essarily having an immediate application inmind. For example, the study of the generalproperties of vectors, considered purely asentities with certain properties, could beconsidered as a branch of pure mathemat-ics. The use of vector algebra in mechanicsto solve a problem on forces or relative ve-locity is a branch of applied mathematics.Pure mathematics, then, deals with ab-stract entities, without any necessary refer-ence to physical applications in the ‘realworld’.

pyramid A solid figure in which one ofthe faces, the base, is a polygon and theothers are triangles with the same vertex. Ifthe base has a center of symmetry, a line

protractor

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apex

triangular base

apex

square base

Pyramids

from the vertex to the center is the axis ofthe pyramid. If this axis is at right angles tothe base the pyramid is a right pyramid;otherwise it is an oblique pyramid. A regu-lar pyramid is one in which the base is aregular polygon and the axis is at right an-gles to the base. In a regular pyramid all thelateral faces are congruent isosceles trian-gles making the same angle with the base.A square pyramid has a square base andfour congruent triangular faces. The vol-ume of a pyramid is one third of the area ofthe base multiplied by the perpendiculardistance from the vertex to the base.

Pythagoras’ theorem /pÿ-thag-ŏ-răs-iz/A relationship between the lengths of thesides in a right-angled triangle. The squareof the hypotenuse (the side opposite theright angle) is equal to the sum of the

squares of the other two sides. There aremany proofs of Pythagoras’ theorem. It isnamed for the Greek mathematician andphilosopher Pythagoras (c. 580 BC–c. 500BC).

Pythagorean triple /pÿ-thag-ŏ-ree-ăn/ Aset of three positive integers a, b, c that cancorrespond to the lengths of sides in right-angled triangles and hence, by PYTHAGO-RAS’ THEOREM satisfy the relation a2 + b2 =c2. The most familiar example of aPythagorean triple is (3, 4, 5). Other exam-ples of Pythagorean triples include (5, 12,13), (7, 24, 25) and (8, 15, 17). It was es-tablished by the ancient Greeks that an in-finite number of Pythagorean triples exist.If (a, b, c) is a Pythagorean triple then (ma,mb, mc), where m is a positive integer isalso a Pythagorean triple.

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Pythagorean triple

ccc

bbb

aaa

Pythagoras’ theorem: c2 = a2 + b2.

quadrangular prism /kwod-rang-gyŭ-ler/See prism.

quadrant /kwod-rănt/ 1. One of four divi-sions of a plane. In rectangular CARTESIAN

COORDINATES, the first quadrant is the areato the right of the y-axis and above the x-axis, that is, where both x and y arepositive. The second quadrant is the area to the left of the y-axis and above the x-axis, where x is negative and y is posi-tive. The third quadrant is below the x-axis and to the left of the y-axis, whereboth x and y are negative. The fourthquadrant is below the x-axis and to theright of the y-axis, where x is positive andy is negative. In polar coordinates, the first,second, third, and fourth quadrants occurwhen the direction angle, θ, is 0 to 90°(0 to π/2); 90° to 180° (π/2 to π); 180° to270° (π to 3π/2); and 270° to 360° (3π/2 to2π), respectively. See also polar coordi-nates.2. A quarter of a circle, bounded by two

perpendicular radii and a quarter of the cir-cumference.3. A unit of plane angle equal to 90 degrees(π/2 radians). A quadrant is a right angle.

quadrantal spherical triangle /kwod-ran-tăl/ See spherical triangle.

quadratic equation /kwod-rat-ik/ Apolynomial equation in which the highestpower of the unknown variable is two. Thegeneral form of a quadratic equation in thevariable x is

ax2 + bx + c = 0where a, b, and c are constants. It is alsosometimes written in the reduced form

x2 + bx/a + c/a = 0In general, there are two values of x thatsatisfy the equation. These solutions (orroots), are given by the formula

x = [–b ± √(b2 – 4ac)]/2aThe quantity b2 – 4ac is called the DISCRIM-INANT. If it is a positive number, there aretwo real roots. If it is zero, there are two

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Q

y

x

1stquadrant

4thquadrant

3rdquadrant

2ndquadrant

O•

Quadrant: the four quadrants of a Cartesian coordinate system.

equal roots. If it is negative there are noreal roots. The Cartesian coordinate graphof a quadratic function

y = ax2 + bx + cis a parabola and the points where itcrosses the x-axis are solutions to

ax2 + bx + c = 0If it crosses the axis twice there are two realroots, if it touches the axis at a turningpoint the roots are equal, and if it does notcross it at all there are no real roots. In thislast case, where the discriminant is nega-tive, the roots are two conjugate complexnumbers.

quadratic graph A graph of the curve y =ax2 + bx + c, where a, b, and c are con-stants. If a is a positive number then thegraph has a minimum when x = –b/2a. Ithas a maximum when x = –b/2a if a is anegative number. There is only one sta-tionary point in a quadratic graph. Thiscan either be a minimum or a maximum, asspecified. See also quadratic equation.

quadric cone /kwod-rik/ A type ofQUADRIC SURFACE that is described by theequation:

x2/a2 + y2/b2 = z2/c2,using a particular coordinate system. If across-section of this surface is taken paral-lel to the xy-plane an ellipse results in gen-eral. In the specific case of a = b, a circleresults. Cross-sections parallel to theplanes of the other axes result in hyperbo-las.

quadric surface See conicoid.

quadrilateral /kwod-ră-lat-ĕ-răl/ A planefigure with four straight sides. For exam-ple, squares, kites, rhombuses, and trapez-iums are all quadrilaterals. A square is aregular quadrilateral.

quantifier /kwon-tă-fÿ-er/ See existentialquantifier; universal quantifier.

quantum computer A computer that op-erates by using the principles of QUANTUM

MECHANICS. In particular, a quantum com-puter could process alternative pathwayssimultaneously. It is possible that for manyproblems, such as factorizing large inte-gers, a quantum computer could performcalculations much more quickly than aconventional computer. At the time ofwriting, quantum computers have not beendeveloped to the point of being technolog-ically useful. Several different quantummechanical systems have been proposedand used as quantum computers but theyall have practical difficulties such as re-quiring very low temperatures for their op-eration.

quantum mechanics The theory thatgoverns the behavior of particles at theatomic and sub-atomic level. It bears thesame relationship to Newtonian mechanicsas wave optics does to geometrical optics.The development of quantum mechanicshas had a substantial influence on several

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quantum mechanics

squarerectangle

kite

rhombus

quadrilateral

parallelogram

Quadrilateral: six types of quadrilateral.

branches of mathematics, notably the the-ory of operators and group theory. Quan-tum mechanics can be formulatedmathematically in several ways includingthe use of differential equations, matrices,variational principles, and informationtheory. Characteristic features of quantummechanics are that many quantities have aset of discrete values rather than a range ofcontinuous values and that particles suchas electrons cannot be ascribed preciselydefined trajectories like the orbits of plan-ets. A further feature is the existence ofnon-commutative operators, i.e. pairs ofoperators in which the final result of theoperations depends on the order in whichthe operators operate. At the time of writ-ing, there is a great deal of activity in de-veloping quantum, i.e. discrete andnon-commutative, versions of mathematicswhich have been previously formulated forcontinuous, commutative quantities. Someof these branches of ‘quantum mathemat-ics’ may well turn out to be important inphysics.

quart A unit of capacity equal to twopints. In the USA a dry quart is equal totwo US dry pints.

quartic equation /kwor-tik/ A polyno-mial equation in which the highest powerof the unknown variable is four. The gen-eral form of a quartic equation in a vari-able x is

ax4 + bx3 + cx2 + dx + e = 0where a, b, c, d, and e are constants. It isalso sometimes written in the reduced form

x4 + bx3/a + cx2/a + dx/a + e/a = 0In general, there are four values of x thatsatisfy a quartic equation. For example,

2x4 – 9x3 + 4x2 + 21x – 18 = 0can be factorized to

(2x + 3)(x – 1)(x – 2)(x – 3) = 0and its solutions (or roots) are –3/2, 1, 2,and 3. On a Cartesian coordinate graph,the curve

y = 2x4 – 9x3 + 4x2 + 21x – 18 = 0crosses the x-axis at x = –3/2; x = 1; x = 2;and x = 3. Compare cubic equation; qua-dratic equation.

quartile One of the three points that di-vide a set of data arranged in numericalorder into four equal parts. The lower (orfirst) quartile, Q1, is the 25th PERCENTILE

(P25). The middle (or second) quartile, Q2,is the MEDIAN (P50). The upper (or third)quartile, Q3, is the 75th percentile (P75).

quasicrystalline symmetry /kway-sÿ-kriss-tă-lin, -lÿn, kway-zÿ-, kwah-see-,kwah-zee-/ The symmetry that is the three-dimensional analog of a PENROSE PATTERN,i.e. there is order but not the periodicity ofcrystals. Quasicrystalline symmetry is real-ized in certain solids, called quasicrystals,such as AlMn. This symmetry is made ap-parent by the fact that quasicrystals giveclear x-ray diffraction images, characteris-tic of there being order. Quasicrystals suchas AlMn have the point group symmetry ofan ICOSAHEDRON, a type of point groupsymmetry that is incompatible with crys-talline symmetry.

quaternions /kwă-ter-nee-ŏnz/ General-ized complex numbers invented by Hamil-ton. A quaternion is of the form a + bi + cj+ dk, where i2 = j2 = k2 = –1 and ij = –ji = kand a, b, c, d are real numbers. The moststriking feature of quaternions is that mul-tiplication is not commutative. They haveapplications in the study of the rotations ofrigid bodies in space.

quintic equation /kwin-tik/ A polynomialequation in which the highest power of theunknown variable is five. The general formof a quintic equation in a variable x is:

ax5 + bx4 + cx3 + dx2 + ex + f = 0where a, b, c, d, e, and f are constants. It isalso sometimes written in the reduced form

x5 + bx4/a + cx3/a + dx2/a+ ex/a + f/a = 0

In general, there are five values of x thatsatisfy a quintic equation. For example,

2x5 – 17x4 + 40x3 + 5x2 – 102x + 72 =0

can be factorized to(2x + 3)(x – 1)(x – 2)(x – 3)(x – 4) = 0

and its solutions (or roots) are –3/2, 1, 2, 3,and 4. On a Cartesian coordinate graph,the curve

y = 2x5 – 17x4 + 40x3 +

quart

180

5x2 – 102x + 72crosses the x-axis at x = –3/2; x = 1; x = 2;x = 3; and x = 4. Compare cubic equation;quadratic equation; quartic equation.

quotient /kwoh-shĕnt/ The result of divid-ing one number by another. There may or

may not be a remainder. For example, 16/3gives a quotient of 5 and a remainder of 1.

qwerty /kwer-tee/ Describing the standardlayout of ALPHANUMERIC characters on atypewriter or computer keyboard (namedfor the first six letters of the top letter row).

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qwerty

radial /ray-dee-ăl/ Along the direction ofthe radius.

radial symmetry Symmetry about a lineor plane through the center of a shape. Seealso bilateral symmetry; symmetrical.

radian Symbol: rad The SI unit for mea-suring plane angle. It is the angle subtendedat the center of a circle by an arc equal inlength to the radius of the circle. π radians= 180°.

radical /rad-ă-kăl/ An expression for aroot. For example, √2, where √ is the radi-cal sign.

radical axis The radical axis of two cir-cles

x2 + y2 + 2a1x + 2b1y + c1 = 0and

x2 + y2 + 2a2x + 2b2y + c2 = 0is the straight line obtained by eliminatingthe square terms between the equations ofthe circles, i.e.

2(a1 – a2)x + 2(b1 – b2)y +(c1 – c2) = 0

When the circles intersect, the radical axispasses through their two points of intersec-tion.

radius /ray-dee-ŭs/ (pl. radii or radiuses)The distance from the center of a circle toany point on its circumference or from thecenter of a sphere to its surface. In polarcoordinates, a radius r (distance from afixed origin) is used with angular positionθ to specify the positions of points.

radius of convergence For a power se-ries a0 + a1(x – a) + a2(x – a)2 + … + an(x –a)n + … there is a value R such that the

series converges for values of |x − a| < R.Here, R is the radius of convergence of thepower series.

radius of curvature See curvature.

radius of gyration Symbol: k For a bodyof mass m and moment of inertia I aboutan axis, the radius of gyration about thataxis is given by

k2 = I/mIn other words, a point mass m rotating ata distance k from the axis would have thesame moment of inertia as the body.

radius vector The vector that representsthe distance and direction of a point fromthe origin in a polar coordinate system.

radix /ray-diks/ (pl. radices or radixes)The base number of any counting system,also known as the base. For example, thedecimal system has the radix 10, whereasthe radix of the binomial system is 2. Seealso base.

RAM Random-access memory. See ran-dom access.

random access A method of organizinginformation in a computer storage deviceso that one piece of information may bereached directly in about the same time asany other. Main store and disk units oper-ate by random access and are thus knownas random-access memory (RAM). In con-trast a magnetic tape unit operates moreslowly by serial access: a particular piece ofinformation can only be retrieved by work-ing through the preceding blocks of dataon the tape.

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R

random error See error.

randomness The lack of a pattern ororder in either a physical system or a set ofnumbers. Precise mathematical analysis ofthe concept of randomness is closely asso-ciated with CHAOS THEORY, ENTROPY, andINFORMATION THEORY.

random number table A table consistingof a sequence of randomly chosen digitsfrom 0 to 9, where each digit has a proba-bility of 0.1 of appearing in a particular po-sition, and choices for different positionsare independent. Random numbers areused in statistical random sampling.

random sampling See sampling.

random variable (chance variable; sto-chastic variable) A quantity that can takeany one of a number of unpredicted values.A discrete random variable, X, has a defi-nite set of possible values x1, x2, x3, … xn,with corresponding probabilities p1, p2, p3,… pn. Since X must take one of the valuesin this set,

p1 + p2 + … + pn = 1If X is a continuous random variable, it

can take any value over a continuousrange. The probabilities of a particularvalue x occurring is given by a probabilitydensity function f(x). On a graph of f(x)against x, the area under the curve betweentwo values a and b is the probability that Xlies between a and b. The total area underthe curve is 1.

random walk A succession of move-ments along line segments where the direc-tion and the length of each move israndomly determined. The problem is todetermine the probable location of a pointsubject to such random motions given theprobability of moving some distance insome direction, where the probabilities arethe same at each step. Random walks canbe used to obtain probability distributionsto practical problems. Consider, for exam-ple, a drunk man moving a distance of oneunit in unit time, the direction of motionbeing random at each step. The problem isto find the probability distribution of the

distance of the point from the startingpoint after some fixed time. Technically arandom walk is a sequence

Sn = X1 + X2 + … + Xnwhere Xi is a sequence of independentrandom variables.

range 1. The difference between thelargest and smallest values in a set of data.It is a measure of dispersion. In terms ofpercentiles, the range is (P100 – P0). Com-pare interquartile range.2. A set of numbers or quantities that formpossible results of a mapping. In algebra,the range of a function f(x) is the set of val-ues that f(x) can take for all possible valuesof x. For example, if f(x) is taking thesquare root of positive rational numbers,then the range would be the set of realnumbers. See also domain.

rank A method of ordering a set of objectsaccording to the magnitude or importanceof a variable measured on them, e.g.arranging ten people in order of height. Ifthe objects are ranked using two differentvariables, the degree of association be-tween the two rankings is given by the co-efficient of rank correlation. See alsoKendall’s method.

raster graphics See computer graphics.

rate of change The rate at which onequantity y changes with respect to anotherquantity x. If y is written as y = f(x) and iff(x) is a differentiable function, the rate ofchange of y with respect to x is describedby differential calculus. This rate of changeis denoted by dy/dx or f′(x). The value ofthe rate of change varies with x since dy/dxis a function of x. The value of dy/dx at aspecific value of x is found by expressingdy/dx as a function of x and substitutingthe value of x into the function for the rateof change.

Frequently in physical systems the rateof change of a function with respect to timeis of interest. For example, the velocity of abody is the rate of change of the displace-ment of the body with respect to time andthe acceleration of a body is the rate of

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rate of change

change of the velocity of the body with re-spect to time.

ratio /ray-shee-oh/ One number or quan-tity divided by another. The ratio of twovariable quantities x and y, written as x/yor x:y, is constant if y is proportional to x.See also fraction.

rational function A REAL FUNCTION, f(x),of a real variable x in some domain, whichis frequently taken to be the set of all realnumbers, that can be expressed as the ratioof two polynomial functions g(x) and h(x).In this definition it is assumed that the twopolynomials do not have a common factorwith a degree that is greater than or equalto 1. A rational function defined in thisway is continuous except for values of x forwhich the value of the denominator h(x) iszero. The concept of a rational functioncan be extended to a function, f(z), of acomplex variable z.

rationalize /rash-ŏ-nă-lÿz/ To removeradicals (such as square root, or √) from analgebraic expression without changing itsvalue, thus making the equation easier todeal with. For example, by squaring bothsides, the equation √(2x + 3) = x rational-izes to 2x + 3 = x2, equivalent to x2 – 2x –3 = 0.

rationalized units A system of units inwhich the equations have a logical form re-lated to the shape of the system. SI unitsform a rationalized system of units. For ex-ample, in it formulae concerned with circu-lar symmetry contain a factor of 2π; thoseconcerned with radial symmetry contain afactor of 4π.

rational numbers /rash-ŏ-năl/ Symbol: QThe set of numbers that includes integersand fractions. Rational numbers can bewritten down exactly as ratios or as finiteor repeating decimals. For example, ⅓ (=0.333…) and ¼ (= 0.25) are rational. Thesquare root of 2 (= 1.414 213 6…) is not.Compare irrational numbers.

ratio theorem A result concerning vec-tors stating that if two points A and B are

specified by position vectors a and b, rela-tive to some origin O, and the point C di-vides the line AB in the ratio l:m then theposition vector c for C is given by: c = (ma+ lb)/(l + m). By defining l and m so that l +m = 1, the expression c = ma + (1 – m)b isobtained.

ray A set consisting of all the points on agiven line to the left or to the right of agiven point, and including that point itself.

reaction Newton’s third law of forcestates that whenever object A applies aforce on object B, B applies the same forceon A. An old word for force is ‘action’; ‘re-action’ is thus the other member of thepair. Thus in the interaction between twoelectric charges, each exerts a force on theother. Thus, in general, action and reactionhave little meaning. The word ‘reaction’ isstill sometimes used in restricted cases,such as the reaction of a support on the ob-ject it supports. In this case the ‘action’ isthe effect of the weight of the object on thesupport.

reader A device used in a computer sys-tem to sense the information recorded onsome source and convert it into anotherform.

read-only memory (ROM) See store.

read–write head See disk; drum; mag-netic tape.

real analysis The branch of analysis con-cerned with REAL FUNCTIONS.

real axis An axis of the complex plane onwhich the points represent the real num-bers. It is customary to draw the real axisas the x-axis of the complex plane.

real function A function f that mapsmembers of the set R of real numbers (or asubset of R) to members of R. This meansthat if x is a real number in R or its subsetthen f(x) is also a real number.

real line The line that represents the realnumbers on the complex plane. The real

ratio

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line is drawn as a horizontal line. A specificpoint O is chosen to be the origin and an-other specific point A, which is to the rightof O on the horizontal line, is chosen sothat the length of OA represents one unit.All positive real numbers are representedby points that lie to the right of O on thereal line and all negative real numbers arerepresented by points that lie to the left ofO, with the origin being the real numberwith value zero.

real numbers Symbol: R The set of num-bers that includes all rational and irra-tional numbers.

real part Symbol Rez. The part x of acomplex number z defined by z = x + iy,where x and y are both real numbers. Seealso imaginary part.

real time The actual time in which a phys-ical process takes place or in which a phys-ical process, machine, etc., is under thedirect control of a computer. A real-timesystem is able to react sufficiently rapidlyso that it may control a continuing process,making changes or modifications whennecessary. Air-traffic control and airlinereservations require real-time systems.Compare batch processing. See also timesharing.

reciprocal /ri-sip-rŏ-kăl/ The number 1divided by a quantity. For example, thereciprocal of 2 is ½. The reciprocal of (x2 +1) is 1/(x2 + 1). The product of any expres-sion and its reciprocal is 1. For any func-tion, the reciprocal is the multiplicativeinverse.

rectangle /rek-tang-găl/ A plane figurewith four straight sides, two parallel pairsof equal length forming four right angles.The area of a rectangle is the product of thetwo different side lengths, the length timesthe breadth. A rectangle has two axes ofsymmetry, the two lines joining the mid-points of opposite sides. It can also be su-perimposed on itself after rotation through180° (π radians). The two diagonals of arectangle have equal lengths.

rectangular hyperbola /rek-tang-gyŭ-ler/See hyperbola.

rectangular parallelepiped See paral-lelepiped.

rectilinear /rek-tă-lin-ee-er/ Describingmotion in a straight line.

recurring decimal A repeating DECIMAL.

recursion formula A formula that relatessome quantity Qn, where n is a non-nega-tive integer, to quantities such as Qn–1,Qn–2, .... Formulas of this type are used ex-tensively to calculate integrals involvingpowers of trigonometric functions. For ex-ample if In = ∫sinnxdx, there is a recursionformula for In which can be found by inte-grating In by parts. This gives: In =[(n–1)/n]In–2, where n ≥ 2. The cases of I0and I1 are readily found by direct integra-tion. The recursion formula means that Incan be determined for all even and odd val-ues of n. Another example of a recursionformula is the expression Γ(n + 1) = nΓ(n)for the GAMMA FUNCTION.

A recursion formula is sometimes calleda reduction formula.

reduced form (of a polynomial) TheEQUATION of the form

xn + (b/a)xn–1 + (c/a)xn–2 + … = 0that is derived from a POLYNOMIAL of theform

axn + bxn–1 + cxn–2 + … = 0For example,

2x2 – 10x + 12 = 0is equivalent to the reduced form

x2 – 5x + 6 = 0See also quadratic equation.

reductio ad absurdum /ri-duk-tee-oh adăb-ser-dŭm/ A method of proof which pro-ceeds by assuming the falsity of what wewish to prove and showing that it leads to a contradiction. Hence the statementwhose falsity we assumed must be true.The following proof that √2 is irrational isa simple example of proof by this method.

Assume √2 is rational. In that case it canbe expressed in the form a/b where a and bare integers. Assume that this fraction is in

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reductio ad absurdum

its lowest terms and so a and b have nocommon factor. Since a/b = √2 then a2/b2 =2. Hence a2 = 2b2. This means that a2 iseven and hence a itself is even. In that casewe can write a as 2m where m is some in-teger. But then since a2 = 2b2 we have(2m)2 = 2b2, or 4m2 = 2b2. Dividing by 2we get 2m2 = b2. But this means that b2 andhence b is also even. Hence, a and b dohave a common factor, namely 2. But weassumed they had no common factor. Sincewe have reached a contradiction, our start-ing-point – the assumption that √2 is ratio-nal – must be false.

reduction formulae In trigonometry, theequations that express sine, cosine, andtangent functions of an angle in terms of anangle between 0 and 90° (π/2). For exam-ple:

sin(90° + α) = cosαsin(180° + α) = –sinαsin(270° + α) = –cosαcos(90° + α) = –sinαtan(90° + α) = –cotα

re-entrant polygon A many-sided figure(polygon) with one of its internal angles areflex angle (i.e. greater than 180°).

reflection /ri-flek-shŏn/ The geometricaltransformation of a point or a set of pointsfrom one side of a point, line, or plane to asymmetrical position on the other side. Onreflection in a line, the image of a point P

would be point P′ at the same distancefrom the line but on the other side. Theline, the axis of reflection, is the perpendic-ular bisector of the line PP′. In a symmetri-cal plane figure there is an axis ofreflection, also called the axis of symmetry,in which the figure is reflected onto itself.An equilateral triangle, for example, hasthree axes of symmetry. In a circle, any di-ameter is an axis of symmetry. Similarly, asolid may undergo reflection in a plane. Ina sphere, any plane passing through thesphere’s center would be a plane of sym-metry.

In a Cartesian coordinate system, re-flection in the x-axis changes the sign of they-coordinate. A point (a,b) would become(a,–b). Reflection in the y-axis changes thesign of the x-coordinate, making (a,b) be-come (–a,b). In three dimensions, changingthe sign of the z-coordinate is equivalent toreflection in the plane of the x and y axes.Reflection in a point is equivalent to rota-tion through 180°. Each point P is movedto a position P′ so that the point of reflec-tion bisects the line PP′. Reflection in theorigin of plane Cartesian coordinateschanges the signs of all the coordinates. Itis equivalent to reflection in the x-axis fol-lowed by reflection in the y-axis, or viceversa. See also rotation.

reflex /ree-fleks/ Describing an angle thatis greater than 180° (but less than 360°).Compare acute; obtuse.

reduction formulae

186

y

xO

A

B

C

A´

B´

C´

Reflection in the y-axis

reflexive /ri-fleks-iv/ A relation R definedon a given set is said to be reflexive if everymember of the set has this relation to itself.Equality is an example of a reflexive rela-tion.

register See central processor.

regression line A line y = ax + b, calledthe regression line of y on x, which givesthe expected value of a random variable yconditional on the given value of a randomvariable x. The regression line of x on y is not in general the same as that of y on x.If a SCATTER DIAGRAM of data points(x1,y1), …, (xn,yn) is drawn and a linear re-lationship is shown up, the line can bedrawn by hand. The best line is drawnusing the LEAST-SQUARES METHOD. See alsocorrelation.

regular Describing a figure that has allfaces or sides of equal size and shape. Seepolygon; polyhedron.

relation A property that holds for orderedpairs of elements of some set, for examplebeing greater than. We can think of a rela-tion abstractly as the set of all orderedpairs in which the two members have thegiven relation to one another.

relative Expressed as a difference from oras a ratio to, some reference level. Relativedensity, for example, is the mass of a sub-stance per unit volume expressed as a frac-tion of a standard density, such as that ofwater at the same temperature. Compareabsolute.

relative error The ERROR or uncertaintyin a measurement expressed as a fraction ofthe measurement. For example, if, in mea-suring a length of 10 meters, the tape mea-sures only to the nearest centimeter, thenthe measurement might be written as 10 ±0.01 meters. The relative error is 0.01/10 =0.001. Compare absolute error.

relative maximum See local maximum.

relative minimum See local minimum.

relative velocity If two objects are mov-ing at velocities vA and vB in a given direc-tion the velocity of A relative to B is vA – vBin that direction. In general, if two objectsare moving in the same frame at nonrela-tivistic speeds their relative velocity is thevector difference of the two velocities.

relativistic mass /rel-ă-ti-vis-tik/ Themass of an object as measured by an ob-server at rest in a frame of reference inwhich the object is moving with a velocityv. It is given by

m = m0/√(1 – v2/c2)where m0 is the REST MASS, c is the velocityof light in free space, and m is the relativis-tic mass. The equation is a consequence ofthe special theory of relativity, and is in ex-cellent agreement with experiment. No ob-ject can travel at the speed of light in freespace because its mass would then be infi-nite. See also relativity, theory of.

relativistic mechanics A system of me-chanics based on relativity theory. See alsoclassical mechanics.

relativistic speed (relativistic velocity)Any speed (velocity) that is sufficientlyhigh to make the mass of an object signifi-cantly greater than its REST MASS. It is usu-ally expressed as a fraction of c, the speedof light in free space. At a speed of c/2 theRELATIVISTIC MASS of an object is about15% greater than the rest mass.

relativity, theory of /rel-ă-tiv-ă-tee/ Atheory put forward in two parts by AlbertEinstein. The spcecial theory (1905) re-ferred only to nonaccelerated (inertial)frames of reference. The general theory(1915) is also applicable to accelerated sys-tems.

The special theory was based on twopostulates: 1. That physical laws are the same in all in-ertial frames of reference.2. That the speed of light in a vacuum isconstant for all observers, regardless of themotion of the source or observer.The second postulate seems contrary to‘common sense’ ideas of motion. Einsteinwas led to the theory by considering the

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relativity, theory of

problem of the ‘ether’ and the relation be-tween electric and magnetic fields in rela-tive motion. The theory accounts for thenegative result observed in the MICHELSON–MORLEY EXPERIMENT and shows that theLorentz–Fitzgerald contraction is only anapparent effect of motion on an object rel-ative to an observer, not a ‘real’ contrac-tion. It leads to the result that the mass ofan object moving at a speed v relative to anobserver is given by:

m = m0/√(1 – v2/c2)where c is the speed of light and m0 themass of the object when at rest relative tothe observer. The increase in mass is signif-icant at high speeds. Another consequenceof the theory is that an object has an energycontent by virtue of its mass, and similarlythat energy has inertia. Mass and energyare related by the famous equation E =mc2.

The general theory of relativity seeks toexplain the difference between acceleratedand nonaccelerated systems and the natureof the forces acting in both of them. For ex-ample, a person in a spacecraft far out inspace would not be subject to gravitationalforces. If the craft were rotating, he wouldbe pressed against the walls of the craft andwould consider that he had weight. Therewould not be any difference between thisforce and the force of gravity. To an out-side observer the force is simply a result ofthe tendency to continue in a straight line;i.e. his inertia. This type of analysis offorces led Einstein to a principle of equiva-lence that inertial forces and gravitationalforces are equivalent, and that gravitationcan be a consequence of the geometricalproperties of space. He visualized a four-dimensional space–time continuum inwhich the presence of a mass affects thegeometry – the space-time is ‘curved’ by themass with the geometry being non-Euclid-ean.

remainder The number left when onenumber is divided into another. Dividing12 into 57 gives 4 remainder 9 (4 × 12 = 48;57 – 48 = 9).

remainder theorem The theorem ex-pressed by the equation

f(x) = (x – a)g(x) + f(a)This means that if a polynomial in x, f(x),is divided by (x – a), where a is a constant,the remainder term is equal to the value ofthe polynomial when x = a. For example, if

2x3 + 3x2 – x – 4is divided by (x – 4), then the remainderterm is

f(4) = 128 + 48 – 4 – 4 = 168The remainder theorem is useful for find-ing the factors of a polynomial. In this ex-ample,

f(1) = 2 + 3 – 1 – 4 = 0Thus, there is no remainder so (x – 1) is afactor.

repeated root A root of an equation thatoccurs more than once.

repeating decimal See decimal.

representation of a group A set of oper-ators that correspond to the elements of theGROUP. The operators are defined in a VEC-TOR SPACE V. The dimension of the rep-resentation is the dimensionality of V.Frequently, the set of operators is ex-pressed in terms of matrices, with this typeof representation being called a matrix rep-resentation of the group. Group represen-tations are of great importance in QUAN-TUM MECHANICS. In particular, irreduciblerepresentations, i.e. representations thatcannot be reduced to representations oflower dimensions, characterize quantummechanical systems since the quantumnumbers that characterize the energy levelsof a system correspond to the irreduciblerepresentations of the group that describesthe symmetry of the system. For instance,the irreducible representations of the ROTA-TION GROUP characterize the energy levelsof atoms and the irreducible representa-tions of POINT GROUPS characterize the en-ergy levels of molecules.

representative fraction A fraction usedto express the SCALE of a map in which thenumerator represents a distance on themap and the denominator represents thecorresponding distance on the ground. Asa fraction is a ratio, the units of the numer-ator and denominator must be the same.

remainder

188

For example, a scale of 1 cm = 1 km wouldbe given as a representative fraction of1/100 000, because there are 100 000 cmin 1 km.

residue /rez-ă-dew/ If there exists an xsuch that xn ≡ a(mod p), i.e. xn is congruentto a modulo p, then a is called a residue ofp of order n.

resolution of vectors The determinationof the components of a vector in two givendirections at 90°. The term is sometimesused in relation to finding any pair of com-ponents (not necessarily at 90° to eachother).

resonance The large-amplitude vibrationof an object or system when given impulsesat its natural frequency. For instance, apendulum swings with a natural frequencythat depends on its length. If it is given aperiodic ‘push’ at this frequency – for ex-ample, at each maximum of a complete os-cillation – the amplitude is increased withlittle effort. Much more effort would be re-quired to produce a swing of the same am-plitude at a different frequency.

restitution, coefficient of /res-tă-tew-shŏn/ Symbol: e For the impact of twobodies, the elasticity of the collision is mea-sured by the coefficient of restitution. It isthe relative velocity after collision dividedby the relative velocity before collision(with the velocities measured along the lineof centers). For spheres A and B:

vA′ – vB′ = e(vA – vB)v indicates velocity before collision; v′ ve-locity after collision. Kinetic energy is con-served only in a perfectly elastic collision.

rest mass Symbol: m0 The mass of an ob-ject at rest as measured by an observer atrest in the same frame of reference. See alsorelativistic mass.

resultant /ri-zul-tănt/ 1. A vector with thesame effect as a number of vectors. Thus,the resultant of a set of forces is a force thathas the same effect; it is equal in magnitudeand opposite in direction to the equilib-rium. Depending on the circumstances, the

resultant of a set of vectors can be found bydifferent methods. See parallel forces; par-allelogram of vectors; principle of mo-ments.2. See eliminant.

revolution, solid of A solid generated byrevolving a plane area about a line calledthe axis of revolution. For example, rotat-ing a rectangle about an axis joining themidpoints of two opposite sides produces acylinder as the solid of revolution.

rhombic dodecahedron /rom-bik/ Atype of polyhedron in which there are 12faces, each of which is a RHOMBUS.

rhombohedron /rom-bŏ-hee-drŏn/ Asolid figure bounded by six faces, each onea parallelogram, with opposite faces con-gruent.

rhomboid /rom-boid/ A parallelogramthat is neither a rhombus nor a rectangle.See parallelogram.

rhombus /rom-bŭs/ (pl. rhombuses orrhombi) A plane figure with four straightsides of equal length; i.e. a parallelogramwith equal sides. Its area is equal to half theproduct of the lengths of its two diagonals,which bisect each other perpendicularly.The rhombus is symmetrical about both ofits diagonals and also has rotational sym-metry, in that it can be superimposed onitself after rotation through 180° (π radi-ans).

Riemannian geometry /ree-mah-nee-ăn/A type of geometry that describes thehigher-dimensional analogs of curved sur-faces. When extended from space to space–time, Riemannian geometry can be used todescribe the effect of gravitation in thegeneral theory of relativity. This type ofgeometry is named for the German mathe-matician (Georg Friedrich) Bernhard Rie-mann (1826–66).

Riemann integral /ree-mahn/ See definiteintegral; Riemann sum.

Riemann sum The series that approxi-

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Riemann sum

mates the area between the curve of a func-tion f(x) and the x-axis:

i = 1∑n

f(ξi)∆xi

where ∆x is an increment of x, ξi is anyvalue of f(x) within that interval, and n isthe number of intervals. The definite (orRiemann) integral is the limit of the sum asn becomes infinitely large and ∆x infinites-imally small.

Riemann zeta function /zay-tă/ A func-tion of a complex variable z = x + iy, wherex and y are real numbers, defined by the in-finite series:

ζ(z) = n = 1∑∞

n–z

The Riemann zeta function can also be de-fined for real numbers. There is a conjec-ture concerning the Riemann zeta functionfor a complex variable z that has remainedunproved since Riemann postulated it inthe nineteenth century. If Riemann’s con-jecture is correct it would establish a firmresult about the distribution of prime num-bers.

right angle An angle that is 90° or π/2 ra-dians. It is the angle between two lines orplanes that are perpendicular to eachother. The corner of a square, for example,is a right angle.

right-handed system A way of specify-

ing a coordinate system such that rotatingthe x-axis by 90° into the y-axis corre-sponds to rotating a right-handed screw inthe positive z direction.

When discussing VECTOR PRODUCTS it isnecessary to specify whether a right-handed or left-handed coordinate system isbeing used.

right solid A solid geometrical figure thatis upright; for example, a cone, cylinder,pyrimid, or prism that has an axis at rightangles to the base. Compare oblique solid.

rigid body In mechanics, a body forwhich any change of shape produced byforces on the body can be neglected in thecalculations.

ring A set of entities with two binary op-erations called addition and multiplicationand denoted by + and • respectively, suchthat: 1. the set is a commutative group under ad-dition;2. for every pair of elements a,b, in the ringthe product a•b is unique, multiplication isassociative, i.e. (a•b)•c = a•(b•c), and mul-tiplication is distributive with respect toaddition, i.e. a•(b + c) = a•b + a•c and (b +c)•a = b•a + c•a for each a, b, and c in theset.

If multiplication is also commutative,the ring is called a commutative ring. Forexample, the set of real numbers, the set ofintegers, and the set of rational numbers

Riemann zeta function

190

z

x

y

k

i

j

Right-handed system

z

y

x

k

j

i

Left-handed system

are rings with respect to ordinary additionand multiplication.

robot A feedback-controlled mechanicaldevice. Robotics is the study of the design,applications, and control and sensory sys-tems of robots; for example, the design ofrobot arms that can approach an objectfrom any orientation and grip it. A robot’scontrol system may be simple and consistof only a sequencing device so that the de-vice moves in a repetitive pattern, or moresophisticated so that the robot’s move-ments are generated by computer fromdata about the environment. The robot’ssensory system gathers information neededby the control system, usually visually byusing a television camera.

rod An object that is considered to have alength but no breadth or depth, with all themass of the object concentrated along thelength. Although this definition is an ideal-ization it is a reasonably good approxima-tion for objects in which the length is muchbigger than both the breadth and depth.The pole of a broom can be modelled fairlyaccurately by regarding it as a rod. In a uni-form rod equal lengths of the rod haveequal masses. A broom pole is an exampleof a uniform rod if it is made of the samewood and has the same cross-sectionalarea all along the length of the rod. A non-uniform rod is a rod in which equal lengthsof the rod do not have equal masses. An ex-ample would be an object in two connected

sections made of different woods. A lightrod is a rod in which the mass of the rod isneglected. The concept of a light rod is use-ful when considering two bodies connectedby a rod, with the masses of both the bod-ies being much greater than the mass of therod.

Rolle’s theorem /rol-ĕz/ A curve that in-tersects the x-axis at two points a and b, iscontinuous, and has a tangent at everypoint between a and b, must have at leastone point in this interval at which the tan-gent to the curve is horizontal. For a curvey = f(x), it follows from Rolle’s theoremthat the function f(x) has a turning point (amaximum or minimum value) between f(a)and f(b), where the derivative f′(x) = 0. Thetheorem is named for the French mathe-matician Michel Rolle (1652–1719). Seealso turning point.

rolling friction See friction.

Roman numerals The system of writingintegers that was used by the ancient Ro-mans, in which I denotes 1, V denotes 5, Xdenotes 10, L denotes 50, C denotes 100, Ddenotes 500, and M denotes 1000. The in-tegers are written using the following rules:(1) the values of the letters are added if aletter is repeated or immediately followedby a letter of lesser value;(2) the value of the letter of smallest valueis subtracted from the value of the letter of

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Roman numerals

y

O a bx

Chorizontaltangent at C

•

Rolle’s theorem for a function f(x) that is continuous between x = a and x = band for which f(a) = f(b) = 0

greater value when a letter is immediatelyfollowed by a letter of greater value.

There is no symbol for zero. The inte-gers from 1 to 10 are written I, II, III, IV,V, VI, VII, VIII, IX, X; and, for example,1987 is written MCMLXXXVII.

root In an equation, a value of the inde-pendent variable that satisfies the equa-tion. In general, the degree of aPOLYNOMIAL is equal to the number ofroots. A QUADRATIC EQUATION (one of de-gree two) has two roots, although in somecircumstances they may be equal. For anumber a, an nth root of a is a number thatsatisfies the equation

xn = aSee also discriminant.

root-mean-square (rms) For a number ofvalues, an average equal to the square rootof (the sum of the squares of the values di-vided by the number of values). For exam-ple, the rms of 2, 3, 4, 5 is √(54/4) = 3.674.

roots and coefficients Relations be-tween the roots of a polynomial equationand the coefficients of that equation. If theroots of the quadratic equation ax2 + bx +c = 0 are denoted by α and β then one hasthe relations α + β = –b/a and αβ = c/a. Ifthe roots of the cubic equation ax3 + bx2 +cx + d = 0 are written α, β, and γ, then b/a= –(α + β + γ), c/a = αβ + βγ + γα, d/a =–αβγ. If one uses the notation α + β + γ =Σα and αβ + βγ + γα = Σαβ these results canbe summarized in the form:

Σα = –b/a, Σαβ = c/a. If the roots of the quartic equation ax4 +bx3 + cx2 + dx + e = 0 are denoted by α, β,γ and δ, then using the Σ notation, one hasthe relations:

Σα = –b/a, Σαβ = c/a, Σαβγ = –d/a, αβγδ = e/a.

If the roots of the quintic equation ax5 +bx4 + cx3 + dx2 + ex + f = 0 are written α,β, γ, δ, ε, then Σα = –b/a, Σαβ = c/a, Σαβγ =–d/a, Σαβγδ = e/a, αβγδε = –f/a. This pat-tern of relations between roots and coeffi-cients extends to higher degree.

If the coefficients are all taken to be realnumbers then these results mean that thesum of the roots and the product of the

roots are both real numbers, even if theroots are complex numbers. This is the casebecause complex roots occur in pairs thatare complex conjugates of each other. Thesolutions of cubic and higher power equa-tions cannot, in general, be found by usingthe relationships between roots and coeffi-cients.

rose A curve obtained by plotting theequation

r = asinnθin polar coordinates (a is a real-numberconstant and n is an integer constant). Ithas a number of petal-shaped loops, orleafs. When n is even there are 2n loopsand when n is odd there are n loops. Forexample, the graph of r = asin2θ is a four-leafed rose.

rotation A geometrical transformation inwhich a figure is moved rigidly around afixed point. If the point, the center of rota-tion, is labelled O, then for any point P inthe figure, moving to point P′ after rota-tion, the angle POP′ is the same for allpoints in the figure. This angle is the angleof rotation. Some figures are unchanged bycertain rotations. A circle is not affected byany rotation about its center. A squaredoes not change if it is rotated through 90°about the point at which its diagonalscross. Similarly an equilateral triangle isunchanged by rotation through 120° aboutits centroid. These properties are known asthe rotational symmetry of the figure. Seealso rotation of axes; transformation.

rotational motion Motion of a bodyturning about an axis. The physical quan-tities and laws used to describe linear mo-tion all have rotational analogs; theequations of rotational motion are theanalogs of the equations of motion (linear).

As well as the kinematic equations, theequations of rotational motion include T =Iα, the analog of F = ma. Here T is theturning-force, or torque (the analog offorce), I is the moment of inertia (analo-gous to mass), and α is the angular acceler-ation (analogous to linear acceleration).

root

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The kinematic equations relate the an-gular velocity, ω1, of the object at the startof timing to its angular velocity, ω2, atsome later time, t, and thus to the angulardisplacement θ. They are:

ω = ω1 + αtθ = (ω1 + ω2)/2tθ = ω1t + αt2/2θ = ω2t – αt2/2

ω22 = ω1

2 + 2αθ

rotation group The group consisting ofthe set of all possible rotations about anaxis. This group is a continuous group, i.e.it has an infinite number of members. Therotation group in two dimensions is anABELIAN GROUP but the rotation group inthree dimensions is a non-Abelian group.In physical systems the rotation group isclosely associated with the angular mo-mentum of the system. There are many ap-plications of the rotation group in thequantum theory of atoms, molecules, andatomic nuclei.

rotation of axes In coordinate geometry,the shifting of the reference axes so thatthey are rotated with respect to the originalaxes of the system by an angle (θ). If thenew axes are x′ and y′ and the original axesx and y, then the coordinates (x,y) of apoint with the original axes are related tothe new coordinates (x′,y′) by:

x = x′cosθ – y′sinθy = x′sinθ – y′cosθ

rough In mechanics, describing a systemin which frictional effects have to be takeninto consideration in the calculations.

rounding (rounding off) The process ofadjusting the least significant digit or digitsin a number after a required number of dig-its has been truncated (dropped). This re-duces the error arising from truncation butstill leaves a rounding error so that the ac-curacy of, say, the result of a calculationwill be decreased. For example, the num-ber 2.871 329 71 could be truncated to2.871 32 but would be rounded to2.871 33.

routine A sequence of instructions used incomputer programming. It may be a shortprogram or sometimes part of a program.See also subroutine.

row matrix See row vector.

row vector (row matrix) A number, (n),of quantities arranged in a row; i.e. a 1 × nmatrix. For example, the coordinates of apoint in a Cartesian coordinate systemwith three axes is a 1 × 3 row vector,(x,y,z).

Runge–Kutta method /rûng-ĕ kût-ă/ Aniterative technique for solving ordinary DIF-FERENTIAL EQUATIONS, used in computeranalysis. See also iteration. The method isnamed for the German mathematiciansCarl Runge (1856–1927) and MartinKutta (1867–1944).

Russell’s paradox /russ-ĕlz/ A PARADOX

at the foundations of set theory which waspointed out by the British philosopherBertrand Russell (1872–1970) in 1901.

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Russell’s paradox

saddle point A stationary point on acurved surface, representing a function oftwo variables, f(x,y), that is not a turningpoint, i.e. it is neither a maximum nor aminimum value of the function. At a saddlepoint, the partial derivatives ∂f/∂x and∂f/∂y are both zero, but do not change sign.The tangent plane to the surface at the sad-dle point is horizontal. Around the saddlepoint the surface is partly above and partlybelow this tangent plane.

sample space See probability.

sampling The selection of a representa-tive subset from a whole population.Analysis of the sample gives informationabout the whole population. This is calledstatistical inference. For example, popula-tion parameters (such as the populationmean and variance) can be estimated usingsample statistics (such as the sample meanand variance). Significance (or hypothesis)tests are used to test whether observed dif-ferences between two samples are due tochance variation or are significant, as in

testing a new production process againstan old. The population can be finite or in-finite. In sampling with replacement, eachindividual chosen is returned to the popu-lation before the next choice is made. Inrandom sampling every member of thepopulation has an equal chance of beingchosen. In stratified random sampling thepopulation is divided into strata and therandom samples drawn from each arepooled. In systematic sampling the popula-tion is ordered, the first individual chosenat random and further individuals chosenat specified intervals, for example, every100th person on the electoral roll. If a ran-dom sample of size n is the set of numericalvalues x1,x2,…xn, the sample mean is:

1∑n -x = xi/n

The sample variance is:∑(xi – -x)2/(n – 1)

∑(xi – -x)2/nfor a normal distribution. If µ is the popu-lation mean, the sample variance is:

∑(xi – µ)/n

194

S

x

y

z

saddle point

Saddle point on a surface

sampling distribution The distributionof a sample statistic. For example, whendifferent samples of size n are taken fromthe same population the means of eachsample form a sampling distribution. If thepopulation is infinite (or very large) andsampling is with replacement, the mean ofthe sample means is µx = µ and the stan-dard deviation of the sample means is σx =σ/√n, where µ and σ are the populationmean and standard deviation. When n ≥30, sampling distributions are approxi-mately normal and large-sampling theoryis used. When n < 30, exact sample theoryis used. See also sampling.

satisfy To be a solution of. For example,3 and –3 as values of x satisfy the equationx2 = 9.

scalar /skay-ler/ A number or a measure inwhich direction is unimportant or mean-ingless. For instance, distance is a scalarquantity, whereas displacement is a vector.Mass, temperature, and time are scalars –they are each quoted as a pure numberwith a unit. See also vector.

scalar product A multiplication of twovectors to give a scalar. The scalar productof A and B is defined by A.B = ABcosθ,where A and B are the magnitudes of A andB and θ is the angle between the vectors.An example is a force F displaced s. Herethe scalar product is energy transferred (orwork done):

W = F.sW = Fscosθ

where θ is the angle between the line of ac-tion of the force and the displacement. Ascalar product is indicated by a dot be-tween the vectors and is sometimes called adot product. The scalar product is commu-tative

A.B = B.Aand is distributive with respect to vectoraddition

A.(B + C) = A.B + A.CIf A is perpendicular to B, A.B = 0. In two-dimensional Cartesian coordinates withunit vectors i and j in the x- and y-direc-tions respectively,

A.B = (a1i + a2j).(b1i + b2j) =

a1b1 + a2b2See also vector product.

scalar projection The length of an or-thogonal projection of one vector on an-other. For example, the scalar projection ofA on B is Acosθ, or (A.B)/b where θ is thesmaller angle between A and B and b is theunit vector in the direction of B. Comparevector projection.

scale 1. The markings on the axes of agraph, or on a measuring instrument, thatcorrespond to values of a quantity. Eachunit of length on a linear scale representsthe same interval. For example, a ther-mometer that has markings 1 millimeterapart to represent 1°C temperature inter-vals has a linear scale. See also logarithmicscale.2. The ratio of the length of a line betweentwo points on a map to the distance repre-sented. For example, a map in which twopoints 5 kilometers apart are shown 5 cen-timeters apart has a scale of 1/100 000.

scale factor The multiplying factor foreach linear measurement of an object whenit is to be enlarged about a given center ofenlargement. A scale factor can be positiveor negative. If the scale factor is positive,the image is on the same side of the centerof enlargement as the object. If the scalefactor is negative, the image will be on theopposite side of the center of enlargementand will be inverted. Fractional scale fac-tors can be used – these give images thatare smaller than the objects.

scalene /skay-leen/ Denoting a trianglewith three unequal sides.

scanner A computer input device thatproduces a digital electronic version of animage (or text), so that it can be manipu-lated using the computer.

scatter diagram (Galton graph) A graph-ical representation of data from a bivariatedistribution (x,y). The variables are mea-sured on n individuals giving data (x2,y1),…, (xn,yn); e.g. xi and yi are the height andweight of the ith individual. If yi is plotted

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scatter diagram

against xi the resultant scatter diagram willindicate any relationship between x and yby showing if a smooth curve can be drawnthrough the points. If the points seem to lienear a straight line they are said to be lin-early correlated. If they lie near anothertype of curve they are non-linearly corre-lated. Otherwise they are uncorrelated. Seealso regression line.

Schwarz inequality /shworts, shvarts/ Aninequality that applies to both series andintegrals. In the case of series the Schwarzinequality is frequently known as theCAUCHY INEQUALITY. For functions f(x) andg(x) the Schwarz inequality states that fordefinite integrals∫f(x)g(x)dx ≤ ∫f2(x)dx ∫g2(x)dxThe case of equality holds if and only ifg(x) = cf(x), where c is a constant, i.e. g(x)is directly proportional to f(x).

The Schwarz inequality can be deducedfrom the Cauchy inequality by taking theintegrals as limits of series.

scientific notation (standard form) Anumber written as the product of a numberbetween 1 and 10 with a power of 10. Forexample, 2342.6 in scientific notation is2.342 6 × 103, and 0.0042 is written as 4.2 × 10–3.

screw 1. A type of machine, related to the inclined plane, and, in practice, to thesecond-order lever. The efficiency of screwsystems is very low because of friction.

Even so, the force ratio (F2/F1) can be veryhigh. The distance ratio is given by 2πr/p,where r is the radius and p the pitch of thescrew (the angle between the thread and aplane at right angles to the barrel of thescrew). 2. A symmetry that can occur in a crystal,formed by a combination of a translationand a rotation. See also space group.

scruple A unit of mass equal to 20 grains.It is equivalent to 1.295 978 grams.

sec /sek/ See secant.

secant /see-kănt/ 1. A line that intersects agiven curve. The intercept is a chord of thecurve.2. (sec) A trigonometric function of anangle equal to the reciprocal of its cosine;i.e. secα = 1/cosα. See also trigonometry.

sech /sech, shek, sek-aych/ A hyperbolicsecant. See hyperbolic functions.

second 1. Symbol: s The SI unit of time. Itis defined as the duration of 9 192 631 770cycles of a particular wavelength of radia-tion corresponding to a transition betweentwo hyperfine levels in the ground state ofthe cesium-133 atom.2. A unit of plane angle equal to one sixti-eth of a minute.

second-order determinant See determi-nant.

Schwarz inequality

196

y

x

y

x

Scatter diagrams: the left diagram shows weak correlation; the right shows strong correlation.

second-order differential equation ADIFFERENTIAL EQUATION in which the high-est derivative of the dependent variable is asecond derivative.

section See cross-section.

sector Part of a circle formed between tworadii and the circumference. The area of asector is ½r2θ, where r is the radius and θis the angle, in radians, formed at the cen-ter of the circle between the two radii. Aminor sector has θ less than π; a major sec-tor has θ greater than π.

segment Part of a line or curve betweentwo points, part of a plane figure cut off bya straight line, or part of a solid cut off bya plane. For example, on a graph, a linesegment may show the values of a functionwithin a certain range. The area between achord of a circle and the corresponding arcis a segment of the circle. A cut through acube parallel to one of its faces forms twocuboid segments.

semantics /si-man-tiks/ In computing orlogic, the semantics of a language or systemis the meaning of particular forms or ex-pressions in the language or system. Com-pare syntax.

semicircle // Half a circle, bounded by adiameter and half of the circumference.

semiconductor memory See store.

semigroup /sem-ee-groop/ A set G with abinary operation, •, that maps G×G, theset of ordered pairs of members of G, intoG, and that is associative, i.e. a•(b•c) =(a•b)•c for all a, b, and c in G. A semi-group is Abelian or commutative if a•b =b•a for all a,b in G. Sometimes a cancella-tion law is assumed, i.e. x = y if there is anelement z such that xz = yz or zx = zy.Compare group.

semilogarithmic graph /sem-ee-lôg-ă-rith-mik/ See log-linear graph.

semi-regular polyhedron A POLYHE-DRON that is bounded by regular polygons

but, unlike regular polyhedra (Platonicsolids), there is more than one kind of poly-gon. Generally, there are two types of poly-gon. There are 13 semi-regular polyhedra.All of them can be inscribed inside asphere. An example of a semi-regular poly-hedron is the polyhedron with faces of 12pentagons and 20 hexagons arranged likethe panels on a soccer ball. Semi-regularpolyhedra are also referred to asArchimedean polyhedra or Archimedeansolids. See also tetrakaidekahedron.

separation of variables A method ofsolving ordinary differential equations. Ina first-order DIFFERENTIAL EQUATION,

dy/dx = F(x,y)if F(x,y) can be written as f(x),g(y), thevariables in the function are separable andthe equation can therefore be solved bywriting it as

dy/g(y) = f(x)dxand integrating both sides. For example

dy/dx = x2ycan be written

(1/y)dy = x2dx.

sequence (progression) An ordered set ofnumbers. Each term in a sequence can bewritten as an algebraic function of its posi-tion. For example, in the sequence (2, 4, 6,8, …) the general expression for the nthterm is an = 2n. A finite sequence has a def-inite number of terms. An infinite sequencehas an infinite number of terms. Compareseries. See also arithmetic sequence; con-vergent sequence; divergent sequence; geo-metric sequence.

serial access See random access.

series The sum of an ordered set of num-bers. Each term in the series can be writtenas an algebraic function of its position. Forexample, in the series 2 + 4 + 6 + 8 … thegeneral expression for the nth term, an, is2n. A finite series has a finite fixed numberof terms. An infinite series has an infinitenumber of terms. A series with m terms, orthe sum of the first m terms of an infiniteseries, can be written as Sm or

∑n=1

m an

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series

Compare sequence. See also arithmetic se-ries; convergent series; divergent series;geometric series.

set Any collection of things or numbersthat belong to a well-defined category. Forexample, ‘dog’ is a member, or element, ofthe set of ‘types of four-legged animal’. Theset of ‘days in the week’ has seven ele-ments. In a set notation, this would bewritten as Monday, Tuesday, …. Thiskind of set is a finite set. Some sets, such asthe set of natural numbers, N = 1, 2, 3, …, have an infinite number of elements. Aline segment also is an infinite set of points.

Another way of writing a set of num-bers is by defining it algebraically. The setof all numbers between 0 and 10 could bewritten as x:0<x<10. That is, all values ofa variable, x, such that x is greater thanzero and less than ten. See also Venn dia-gram.

set square A drawing instrument, consist-ing of a flat right-angled triangular shape,used for drawing right angles and angles of30°, 45°, and 60°. Some types have a mov-ing part with a scale so that other anglescan be drawn.

sexagesimal /sex-ă-jess-ă-măl/ Based onmultiples of 60. The measurement of anangle in degrees, minutes, and seconds, forexample, is a sexagesimal measure, be-cause there are 60 seconds in one minute,and 60 minutes in one degree. A sexagesi-mal number is one that uses 60 as a base in-stead of 10. See also base.

sextant /sex-tănt/ A unit of plane angleequal to 60 degrees (π/3 radians).

sextic /sex-tik/ An equation of degree six,i.e.

ax6 + bx5 + cx4 + dx3 + ex2 + fx + g = 0There is no general method of solution forsuch an equation.

shares See stocks and shares.

sheaf A sheaf of planes is a set of planesthat all pass through a given point, calledthe center of the sheaf. Compare pencil.

shear A TRANSFORMATION of a shape orbody in which a line or plane of the shapeor body is left unchanged. The unchangedline and plane are called the invariant lineand invariant plane respectively. Physi-cally, shear of a body can occur if a force isapplied to the body in which the force ei-ther lies in the plane of an area of a surfaceof the body or a plane which is parallel tosuch a surface.

s.h.m. See simple harmonic motion.

short-wavelength approximation Anapproximate technique for solving differ-ential equations or evaluating integrals in-volving waves in which the starting point isthe solution for the problem in the limit ofthe wavelength going to zero and the ap-proximate solution of the problem for non-zero wavelengths is expressed as anASYMPTOTIC SERIES. For example, this tech-nique can be used to study problems inwave optics, with geometrical optics, i.e.the zero-wavelength limit for light andother electromagnetic waves, being thestarting point. Similarly, many problems inquantum mechanics in which the wave as-pect of particles such as electrons is impor-tant can be studied using this method, withthe zero wavelength limit of quantum me-chanics, i.e. classical mechanics, being thestarting point.

SI See SI units.

side One of the line segments that formsthe boundary of a polygon (a many-sidedplane figure). For example, a triangle hasthree sides, a pentagon has five.

siemens /see-mĕnz/ (mho) Symbol: S TheSI unit of electrical conductance, equal to aconductance of one ohm–1. The unit isnamed for the German electrical engineerErnst Werner von Siemens (1816–92).

sieve of Eratosthenes See Eratosthenes;sieve of.

sigma /sig-mă/ The Greek letter Σ (uppercase) or σ (lower case). In mathematics,upper-case sigma (Σ) is used to denote sum-

set

198

mation; lower-case sigma (σ) often denotesthe standard deviation in statistics.

sign A unit of plane angle equal to 30 de-grees (π/6 radian).

signed number A number that is denotedas positive or negative.

significance test See hypothesis test.

significant figures The number of digitsused to denote an exact value to a specifieddegree of accuracy. For example, 6084.324is a value accurate to seven significant fig-ures. If it is written as approximately 6080,it is accurate to three significant figuresThe final 0 is not significant because it isused only to show the order of magnitudeof the number.

similar Denoting two or more figures thatdiffer in scale but not in shape. The condi-tions for two triangles to be similar are: 1. Three sides of one are proportional tothree sides of the other.2. The angle of one is equal to the angle ofthe other and the sides forming the angle inone are proportional to the same sides inthe other.3. Three angles of one are equal to threeangles of the other.Compare congruent.

simple harmonic motion (s.h.m.) Anymotion that can be drawn as a sine wave.Examples are the simple oscillation (vibra-tion) of a pendulum (with small amplitude)or a sound source and the variation in-volved in a simple wave motion. Simpleharmonic motion is observed when the sys-tem, moved away from the central posi-tion, experiences a restoring force that isproportional to the displacement from thisposition.

The equation of motion for such a sys-tem can be written in the form:

md2x/dt2 = λxλ being a constant. During the motionthere is an exchange of kinetic and poten-tial energy, the sum of the two being con-stant (in the absence of damping). Theperiod (T) is given by

T = 1/fT = 2π/ω

where f is frequency and ω pulsatance.Other relationships are:

x = x0sinωtdx/dt = ±ω√(x0

2 – x2)d2x/dt2 = –ω2x

Here x0 is the maximum displacement; i.e.the amplitude of the vibration. In the caseof angular motion, as for a pendulum, θ isused rather than x.

A simple harmonic motion can be rep-resented by the motion of a point at con-stant speed in a circular path. The foot ofthe perpendicular from the point to an axisthrough a diameter describes a simple har-monic motion. This is used in a method ofrepresenting simple harmonic motions byrotating vectors (called phasors).

simple interest The interest earned oncapital when the interest is withdrawn as itis paid, so that the capital remains fixed. Ifthe amount of money invested (the princi-pal) is denoted by P, the time in years by T,and the percentage rate per annum by R,then the simple interest is PRT/100. Com-pare compound interest.

simple proposition See proposition.

simplify To use mathematical and alge-braic techniques to contract an expression.For example, simplifying the following ex-pressions:

115/184 = 5/82x – 7x + 9x = 4x

15x3y2z/5x2yz = 3xy

simply connected Describing a regionfor which any closed curve lying in the re-gion can be continuously deformed into asingle point without leaving the region, i.e.the region does not have any ‘holes’ in it. Ifa region is not simply connected it is said tobe multiply connected, i.e. the region hasone or more ‘hole’. A doubly-connectedregion has one hole, etc. In general, an n-tuply-connected region has (n–1) holes.

The concepts of simply connected andmultiply connected regions are importantin topology and the theory of functions ofa complex variable. The concepts of simply

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simply connected

and multiply connected regions also haveimportant applications in many branchesof physics.

Simpson’s rule A rule for finding the ap-proximate area under a curve by dividing itinto pairs of vertical columns of equalwidth with bases lying along the horizontalaxis. Each pair of columns is bounded bythe vertical lines from the x-axis to threecorresponding points on the curve and atthe top by a parabola that goes throughthese three points, which approximates thecurve. For example, if the value of f(x) isknown at x = a, x = b, and at a value mid-way between a and b the integral of f(x)dxbetween limits a and b is approximatelyequal to:

hf(a) + 4f[(a+b)/2] + f(b)/3h is half the distance between a and b. Aswith the trapezium rule, which is less accu-rate, better approximations can be ob-tained by subdividing the area into 4, 6, 8,… columns until further subdivision makesno significant difference to the result. Therule is named for the British mathematicianThomas Simpson (1710–61). Comparetrapezium rule. See also numerical integra-tion.

simultaneous equations A set of two ormore equations that together specify con-ditions for two or more variables. If thenumber of unknown variables is the sameas the number of equations, then there is a

unique value for each variable that satisfiesall the equations. For example, the equa-tions

x + 2y = 6and

3x + 4y = 9have the solution x = –3; y = –4.5. Themethod of solving simultaneous equationsis to eliminate one of the variables byadding or subtracting the equations. Forexample, multiplying the first equationabove by 2 and subtracting it from the sec-ond gives:

3x + 4y – 2x – 4y = 9 – 12i.e. x = –3. Substituting this into eitherequation gives the value of y. Simultaneousequations can also be solved graphically.On a Cartesian coordinate graph, eachequation would be shown as a straight lineand the point at which the two cross is, inthis case, (–3,–4.5). See also substitution;inverse of a matrix.

sin /sÿn/ See sine.

sine /sÿn/ (sin) A trigonometric function ofan angle. The sine of an angle α (sinα) in aright-angled triangle is the ratio of the sideopposite the angle to the hypotenuse. Thisdefinition applies only to angles between0° and 90° (0 and π/2 radian).

More generally, in rectangular Carte-sian coordinates, the y-coordinate of anypoint on the circumference of a circle of ra-dius r centred at the origin is rsinα, where

Simpson’s rule

200

yparabola approximatingcurve

f(a)f(b)

y = f(x)

xa b

O

f((a + b)/2)

Simpson’s rule

α is the angle between the x-axis and theradius to that point. In other words, thesine function depends on the vertical com-ponent of a point on a circle. Sinα is zerowhen α is 0°, rises to 1 when α = 90° (π/2)falls again to zero when α = 180° (π), be-comes negative and reaches –1 at α = 270°(3π/2), and then returns to zero at α = 360°(2π). This cycle is repeated every completerevolution. The sine function has the fol-lowing properties:

sinα = sin(α + 360°)sinα = –sin(180° + α)

sin(90° – α) = sin(90° + α)The sine function can also be defined as aninfinite series. In the range between 1 and–1:

sinx = x/1! – x3/3! + x5/5! – …See also trigonometry.

sine rule In any triangle, the ratio of theside length to the sine of the angle oppositethat side is the same for all three sides.Thus, in a triangle with sides of lengths a,b, and c and angles α, β, and γ (α oppositea, β opposite b, and γ opposite c):

a/sinα = b/sinβ = c/sinγ

sine wave The waveform resulting fromplotting the sine of an angle against theangle. Any motion for which distance plot-ted against time gives a sine wave is a sim-ple harmonic motion.

singular matrix A square matrix that hasa determinant equal to zero and that has noinverse matrix. See also determinant.

singular point A point on a curve y = f(x)at which the derivative dy/dx has the inde-terminate form 0/0. The singular points ona curve are found by writing the derivativein the form

dy/dx = g(x)/h(x)

and then finding the values of x for whichg(x) and h(x) are both zero.

sinh /shÿn, sinsh, sÿn-aych/ A hyperbolicsine. See hyperbolic functions.

sinusoidal /sÿ-ŭ-soid-ăl/ Describing aquantity that has a waveform that is a sinewave.

SI units (Système International d’Unités)The internationally adopted system ofunits used for scientific purposes. It is com-prised of seven base units (the meter, kilo-gram, second, kelvin, ampere, mole, andcandela) and two supplementary units (theradian and steradian). Derived units areformed by multiplication and/or divisionof base units; a number have special names.Standard prefixes are used for multiplesand submultiples of SI units. The SI sys-

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SI units

cosine

π 2π 3π 4π

+1

0

–1–1–1

x

Sine curve: a graph of y = sin x, with x in radians.

2 14 2

= (2 x 2) – (4 x 1) = 0

2 14 2

Singular 2 x 2 matrix

tem is a coherent rationalized system ofunits.

skew lines Lines in space that are not par-allel and do not intersect. Whether a pair oflines in space are skew lines rather than theother two possibilities for lines in space, i.e.parallel lines and lines that are not paralleland intersect, can be determined from theequations for the lines. If the direction ra-tios of the two lines are equal then the twolines are parallel. If the equations of thetwo lines are: (x–x1)/a1 = (y–y1)/b1 =(z–z1)/c1 = m, and (x–x1)/a2 = (y–y2)/b2 =(z–z2/c2 = n, where a1, b1, c1, m, a2, b2, c2and n are constants, then these two linescan only intersect if these are unique valuesof m and n such that x = x3, y = y3, z = z3satisfy the equations for both lines. If suchunique values of m and n cannot be foundthe lines are skew lines.

skewness A measure of the degree ofasymmetry of a distribution. If the fre-quency curve has a long tail to the rightand a short one to the left, it is calledskewed to the right or positively skewed. Ifthe opposite is true, the curve is skewed tothe left, or negatively skewed. Skewness ismeasured by either Pearson’s first measureof skewness (mean minus mode) divided bystandard deviation, or the equivalent Pear-son’s second measure of skewness dividedby standard deviation.

skew-symmetric matrix A matrix Awhich has its transpose equal to –A, i.e. ifthe entries of A are denoted by aij then aij =–aji for all i and j in the matrix. It followsfrom the definition of a skew-symmetricmatrix that the entries in the main diagonalof the matrix, i.e. all entries of the form aii,have to be zero.

slant height 1. The length of an elementof a right cone; i.e. the distance from thevertex to the directrix.2. The altitude of the faces of a right pyra-mid.

slide rule A calculating device on whichsliding logarithmic scales are used to mul-tiply and divide numbers. Most slide rules

also have fixed scales showing squares,cubes, and trigonometric functions. Theaccuracy of the slide rule is usually to threesignificant figures. Slide rules have gener-ally been replaced by electronic calcula-tors.

sliding friction See friction.

slope See gradient.

small circle A circle, drawn on the surfaceof a sphere, whose centre is not at the cen-tre of the sphere. Compare great circle.

smooth In mechanics, describing a systemin which friction can be neglected in thecalculations.

snowflake curve See fractal.

software The PROGRAMS that can be runon a computer, together with any associ-ated written documentation. A softwarepackage is a professionally written pro-gram or group of programs that is designedto perform some commonly required task,such as statistical analysis or graph plot-ting. The availability of software packagesmeans that common tasks need not be pro-grammed over and over again. Comparehardware. See also program.

solenoidal vector /soh-lĕ-noi-dăl/ A vec-tor V for which divV = 0. It is possible towrite a solenoidal vector in the form V =curlA, where A is a vector called the vectorpotential. If it is known that V is a sole-noidal vector it is possible to construct aninfinite number of A’s that satisfy V =curlA. The concept of a solenoidal vector isused extensively in the theory of electro-magnetism.

solid A three-dimensional shape or object,such as a sphere or a cube.

solid angle Symbol: Ω The three-dimen-sional analog of angle; the region sub-tended at a point by a surface (rather thanby a line). The unit is the steradian (sr),which is defined analogously to the radian– the solid angle subtending unit area at

skew lines

202

203

solid angle

BASE AND DIMENSIONLESS SI UNITS

Physical quantity Name of SI unit Symbol for SI unitlength meter mmass kilogram(me) kgtime second selectric current ampere Athermodynamic temperature kelvin Kluminous intensity candela cdamount of substance mole mol*plane angle radian rad*solid angle steradian sr*supplementary units

DERIVED SI UNITS WITH SPECIAL NAMES

Physical quantity Name of SI unit Symbol for SI unitfrequency hertz Hzenergy joule Jforce newton Npower watt Wpressure pascal Paelectric charge coulomb Celectric potential difference volt Velectric resistance ohm Ωelectric conductance siemens Selectric capacitance farad Fmagnetic flux weber Wbinductance henry Hmagnetic flux density tesla Tluminous flux lumen lmilluminance (illumination) lux lxabsorbed dose gray Gyactivity becquerel Bqdose equivalent sievert Sv

DECIMAL MULTIPLES AND SUBMULTIPLES USED WITH SI UNITS

Submultiple Prefix Symbol Multiple Prefix Symbol10–1 deci- d 101 deca- da10–2 centi- c 102 hecto- h10–3 milli- m 103 kilo- k10–6 micro- µ 106 mega- M10–9 nano- n 109 giga- G10–12 pico- p 1012 tera- T10–15 femto- f 1015 peta- P10–18 atto- a 1018 exa- E10–21 zepto- z 1021 zetta- Z10–24 yocto- y 1024 yotta- Y

unit distance. As the area of a sphere is4πr2, the solid angle at its center is 4π stera-dians. See illustration overleaf.

solid geometry The study of geometricfigures in three dimensions.

solid of revolution A solid figure thatcan be produced by revolution of a line orcurve (the generator) about a fixed axis.For instance, rotating a circle about a di-ameter generates a sphere. Rotating a circleabout an axis that does not cut the circlegenerates a torus.

solid-state memory See store.

soliton /sol-ă-ton/ A type of kink solutionthat can occur as a solution to certain non-linear equations for propagation. A char-acteristic feature of solitons is that they arevery stable and can pass through eachother unchanged. Since solitons are veryatypical of solutions to nonlinear equa-tions there has to be some mathematicalstructure in the nonlinear equation to per-mit their existence, and they are frequentlyassociated with the TOPOLOGY of a system.There are many physical applications ofsolitons.

solution A value of a variable that satis-fies an algebraic equation. For example,the solution of 2x + 4 = 12 is x = 4. Anequation may have more than one solu-

tion; for example, x2 = 16 has two; x = –4and x = +4.

solution of triangles Calculating the un-known sides and angles in triangles. Be-cause the sum of angles in a triangle isalways 180°, the third angle can be calcu-lated if two are known. All the sides andangles can be calculated when two sidesand the angle between them are known,but if two sides and another angle areknown there are two possible solutions.Any two angles and one side are sufficientto solve a triangle. See also trigonometry.

source language (source program) Seeprogram.

space curve A curved line in a volume,defined in three-dimensional Cartesian co-ordinates by three functions:

x = f(t)y = g(t)z = h(t)

or by two equations of the form:F(x,y,z) = 0G(x,y,z) = 0

space group The set of all symmetry op-erations of a crystal, i.e. the possible rota-tions, reflections, and translations. Thereare 230 space groups in three dimensions.Space groups can also be analyzed in twodimensions (ornamental or ‘wallpaper’symmetry) and in higher dimensions.

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204

P

r

Ω x r 2

SSSΩ

Solid angle: the surface S subtends a solid angle Ω in steradians at point P. An area that formspart of the surface of a sphere of radius r, center P, and subtends the same solid angle Ω at P is

equal to Ωr2.

Of the 230 space groups 73 of thesegroups consist only of point group opera-tions and the translations of the crystallattice. Such space groups are called sym-morphic. The remaining 157 space groupsconsist of point-group operations andtranslations which are not individually inthe space group. Those space groups con-tain GLIDE planes and SCREW axes, i.e.symmetry operations involving a trans-lation which is not a translation of the crys-tal. Such space groups are said to benon-symmorphic.

The concept of a space group can be ex-tended from crystals to PENROSE PATTERNS

and QUASICRYSTALLINE SYMMETRY. Oneway in which this can be done is by pro-jecting down to the Penrose pattern or qua-sicrystal from the space group of a crystalin a higher dimension.

space–time In Newtonian (pre-relativity)physics, space and time are separate andabsolute quantities; that is they are thesame for all observers in any FRAME OF REF-ERENCE. An event seen in one frame is alsoseen in the same place and at the same timeby another observer in a different frame.

After Einstein had proposed his theoryof relativity, Minkowski suggested thatsince space and time could no longer be re-

garded as separate continua, they shouldbe replaced by a single continuum of fourdimensions, called space–time. Inspace–time the history of an object’s mo-tion in the course of time is represented bya line called the world curve. See also rela-tivity, theory of.

Spearman’s method /speer-mănz/ A wayof measuring the degree of association be-tween two rankings of n objects using twodifferent variables x and y which give data(x1,y1), …, (xn,yn). The objects are rankedusing first the x’s and then the y’s, and thedifference, D, between the RANKS calcu-lated for each object.

Spearman’s coefficient of rank correla-tion

ρ = 1–(6ΣD2/[n(n2 – 1)])The method is named for the British

behavioral scientist Charles Spearman(1863–1945).

special theory See relativity.

spectrum /spek-trŭm/ (pl. spectra) Thespectrum of an operator A is the set of allcomplex numbers λ such that the operatorA – λI, where I is the identity operator, i.e.I(x) ≡ x, does not have an inverse. For ex-ample, if A is a matrix, its spectrum is the

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spectrum

•P(r, , φ)

r

O

φ

Spherical polar coordinates of point P are (r, θ, φ).

set of its eigenvalues since if λ is an eigen-value of A, det(A – λI) = 0 and A – λI istherefore not invertible.

speed Symbol: c Distance moved per unittime: c = d/t. Speed is a scalar quantity; theequivalent vector is velocity – a vectorquantity equal to displacement per unittime. Usage can be confusing and it is com-mon to meet the word ‘velocity’ where‘speed’ is more correct. For instance c0 isthe speed of light in free space, not its ve-locity.

sphere A closed surface consisting of thelocus of points in space that are at a fixeddistance, the radius r, from a fixed point,the center. A sphere is generated when acircle is turned through a complete revolu-tion about an axis that is one of its diame-ters. The plane cross-sections of a sphereare all circles. The sphere is symmetricalabout any plane that passes through itscenter and the two mirror-image shapes oneach side are called hemispheres. In Carte-sian coordinates, the equation of a sphereof radius r with its center at the origin is

x2 + y2 + z2 = r2

The volume of a sphere is 4πr3/3 and itssurface area is 4πr2.

spherical harmonic A type of functionthat frequently occurs as a solution of dif-ferential equations for systems that havespherical symmetry. Spherical harmonicscan be regarded as the analog for thesphere of CIRCULAR FUNCTIONS for the cir-cle. They occur in the solutions of the equa-tions of quantum mechanics for electronsin atoms.

spherical polar coordinates A methodof defining the position of a point in spaceby its radial distance, r, from a fixed point,the origin O, and its angular position onthe surface of a sphere centred at O. Theangular position is given by two angles θand φ. θ is the angle that the radius vectormakes with a vertical axis through O (fromthe south pole to the north pole). It is calledthe co-latitude. For points on the verticalaxis above O, θ = 0. For points lying in the‘equatorial’ horizontal plane, θ = 90°. For

points on the vertical axis below O, θ =180°. φ is the angle that the radius vectormakes with an axis in the equatorial plane.It is called the azimuth. For all points lyingin the axial plane, that is, on, verticallyabove, or vertically below this axis, φ = 0on the positive side of O and φ = 180° onthe negative side. This plane correspondsto the plane y = 0 in rectangular CARTESIAN

COORDINATES. For points in the verticalplane at 90° to this, (x = 0 in rectangularCartesian coordinates) φ = 90° in the posi-tive half-plane and 270° in the negativehalf-plane. For a point P(r,θ,φ), the corre-sponding rectangular Cartesian coordi-nates (x,y,z) are:

x =rcosφsinθy = rsinφsinθ

z =rcosθCompare cylindrical polar coordinates. Seealso polar coordinates.

spherical sector The solid generated byrotating a sector of a circle about a diame-ter of the circle. The volume of a sphericalsector generated by a sector of altitude h(parallel to the axis of rotation) in a circleof radius r is

(⅔)πr2h

spherical segment A solid formed by cut-ting in one or two parallel planes througha sphere. The volume of a spherical seg-ment bounded by circular plane cross-sec-tions of radii r1 and r2 a distance h apart,is:

πh(3r12 + 3r2

2 + h2)/6If the segment is bounded by one one planeof radius r and the curved surface of thesphere, then the volume is:

πh(3r2 + h2)/6

spherical triangle A three-sided figure onthe surface of a sphere, bounded by threegreat circles. A right spherical triangle hasat least one right angle. A birectangularspherical triangle has two right angles anda trirectangular spherical triangle has threeright angles. If one of the sides of a spheri-cal triangle subtends an angle of 90° at thecenter of the sphere, then it is called aquadrantal spherical triangle. An obliquespherical triangle has no right angles.

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207

spherical triangle

small circle

greatcircle

great and small circles on a sphere

spherical wedge

spherical wedge

trirectangular

spherical triangle

spherical triangles

right

birectangular

Spherical trigonometry

spherical trigonometry The study andsolution of spherical triangles.

spheroid /sfeer-oid/ A body or curved sur-face that is similar to a sphere but is length-ened or shortened in one direction. Seeellipsoid.

spiral A plane curve formed by a pointwinding about a fixed point at an increas-ing distance from it. There are many kindsof spirals, e.g. the Archimedean spiral isgiven by r = aθ, the logarithmic spiral isgiven by log(r) = aθ, and the hyperbolic spi-ral is given by rθ = a, where in each case ais a constant, and r and θ are polar coordi-nates.

spline A smooth curve passing through afixed number of points. Splines are used incomputer graphics.

spur See square matrix.

square 1. The second power of a numberor variable. The square of x is x × x = x2 (xsquared). The square of the square root ofa number is equal to that number.2. In geometry, a plane figure with fourequal straight sides and right angles be-tween the sides. Its area is the length of oneof the sides squared. A square has four axesof symmetry – the two diagonals, whichare of equal length and bisect each otherperpendicularly, and the two lines joiningthe mid-points of opposite sides. It can besuperimposed on itself after rotationthrough 90°.

square matrix A matrix that has thesame number of rows and columns, that is,a square array of numbers. The diagonalfrom the top left to the bottom right of asquare matrix is called the leading diagonal(or principal diagonal). The sum of the ele-

ments in this diagonal is called the trace (orspur) of the matrix.

square pyramid See pyramid.

square root For any given number, an-other number that when multiplied by it-self equals the given number. It is denotedby the symbol √ or the index (power) ½.For example, the square root of 25, written√25 = 5.

squaring the circle The attempt to con-struct a square that has the same area as aparticular circle, using a ruler and com-passes. An exact solution is impossible be-cause there is no exact length for the edge,which is a multiple of the transcendentalnumber √π.

stability A measure of how hard it is todisplace an object or system from equilib-rium.

Three cases are met in statics differingin the effect on the center of mass of a smalldisplacement. They are: 1. Stable equilibrium – the system returnsto its original state when the displacingforce is removed.2. Unstable equilibrium – the systemmoves away from the original state whendisplaced a small distance.3. Metastable or neutral equilibrium –when displaced a small distance, the sys-tem is at equilibrium in its new position.An object’s stability is improved by: (a)lowering the center of mass; or (b) increas-ing the area of support; or by both.

stable equilibrium See stability.

standard Established as a reference.1. Writing an equation in standard formenables comparison with other equationsof the same type. For example,

x2/42 + y2/22 = 1and

x2/32 + y2/52 = 1are equations of hyperbolas in rectangularCartesian coordinates, both written instandard form.

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208

1 2 43 5 114 8 9

Square 3 x 3 matrix: the trace is 1 + 5 + 9 = 15.

2. A standard measuring instrument is oneagainst which other instruments are cali-brated.3. Standard form of a number. See scien-tific notation.

standard deviation A measure of the dis-persion of a statistical sample, equal to thesquare root of the variance. In a sample ofn observations, x1,x2,x3, … xn, the samplestandard deviation is:

s = [1∑n

(xi – -x)2 / (n – 1)]

where x- is the sample mean. If the mean µof the whole population from which thesample is taken is known, then

s = [1∑n

(xi – µ)2/n]

standard form See scientific notation.

standard form of a number See scien-tific notation.

standard pressure An internationallyagreed value; a barometric height of760 mmHg at 0°C; 101 325 Pa (approxi-mately 100 kPa).

This is sometimes called the atmosphere(used as a unit of pressure). The bar, used

mainly when discussing the weather, is100 kPa exactly. See also STP.

standard temperature An internation-ally agreed value for which many measure-ments are quoted. It is the meltingtemperature of water, 0°C (273.15 K). Seealso STP.

standing wave See stationary wave.

static friction See friction.

static pressure The pressure on a surfacedue to a second solid surface or to a fluidthat is not flowing.

statics A branch of MECHANICS dealingwith the forces on an object or in a systemin equilibrium. In such cases there is no re-sultant force or torque and therefore no re-sultant acceleration.

stationary point A point on a curved lineat which the slope of the tangent to thecurve is zero. All turning points (maximumpoints and minimum points) are stationarypoints. In this case the slope of the tangentpasses through zero and changes its sign.Some stationary points are not turningpoints. In such cases, the curve levels outand then continues to increase or decrease

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stationary point

•

•

•

•

•

y

x0

absolute maximum

y = f(x)

local maximum

local minimum

absolute minimum

horizontalinflection

point

Five kinds of stationary points for a function

as before. At a stationary point the deriva-tive dy/dx of y = f(x) vanishes (is zero). Ata maximum, the second derivative,d2y/dx2, is negative; at a minimum it is pos-itive. At a horizontal inflection point thesecond derivative is zero. Not all inflectionpoints have dy/dx = 0; i.e. not all are sta-tionary points.

At a stationary point on a curved sur-face, representing a function of two vari-ables, f(x,y), the partial derivatives ∂f/∂xand ∂f/∂y are both zero. This may be amaximum point, a minimum point, or aSADDLE POINT.

stationary wave (standing wave) The in-terference effect resulting from two wavesof the same type moving with the same fre-quency through the same region. The effectis most often caused when a wave is re-flected back along its own path. The re-sulting interference pattern is a stationarywave pattern. Here, some points alwaysshow maximum amplitude; others showminimum amplitude. They are called an-tinodes and nodes respectively. The dis-tance between neighboring node andantinode is a quarter of a wavelength.

statistical inference See sampling.

statistical mechanics The evaluation ofdata that applies to a large number of enti-ties by the use of statistics. It finds its mainapplications in chemistry and physics.

statistics The methods of planning exper-iments, obtaining data, analyzing it, draw-ing conclusions from it, and makingdecisions on the basis of the analysis. Instatistical inference, conclusions about apopulation are inferred from analysis of asample. In descriptive statistics, data issummarized but no inferences are made.

step function A function θ(x) defined by:θ(x) = 1, for x > 0 and θ(x) = 0, for x < 0.In physical problems the variable x issometimes the time t, with the step func-tion representing a sudden ‘turning on’ ofsome effect at t = 0. The step function isalso sometimes known as Heaviside unitfunction. The derivative of the step func-

tion can be identified with the DIRAC DELTA

FUNCTION.

steradian /sti-ray-dee-ăn/ Symbol: sr TheSI unit of solid angle. The surface of asphere, for example, subtends a solid angleof 4π at its center. The solid angle of a coneis the area intercepted by the cone on thesurface of a sphere of unit radius.

stereographic projection /ste-ree-oh-graf-ik, steer-ee-/ A geometrical transfor-mation of a sphere onto a plane. A point istaken on the surface of the sphere – thepole of the PROJECTION. The projection ofpoints on the sphere onto a plane is ob-tained by taking straight lines from thepole through the points, and continuingthem to the plane. The plane taken doesnot pass through the pole and is perpendic-ular to the diameter of the sphere throughthe pole.

Stirling approximation /ster-ling/ Anapproximate expression for the value ofthe factorial of a number or the GAMMA

FUNCTION. The expression for n! is n! ~ nn

exp(–n)√(2πn), with the expression for Γ(n+ 1) being Γ(n + 1) ∼ nnexp(–n)√(2πn),where ∼ denotes ‘is asymptotic to’. Thismeans that the ratio n!/[nn exp(–n)√(2πn)]tends to 1 as n→∞, i.e. Stirling’s approxi-mation becomes a better approximation asn increases, with the ratio of the error tothe value of n! tending to zero as n→∞. Theapproximation is named for the Britishmathematician James Stirling (1692–1770).

stochastic process /stoh-kas-tik/ Aprocess that generates a series of randomvalues of a variable and builds up a partic-ular statistical distribution from these. Forexample, the POISSON DISTRIBUTION can bebuilt up by a stochastic process that startswith values taken from tables of randomnumbers.

Stokes’ theorem A result in vector calcu-lus that states that the surface integral ofthe curl of a vector function is equal to theline integral of that vector function arounda closed curve. If the vector function is de-

stationary wave

210

noted V, Stokes’ theorem can be written inthe form:

scurlv•dS - v•dl

where S represents surface and l representsline. Stokes’ theorem has many physicalapplications, particularly to the theory ofelectricity and magnetism. There are exten-sions of Stokes’ theorem to higher dimen-sions. The theorem is named for the Britishmathematician Sir George Gabriel Stokes(1819–1903).

store (memory) A system or device used incomputing to hold information (programsand data) in such a way that any piece ofinformation can automatically be retrievedby the computer as required. The mainstore (or internal store) of a computer isunder the direct control of the centralprocessor. It is the area in which programs,or parts of programs, are stored while theyare being run on the computer. Data andprogram instructions can be extracted ex-tremely rapidly by random access. Themain store is supplemented by backingstore, in which information can be perma-nently stored. The two basic forms of back-ing store are those in which magnetic tapeis used (i.e. magnetic tape units) and thosein which disks, or some other random-ac-cess device are used.

The main store is divided into a hugenumber of locations, each able to hold oneunit of information, i.e. a word or a byte.The number of locations, i.e. the number ofwords or bytes that can be stored, gives thecapacity of the store. Each location is iden-tified by a serial number, known as its ad-dress.

There are many different ways in whichmemory can be classified. Random-accessmemory (RAM) and serial-access memorydiffer in the manner in which informationis extracted from a store. With volatilememory, stored information is lost whenthe power supply is switched off, unlikenonvolatile memory. With read-only mem-ory (ROM), information is stored perma-nently or semipermanently; it cannot bealtered by programmed instructions but

can in some types be changed by specialtechniques.

Stores may be magnetic or electronic incharacter. The electronic memory nowwidely used in main store consists of highlycomplex integrated circuits. This semicon-ductor memory (or solid-state memory)stores an immense amount of informationin a very small space; items of informationcan be extracted at very high speed.

STP (NTP) Standard temperature andpressure. Conditions used internationallywhen measuring quantities that vary withboth pressure and temperature (such as thedensity of a gas). The values are101 325 Pa (approximately 100 kPa) and0°C (273.15 K). See also standard pres-sure; standard temperature.

straight line The curve that gives theshortest distance between two points inEuclidean space.

In two dimensions, i.e. a plane, astraight line can be represented usingCartesian coordinates by a linear equationof the form ax +by + c = 0, where a, b, andc are constants and a and b cannot both bezero. The equation for a straight line in twodimensions can be written in several differ-ent forms. If the line has a gradient m andcuts the y-axis at the intercept point (0,c)the equation for the line can be written asy = mx + c. If the line has a gradient m andpasses through the point (x1,y1) then itsequation can be written in the form y–y1 =m(x–x1). If the line passes through thepoints (x1, y1) and (x2, y2) then, if x1 ≠ x2,the equation of the line can be written y–y1= [(y2–y1)/(x2–x1)](x–x1).

In three-dimensional space a straightline can be found by the intersection of twoplanes. This means that the straight linecan be given in terms of two equations: a1x+ b1y + c1z + d1 = 0 and a2x + b2y + c2z +d2 = 0, where these two equations are equa-tions for two planes that intersect. See alsovector equation of a plane.

strain A measure of how a solid body isdeformed when a force is applied to it. Thelinear strain, sometimes called the tensilestrain, is the ratio of the change of the body

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strain

to its original length. The linear strain ap-plies to wires. The bulk strain, sometimescalled the volume strain, is the ratio of thechange in the volume of the body to itsoriginal volume. When a body is subject toa shear force there is another type of straincalled a shear strain, which is the angulardistortion of the body measured in radians.Since linear strain and bulk strain are ratiosof length and volume respectively they aredimensionless. See also Poisson’s ratio,stress, Young modulus.

strange attractor A path in phase spacethat is not closed. Strange attractors arecharacteristic of chaotic behavior. See at-tractor.

stratified random sampling See sam-pling.

stress The force per unit area acting on abody and tending to deform that body. It isa term used when a solid body is subject toforces of tension, compression, or shear.The unit of stress is the pascal (Pa), orNm–2.

If the solid body is stretched the stress istensile stress. If the body is compressed thestress is compressive stress. If the forcetends to shear a body the stress is shearstress. The biggest force per unit area thata solid body can withstand without frac-ture is called the breaking stress of thatbody.

The stress–strain graph of a solid is im-portant both in the theory of solids and inengineering. See also strain; Young modu-lus.

Student’s t-distribution The distribu-tion, written tn, of a random variable

t = (-x – µ)√n/σwhere a random sample of size n is takenfrom a normal population x with mean µand standard deviation σ. n is called thenumber of degrees of freedom. The meanof the distribution is 0 for n > 1, and thevariance is n/(n – 2) for n > 2. When n islarge t has an approximately standard nor-mal distribution. The probability densityfunction, f(t), has a symmetrical graph.

The values tn(α) for which P(t≤tn(α)) = αfor various values of n are available in ta-bles. See also mean, standard deviation;Student’s t-test.

Student’s t-test A hypothesis test foraccepting or rejecting the hypothesis thatthe mean of a normal distribution wih un-known variance is µ0, using a small sample.The statistic t = (-x – µ0)√n/s is computedfrom the data (x1,x2, … xn), where -x is thesample mean, s is the sample standard de-viation, and n < 30. If the hypothesis is truet has a tn–1 distribution. If t lies in the crit-ical region |t| > tn–1(1 – α/2) the hypothesisis rejected at significance level α. See alsohypothesis test; Student’s t-distribution.

subgroup A subgroup S of a GROUP G is asubset of G that is also a group under thesame law of combination of elements as G,i.e. a group whose members are membersof another group.

subject The main independent variable inan algebraic formula. For example, in thefunction

y = f(x) = 2x2 + 3x,y is the subject of the formula.

submatrix /sub-may-triks, -mat-riks/ Amatrix that is obtained from another ma-trix M by deleting some rows and columnsfrom M.

subnormal The projection on the x-axisof a line normal to a curve at point P0(x0,y0) and extending from P0 to the x-axis.The length of the subnormal is my0, wherem is the gradient of the tangent to the curveat P0.

subroutine /sub-roo-teen/ (procedure) Asection of a computer program that per-forms a task that may be required severaltimes in different parts of the program. In-stead of inserting the same sequence of in-structions at a number of different points,control is transferred to the subroutine andwhen the task is complete it is returned tothe main part of the program. See also rou-tine.

strange attractor

212

subscript /sub-skript/ A small letter ornumber written below, and usually to theright of, a letter for various purposes, suchas to identify a particular element of a set,e.g. x1,x2 ∈ X, to denote a constant, e.g.a1,a2, or to distinguish between variables,e.g. f(x1,x2,…,xn). Double subscripts arealso used, for example to write a determi-nant with general terms; aij denotes the el-ement in the ith row and jth column. Seealso superscript.

subset Symbol: ⊂ A set that forms part ofanother set. For example, the set of naturalnumbers, N = 1, 2, 3, 4, … is a subset ofthe set of integers I = … –2, –1, 0, 1, 2, …, written as N ⊂ I. ⊂ stands for the rela-

tionship of inclusion, and N ⊂ I can beread: N is included in I. Another symbolused is ⊃, which means ‘contains as a sub-set’ as in I ⊃ N. See also Venn diagram. Seeillustration overleaf.

substitution /sub-stă-stew-shŏn/ Amethod of solving algebraic equations thatinvolves replacing one variable by anequivalent in terms of another variable.For example, to solve the SIMULTANEOUS

EQUATIONS

x + y = 4and

2x + y = 9we can first write x in terms of y, that is:

x = 4 – y

213

substitution

E

A BBB

B ⊂ A

Subset: the set B (shaded) in the Venn diagram is a subset of A.

P(x0,y0)

normal

xNSTO

tangent

y

subtangent subnormal

Subtangent and subnormal of a curve at a point P (x0,y0).

The substitution of 4 – y for x in the secondequation gives:

2(4 – y) + y = 9or y = –1, and therefore, from the firstequation, x = 5. Another use of substitu-tion of variables is in integration. See alsointegration by substitution.

subtangent /sub-tan-jĕnt/ The projectionon the x-axis of the line tangent to a curveat a point P0 (x0,y0) and extending from P0to the x-axis. The length of the subtangentis y0/m, where m is the gradient of the tan-gent.

subtend /sŭb-tend/ To lie opposite andmark out the limits of a line or angle. Forexample, each arc of a circle subtends aparticular angle at the center of the circle.

subtraction /sŭb-trak-shŏn/ Symbol: –The binary operation of finding the DIFFER-ENCE between two quantities. In arith-metic, unlike addition, the subtraction oftwo numbers is neither commutative (4 – 5≠ 5 – 4) nor associative [2 – (3 – 4) ≠ (2 –3) – 4]. The identity element for arithmeticsubtraction is zero only when it comes onthe right-hand side (5 – 0 = 5, but 0 – 5 ≠5). In vector subtraction, two vectors areplaced tail-to-tail forming two sides of atriangle. The length and direction of thethird side gives the VECTOR DIFFERENCE.Just as the sign of the difference between

two numbers depends on the order of sub-traction, the sense of the vector differencedepends on the sense of the angle betweenthe two vectors. Matrix subtraction, likematrix addition, can only be carried outbetween matrices with the same number ofrows and columns. Compare addition. Seealso matrix.

subtraction of fractions See fractions.

subtrahend /sub-tră-hend/ A quantitythat is to be subtracted from another givenquantity.

successor The successor of a member of aseries is the next member of the series. Inparticular, the successor of an integer is thenext integer, i.e. the successor of n is n + 1.

sufficient condition See condition.

sum The result obtained by adding two ormore quantities together.

summation notation Symbol Σ. A sym-bol used to indicate the sum of a series. Thenumber of terms to be summed is indicatedby putting i, the number of the first term tobe summed, beneath the ∑ sign and thenumber of the last term to be summed isput above the Σ sign. The sum to infinity ofa series is indicated by putting the infinitysign ∞ above the ∑ sign.

sum to infinity In a CONVERGENT SERIES,the value that the sum of the first n terms,Sn approaches as n becomes infinitelylarge.

sup See supremum.

superelastic collision /soo-per-i-las-tik/A collision for which the restitution coeffi-cient is greater than one. In effect the rela-tive velocity of the colliding objects afterthe interaction is greater than that before.The apparent energy gain is the result oftransfer from energy within the collidingobjects. For example, if a collision betweentwo trolleys causes a compressed spring inone to be released against the other, the

subtangent

214

A

BO

Subtend: the arc AB subtends an angle θ at thecenter O of the circle.

collision may be superelastic. See also resti-tution; coefficient of.

superscript /soo-per-skript/ A number orletter written above and to the right or leftof a letter. A superscript usually denotes apower, e.g. x3, or a derivative, e.g. f4(x) =d4f/dx4, or is sometimes used in the samesense as a SUBSCRIPT.

supplementary angles A pair of anglesthat add together to make a straight line(180° or π radians). Compare complemen-tary angles; conjugate angles.

supplementary units The dimensionlessunits – the radian and the steradian – usedalong with base units to form derived units.See SI units.

supremum /soo-pree-mŭm/ (sup) Theleast upper BOUND of a set.

surd /serd/ See irrational number.

surface Any locus of points extending intwo dimensions. It is defined as an area. Asurface may be flat (a plane surface) orcurved and may be finite or infinite. For ex-ample, the plane z = 0 in three-dimensionalCartesian coordinates is flat and infinite;the outside of a sphere is curved and finite.

surface integral An integral in which theintegration takes place along a surface. It ispossible to define a surface integral interms of a parametric representation forthe surface and then show that the value ofthe integral is independent of the paramet-ric representation under general condi-tions. There are many mathematical andphysical applications of surface integrals

including the calculation of surface areas,the center of mass and moment of inertia,the flow of fluids through a surface andapplications of VECTOR CALCULUS to thetheory of electricity and magnetism.

surface of revolution A surface gener-ated by rotating a curve about an axis. Forexample, rotating a parabola about its axisof symmetry produces a PARABOLOID of rev-olution.

syllogism /sil-ŏ-jiz-ăm/ In logic, a deduc-tive argument in which a conclusion is de-rived from two propositions, the majorpremiss and the minor premiss, the conclu-sion necessarily being true if the premissesare true. For example, ‘Tim wants a car ora bicycle’, ‘Tim does not want a car’, there-fore ‘Tim wants a bicycle’. A hypotheticalsyllogism is a particular type of syllogismof the form ‘A implies B’, ‘B implies C’,therefore ‘A implies C’.

symbol A letter or character used to rep-resent an object, operation, quantity, rela-tion, or function. See the appendix for a listof mathematical symbols.

symbolic logic (formal logic) The branchof logic in which arguments, the terms usedin them, the relationships between them,and the various operations that can beperformed on them are all represented bysymbols. The logical properties and IMPLI-CATIONS of arguments can then be moreeasily studied strictly and formally, usingalgebraic techniques, proofs, and theoremsin a mathematically rigorous way. It issometimes called mathematical logic.

The simplest system of symbolic logic ispropositional logic (sometimes calledpropositional calculus) in which letters,e.g. P, Q, R, etc., stand for propositions orstatements, and various special symbolsstand for relationships that can hold be-tween them. See also biconditional; con-junction; disjunction; negation; truth table.

symmetrical Denoting any figure thatcan be divided into two parts that are mir-ror images of each other. The letter A, forexample, is symmetrical, and does not

215

symmetrical

α β

Supplementary angles: α + β = 180°.

change when viewed in a mirror, but theletter R is not. A symmetrical plane figurehas at least one line that is an axis of sym-metry, which divides it into two mirror im-ages.

symmetric matrix /să-met-rik/ A matrixA that has its transpose equal to A, i.e. ifthe entries of A are denoted by aij then aij =aji for all and j in the matrix.

symmetry /sim-ĕ-tree/ The property thatfigures and bodies can have that theseentities appear unchanged after certaintransformations, called symmetry transfor-mations, are performed on them. Examplesof symmetry transformations include rota-tions about a fixed point, reflection abouta mirror plane, and translation along a lat-tice. For example, in a square rotationabout the center of the square by 90° is asymmetry transformation since the squareappears unchanged after the rotation,whereas rotation about the center by 45° isnot a symmetry transformation since theposition of the square is different after thetransformation.

Taking symmetry into account fre-quently simplifies the mathematical analy-sis of a problem considerably and

sometimes enables conclusions to bedrawn without performing a calculation.

Symmetry is systematically analyzed inmathematics (and its applications) usingGROUP theory.

In the specific context of graphs offunctions y = f(x) there are symmetries as-sociated with EVEN FUNCTIONS and ODD

FUNCTIONS, with even functions having asymmetry about the y-axis while odd func-tions have a symmetry about the origin.

syntax /sin-taks/ In logic, syntax concernsthe properties of formal systems which donot depend on what the symbols actuallymean. It deals with the way the symbolscan be connected together and what com-binations of symbols are meaningful. Thesyntax of a formal logical language willspecify precisely and rigorously which for-mulae are well-formed within the lan-guage, but not their intuitive meaning.Compare semantics.

systematic error See error.

systematic sampling See sampling.

Système International d’Unités /see-stem an-tair-nas-yo-nal doo-nee-tay/ See SIunits.

symmetric matrix

216

tan See tangent.

tangent 1. A straight line or a plane thattouches a curve or a surface without cut-ting through it. On a graph, the slope of thetangent to a curve is the slope of the curveat the point of contact. In Cartesian coor-dinates, the slope is the derivative dy/dx. Ifθ is the angle between the x-axis and astraight line to the point (x,y) from the ori-gin, then the trigonometric function tanθ =y/x.2. (tan) A trigonometric function of anangle. The tangent of an angle α in a right-angled triangle is the ratio of the lengths ofthe side opposite to the side adjacent. Thisdefinition applies only to angles between0° and 90° (0 and π/2 radians). More gen-erally, in rectangular Cartesian coordi-nates, with origin O, the ratio of they-coordinate to the x-coordinate of a pointP (x,y) is the tangent of the angle betweenthe line OP and the x-axis. The tangentfunction, like the sine and cosine functions,is periodic, but it repeats itself every 180°and is not continuous. It is zero when α =

0°, and becomes an infinitely large positivenumber as α approaches 90°. At + 90°,tanα jumps from +∞ to –∞ and then rises tozero at α = 180°. See also trigonometry.

tangent plane The plane through a pointP on a smooth surface that is perpendicularto the normal to the surface at P. All thetangent lines at P lie in the tangent plane,with a tangent line at P being a tangent atP to any curve on the surface that goesthrough P.

It is frequently the case that all pointson the surface near P are on the same sideof the tangent plane at P. However, at aSADDLE POINT some of the points close to Pare on one side of the tangent plane whileother points close to P are on the other sideof the tangent plane.

tanh /th’an, tansh, tan-aych/ A hyperbolictangent. See hyperbolic functions.

tautology /taw-tol-ŏ-jee/ In LOGIC, aproposition, statement, or sentence of aform that cannot possibly be false. For ex-

217

T

3 –

2 –

1 – 0

–1 –

–2 –

–3 –

– – – –

–π/2 π/2 π 3π/2x

y

Tangent: graph of y = tan x, with x in radians.

ample, ‘if all pigs eat mice then some pigseat mice’ and ‘if I am coming then I amcoming’ are both true regardless ofwhether the component propositions ‘I amcoming’ and ‘all pigs eat mice’ are true orfalse. More strictly, a tautology is a com-pound proposition that is true no matterwhat truth values are assigned to the sim-ple propositions that it contains. A tautol-ogy is true purely because of the laws oflogic and not because of any fact about theWorld (the laws of thought are tautolo-gies). A tautology therefore contains no in-formation. Compare contradiction.

Taylor series (Taylor expansion) A for-mula for expanding a function, f(x), bywriting it as an infinite series of derivativesfor a fixed value of the variable, x = a:f(x) = f(a) + f′(a)(x – a) +

f″(a)(x – a)2/2! + f′″(a)(x – a)3/3! + …If a = 0, the formula becomes:

f(x) = f′(0) + f′(0)x = f″(0)x2/2! + …The Taylor series is named for the

British mathematician Brook Taylor(1685–1731) and the Maclaurin series, orMaclaurin expansion (a special case of theTaylor series) is named for the Scottishmathematician Colin Maclaurin (1698–1746). See also expansion.

t-distribution See Student’s t-distribu-tion.

tension A force that tends to stretch abody (e.g. a string, rod, wire, etc.).

tensor /ten-ser/ A mathematical entity thatis a generalization of a vector. Tensors areused to describe how all the components ofa quantity in an n-dimensional system be-have under certain transformations, just asa VECTOR can describe a translation fromone point to another in a plane or in space.

tera- Symbol: T A prefix denoting 1012.For example, 1 terawatt (TW) = 1012 watts(W).

terminal A point at which a user can com-municate directly with a computer both forthe input and output of information. It issituated outside the computer system,

often at some distance from it, and islinked to it by electric cable, telephone, orsome other transmission channel. A mouseor keyboard, similar to that on a type-writer, is used to feed information (data) tothe computer. The output can either beprinted out or can be displayed on a screen,as with a VISUAL DISPLAY UNIT. An interac-tive terminal is one connected to a com-puter, which gives an almost immediateresponse to an enquiry from the user. Anintelligent terminal can store informationand perform simple operations on it with-out the assistance of the computer’s centralprocessor. See also input/output.

terminating decimal A DECIMAL that hasa finite number of digits after the decimalpoint, also called a finite decimal.

ternery /ter-nĕ-ree/ Describing a numbersystem to the base (radix) 3. It uses thenumbers 0, 1, and 2 (with place values …243, 81, 27, 9, 3, 1). See also binary.

tesla /tess-lă/ Symbol: T The SI unit ofmagnetic flux density, equal to a flux den-sity of one weber of magnetic flux persquare meter. 1 T = 1 Wb m–2. The unit isnamed for the Croatian–American physi-cist Nikola Tesla (1856–1943).

tessellation /tess-ă-lay-shŏn/ A regularpattern of tiles that can cover a surfacewithout gaps. Regular polygons that will,on their own, completely tessellate a sur-face are equilateral triangles, squares andregular hexagons.

tetrahedral numbers /tet-ră-hee-drăl/For a positive integer n, a number definedas the sum of the first n TRIANGULAR NUM-BERS. This definition means that the nthtetrahedral number is given by (n + 1)(n +2)(n/6). The first four triangular numbersare 1, 4, 10, and 20.

tetrahedron /tet-ră-hee-drŏn/ (triangularpyramid) A solid figure bounded by fourtriangular faces. A regular tetrahedron hasfour congruent equilateral triangles as itsfaces. See also polyhedron; pyramid.

Taylor series

218

tetrakaidekahedron /tet-ră-kÿ-dek-ă-hee-drŏn/ A polyhedron bounded by sixsquares and eight hexagons. It can beformed by truncating the six corners of anoctahedron in a symmetrical way. It is pos-sible to fill the whole of space withouteither gaps or overlaps with tetrakai-dekahedra. The tetrakaidekahedron is anexample of a SEMI-REGULAR POLYHEDRON.

theorem /th’ee-ŏ-rĕm, th’eer-ĕm/ The con-clusion which has been proved in thecourse of an argument upon the basis ofcertain given assumptions. A theorem mustbe a result of some general importance.Compare lemma.

theorem of parallel axes If I0 is the mo-ment of the inertia of an object about anaxis, the moment of inertia I about a par-allel axis is given by:

I = I0md2

where m is the mass of the object and d isthe separation of the axes.

theory of games See game theory.

therm A unit of heat energy equal to 105

British thermal units (1.055 056 joules).

third-order determinant See determi-nant.

thou /th’ow/ See mil.

three-dimensional Having length,breadth, and depth. A three-dimensionalfigure (solid) can be described in a coordi-

nate system using three variables, for ex-ample, three-dimensional Cartesian coor-dinates with an x-axis, y-axis, and z-axis.Compare two-dimensional.

thrust A force tending to compress a body(e.g. a rod or bar) in one direction. Thrustacts in the opposite direction to tension.

time sharing A method of operation incomputer systems in which a number ofjobs are apparently executed simultane-ously instead of one after another (as inBATCH PROCESSING). This is achieved bytransferring each program in turn frombacking store to main store and allowing itto run for a short time.

ton /tun/ 1. A unit of mass equal to 2240pounds (long ton, equivalent to 1016.05kg) or 2000 pounds (short ton, equivalentto 907.18 kg).2. A unit used to express the explosivepower of a nuclear weapon. It is equal toan explosion with an energy equivalent ofone ton of TNT or approximately 5 × 109

joules.

tonne /tun/ (metric ton) Symbol: t A unitof mass equal to 103 kilograms.

topologically equivalent See topology.

topological space /top-ŏ-loj-ă-kăl/ Anon-empty set A together with a fixed col-lection (T) of subsets of A satisfying:1. Ø ∈T, A ∈T;2. if U ∈T and V ∈T then U∩V ∈ T;

219

topological space

A

B

C

D

B

CD

A

Top view

Tetrahedron

3. if Ui ∈T, where Ui is a finite or infinitecollection of sets, then ∩Ui ∈ T.

The set of subsets T is called a TOPOLOGY

for A, and the members of T are called theopen sets of the topological space. Com-pare metric space.

topology /tŏ-pol-ŏ-jee/ A branch of geom-etry concerned with the general propertiesof shapes and space. It can be thought of asthe study of properties that are notchanged by continuous deformations, suchas stretching or twisting. A sphere and anellipsoid are different figures in solid (Eu-clidean) geometry, but in topology they areconsidered equivalent since one can betransformed into the other by a continuousdeformation. A torus, on the other hand, isnot topologically equivalent to a sphere – itwould not be possible to distort a sphereinto a torus without breaking or joiningsurfaces. A torus is thus a different type ofshape to a sphere. Topology studies typesof shapes and their properties. A specialcase of this is the investigation of networksof lines and the properties of knots.

In fact Euler’s study of the KÖNIGSBERG

BRIDGE PROBLEM was one of the earliest re-sults in topology. A modern example is inthe analysis of electrical circuits. A circuitdiagram is not an exact reproduction of thepaths of the wires, but it does show theconnections between different points of thecircuit (i.e. it is topologically equivalent tothe circuit). In printed or integrated circuitsit is important to arrange connections sothat they do not cross.

Topology uses methods of higher alge-bra including group theory and set theory.An important notion is that of sets ofpoints in the neighborhood of a given point(i.e. within a certain distance of the point).An open set is a set of points such that eachpoint in the set has a neighborhood con-taining points in the set. A topologicaltransformation occurs when there is a one-to-one correspondence between points inone figure and points in another so thatopen sets in one correspond to open sets inthe other. If one figure can be transformedinto another by such a transformation, thesets are topologically equivalent.

torque /tork/ Symbol: T A turning force(or MOMENT). The torque of a force Fabout an axis (or point) is Fs, where s is thedistance from the axis to the line of actionof the force. The unit is the newton meter.Note that the unit of work, also the newtonmeter, is called the joule. Torque is not,however, measured in joules. The twophysical quantities are not in fact the same.Work (a scalar) is the scalar product offorce and displacement. Torque is the vec-tor product F ×× s and is a vector at 90° tothe plane of the force and displacement.See also couple.

torr A unit of pressure equal to a pressureof 101 325/760 pascals (133.322 Pa). It isequal to the mmHg. The unit is named forthe Italian physicist Evangelista Torricelli(1608–47).

torsional wave /tor-shŏ-năl/ A wave mo-tion in which the vibrations in the mediumare rotatory simple harmonic motionsaround the direction of energy transfer.

torus /tor-ŭs, toh-rŭs/ (anchor ring) Aclosed curved surface with a hole in it, likea donut. It can be generated by rotating acircle about an axis that lies in the sameplane as the circle but does not cut it. Across-section of the torus in a plane per-pendicular to the axis is two concentric cir-cles. A cross-section in any plane thatcontains the axis is a pair of congruent cir-cles at equal distances on both sides of theaxis. The volume of the torus is 4πdr2 andits surface area is 3π2dr, where r is the ra-dius of the generating circle and d is thedistance of its center from the axis.

total derivative A derivative that can beexpressed in terms of a series of partial de-rivatives. For example, if the function z =f(x,y) is a continuous function of x and y,and both x and y are continuous functionsof another variable t, then the total deriva-tive of z with respect to t is:

dz/dt = (∂z/∂x)(dx/dt) +(∂z/∂y)(dy/dt)

See also chain rule; total differential.

topology

220

total differential An infinitesimal changein a function of one or more variables. It isthe sum of all the partial differentials. Seedifferential.

trace See square matrix.

track See disk; drum; magnetic tape;paper tape.

trajectory /tră-jek-tŏ-ree/ 1. The path of amoving object, e.g. a projectile.2. A curve or surface that satisfies somegiven set of conditions, such as passingthrough a given set of points, or having agiven function as its gradient.

transcendental number /tran-sen-den-tăl/ See irrational number; pi.

transfinite /trans-fÿ-nÿt/ A transfinitenumber is a cardinal or ordinal numberthat is not an integer. See cardinal number;ordinal number; aleph.

Transfinite induction is the process bywhich we may reason that if some proposi-tion is true for the first element of a well-ordered set S, and that if the proposition istrue for a given element it is also true forthe next element, then the proposition istrue for every element of S.

transformation /trans-fer-may-shŏn/ 1.In general, any FUNCTION or mapping thatchanges one quantity into another.2. The changing of an algebraic expressionor equation into an equivalent with a dif-ferent form. For example, the equation

(x – 3)2 = 4x + 2can be transformed into

x2 – 10x + 7 = 0See also changing the subject of a formula.3. In geometry, the changing of one shapeinto another by moving each point in it toa different position by a specified proce-dure. For example, a plane figure may bemoved in relation to two rectangular axes.Another example is when a figure is en-larged. See translation. See also deforma-tion; dilatation; enlargement; projection;rotation.

transformation of coordinates 1.Changing the position of the reference axesin a cordinate system by translation, rota-tion, or both, usually to simplify the equa-tion of a curve. See rotation of axes;translation of axes.2. Changing the type of coordinate systemin which a geometrical figure is described;for example, from Cartesian coordinates toPOLAR COORDINATES.

transitive relation A relation *on a setA such that if a*b and b*c then a*c for alla, b, and c in A. ‘Greater than’, ‘less than’,and ‘equals’ are examples of transitive re-lations. A relation that is not transitive isan intransitive relation.

translation /trans-lay-shŏn, tranz-/ Themoving of a geometrical figure so that onlyits position relative to fixed axes ischanged, but not its orientation, size, orshape. See also translation of axes. See il-lustration overleaf.

translation of axes In coordinate geom-etry, the shifting of the reference axes sothat each axis is parallel to its original po-sition and each point is given a new set ofcoordinates. For example, the origin O of asystem of x- and y-axes, may be shifted tothe point O′, (3, 2,) in the original system.The new axes x′ and y′ are at x = 3 and y =2, respectively. This is sometimes done tosimplify the equation of a curve. The circle(x – 3)2 + (y – 2)2 = 4 can be described bynew coordinates x′ = (x – 3) and y′ = (y –3): (x′)2 + (y′)2 = 4. The origin O′ is then atthe center of the circle. See also rotation ofaxes.

translatory motion /trans-lă-tor-ee, -toh-ree, tranz-/ (translation) Motion involvingchange of position; it compares with rota-tory motion (rotation) and vibratory mo-tion (vibration). Each is associated withkinetic energy. In an object undergoingtranslatory motion, all the points move inparallel paths. Translatory motion is usu-ally described in terms of (linear) speed orvelocity, and acceleration.

221

translatory motion

transpose of a matrix

222

y

P P’

xstretch in x-direction shear in x-direction

y

P’

x

Transformations

i.e. M

The transformation matrices are:

reflection in x-axis

reflection in y-axis

enlargement scale factor k

stretch in x-direction

stretch in y-direction

A transformation can be represented by a2 x 2 matrix.A point (x, y) is tranformed to a point (x’, y’)by multiplying the column vector of (x, y) bya matrix M

shear in x-direction by k

rotation by α(anticlockwise positive)

1 00 –1

–1 0 0 1

k 00 k

k 00 1

1 00 k

1 k0 1

cos α –sin αsin α cos α

xy

transpose of a matrix /trans-pohz/ Thematrix that results from interchanging therows and columns of a given matrix. Thedeterminant of the transpose of a squarematrix is equal to that of the original ma-

trix. The transpose of a row vector is a col-umn vector and vise versa. If two matricesA and B are comformable (can be multi-plied together), then the transpose of thematrix product AB = C is

~C = (

~A

~B) =

~B

~A. In

223

transpose of a matrix

6 –

5 –

4 –

3 –

2 –

1 –

0

– – – – – – –

The translation of a triangle ABCThe translation vector k = 3i + j

A translation by a in the x direction and b in the y directiontransforms a point (x, y) to (x’, y’).

In matrix termsx’y’

xy

ab

= +

1 2 3 4 5 6

y

x

B’

C’

A’

B

C

A

kj

3i

Translation

other words, in taking the transpose of amatrix product the order must be reversed.

transversal /trans-ver-săl, tranz-/ A linethat intersects two or more other lines.

transverse axis /trans-vers, tranz-, trans-vers, tranz-/ See hyperbola.

transverse wave A wave motion in whichthe motion or change is perpendicular tothe direction of energy transfer. Electro-magnetic waves and water waves are ex-amples of transverse waves. Comparelongitudinal waves.

trapezium /tră-pee-zee-ŭm/ (pl. trapezi-ums or trapezia) A quadrilateral, none ofwhose sides are parallel. Note that in theUK the terms ‘trapezium’ and ‘trapezoid’have the opposite meanings to those in theUS.

trapezoid /trap-ĕ-zoid/ A quadrilateralwith two parallel sides. It is sometimes partof the definition that the other sides are notparallel. The parallel sides are called thebases of the trapezium and the perpendicu-lar distance between the bases is called thealtitude. The area of a trapezium is theproduct of the sum of the parallel sidelengths and half the perpendicular distancebetween them.

trapezoid rule A rule for finding the ap-proximate area under a curve by dividing itinto pairs of trapezium-shaped sections,forming vertical columns of equal widthwith bases lying on the horizontal axis. Thetrapezium rule is used as a method of NU-MERICAL INTEGRATION. For example, if thevalue of a function f(x) is known at x = a,x = b, and at a value mid-way between aand b, the integral is approximately:

(h/2)f(a) + 2f[(a + b)/2] + f(b)h is half the distance between a and b. Ifthis does not give a sufficiently accurate re-sult, the area may be subdivided into 4, 6,8, etc., columns until further subdivisionmakes no significant difference to the re-sult. See also Simpson’s rule.

travel graph A graph of displacement

transversal

224

A = a1 b1

a2 b2

a3 b3

a1 a2 a3

b1 b2 b3

→ Ã =

Transpose ~A of a matrix A.

hhh hhh xa bO

y

f(a)

f(b)

f((a + b)/2) y = f(x)••

•

Trapezium rule

(usually plotted on a y-axis) against time(usually plotted on the x-axis) which canbe used to illustrate a journey of a person.The velocity of a person at any time is thegradient of the curve at that time. The con-cept of travel graph can be extended tographs of velocity against time, with accel-eration being the gradient of the curve anddisplacement being the area under thecurve.

traveling wave See wave.

traveling salesman problem The prob-lem of minimizing the cost or time to visita number of towns, with the route fromtown a to town b being given a value of cab.This problem can, in principle, be solvedby calculating the cost or time for all possi-ble routes and seeing which route has the

lowest value. However, for n towns thenumber of routes is (n–1)!/2, which meansthat, except for small values of n, this num-ber becomes too large for any computer tohandle. It is possible to find algorithms thatfind roots that are near to the minimum,but not exactly at the minimum, which per-form the calculation fairly quickly.

traversable network A network that canbe drawn by beginning at a point on thenetwork and not lifting the pen from the paper, without going over any linetwice.

tree diagram A type of diagram used asan aid to solve problems in probability the-ory. The different ways in which an event,such as drawing balls with various coloursout of a bag, can occur, are drawn as

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tree diagram

scaleneisoscelesequilateral

Types of triangle

• •

•

C

DE

A

B

The line joining the mid-points of two sidesof a triangle is parallel to the third side andequal to half of it (AB = 2DE).

C D

A B

Triangles on the same base and between thesame parallels are equal in area (area ABC =area ABD).

Triangles

branches of a tree, with each branch show-ing the probability of that particular eventoccurring.

trial solution A guess at a solution to aproblem such as a differential equation.The trial solution is substituted into theproblem to be solved. If it does not solvethe problem exactly then the trial solutionis improved upon until it is a satisfactorysolution to the problem.

triangle A plane figure with three straightsides. The area of a triangle is half thelength of one side (the base) times thelength of the perpendicular line (the alti-tude) from the base to the opposite vertex.The sum of the internal angles in a triangleis 180° (or π radians). In an equilateral tri-angle, all three sides have the same lengthand all three angles are 60°. An isoscelestriangle has two sides of equal length andtwo equal angles. A scalene triangle has noequal sides or angles. In a right-angled tri-angle, one angle is 90° (or π/2 radians), andthe others therefore add up to 90°. In anacute angled triangle, all the angles are lessthan 90°. In an obtuse-angled triangle, oneof the angles is greater than 90°. See illus-tration overleaf.

triangle inequality For any triangleABC, the length of one side is always lessthan the sum of the other two:

AB < BC + CA

triangle of forces See triangle of vectors.

triangle of vectors A triangle describingthree coplanar VECTORS acting at a pointwith zero resultant. When drawn to scale –shown correctly in size, direction, andsense, but not in position – they form aclosed triangle. Thus three forces acting onan object at equilibrium form a triangle of

forces. Similarly a triangle of velocities canbe constructed.

triangle of velocities See triangle of vec-tors.

triangular matrix A square matrix inwhich either all the elements above theleading diagonal or all the elements belowthe leading diagonal are zero. The determi-nant of a triangular matrix is equal to theproduct of its diagonal elements.

triangular numbers The set of numbers1, 3, 6, 10, … generated by triangular ar-rays of dots. Each array has one more rowthan the preceding one, the additional rowhaving one more dot than the longest in thepreceding array. The nth triangular num-ber is n(n + 1)/2.

triangular prism See prism.

triangular pyramid See tetrahedron.

trichotomy /trÿ-kot-ŏ-mee/ The propertyof an ordering that exactly one of the state-ments x < y, x = y, x > y is true for any xand y. See linearly ordered.

trial solution

226

a11 a12 a13

0 a22 a23

0 0 a33

a11 0 0

a21 a22 0

a31 a32 a33

Triangular matrix: these are 3 x 3 matrices.

•• •

••••

• • •

• • • • • •• • • • •

• • • • • • •

1

3

6

10

15

21

28

Triangular numbers

trigonometric functions /trig-ŏ-nŏ-met-rik/ See trigonometry.

trigonometry /trig-ŏ-nom-ĕ-tree/ Thestudy of the relationships between the sidesand angles in a triangle, in terms of thetrigonometric functions of angles (sine, co-sine, and tangent). Trigonometric func-tions, can be defined by the ratio of sides ina right-angled triangle. In a right-angledtriangle with an angle α, o is the length ofthe side opposite α, a the length of side ad-jacent to α, and h the length of the hy-potenuse. For such a triangle the threetrigonometric functions of α are definedby:

sinα = o/hcosα = a/htanα = o/a

The following relationships hold for allvalues of the variable angle α:

cosα = sin(α + 90°)tanα = sinα/cosα

The trigonometric functions of an anglecan also be defined in terms of a circle(hence they are sometimes called circularfunctions). A circle is taken with its centerat the origin of Cartesian coordinates. If apoint P is taken on this circle, the line OPmakes an angle α with the positive direc-tion of the x-axis. Then, the trigonometricfunctions are:

tanα = y/xsinα = y/OPcosα = x/OP

Here, (x,y) are the coordinates of thepoint P and OP = √(x2 + y2). The signs of xand y are taken into account. For example,for an angle β° between 90° and 180°, ywill be positive and x negative. Then:

tanβ = –tan(180 – β)sinβ = + sin(180 – β)cosβ = –cos(180 – β)

Similar relationships can be written forthe trigonometric functions of angles be-tween 180° and 270° and between 270°and 360°.

The functions secant (sec), cosecant(cosec), and cotangent (cot), which are thereciprocals of the cosine, sine, and tangentfunctions respectively, obey the followingrules for all values of α:

tan2α + 1 = sec2α

1 + cot2α = cosec2αSee also addition formulae; cosine rule;

sine rule.

trillion In the US and Canada, a numberrepresented by 1 followed by 12 zeros(1012). In the UK, 1 followed by 18 zeros(1018).

trinomial /trÿ-noh-mee-ăl/ An algebraicexpression with three variables in it. Forexample, 2x + 2y + z and 3a + b = c are tri-nomials. Compare binomial.

triple integral The result of integratingthe same function three times. For exam-ple, if a function f(x,y,z) is integrated firstwith respect to x, holding y and z constant,and the result is then integrated with re-spect to y, holding x and z constant, andthe resultant double integral is then inte-grated with respect to z holding x and yconstant, the triple integral is

∫∫∫f(x,y,z)dzdydxSee also double integral.

triple product A product of three vectors.See triple scalar product; triple vectorproduct.

triple scalar product A product of threevectors, the result of which is a scalar, de-fined as:

A.(B ×× C) = ABCsinθcosφwhere φ is the angle between A and the vec-tor product (B ×× C), θ being the angle be-tween B and C. The scalar triple product isequal to the volume of the parallelepiped ofwhich A, B, and C form nonparallel edges.If A, B, and C are coplanar, their triplescalar product is zero.

triple vector product A product of threevectors, the result of which is a vector. It isa vector product of two vectors, one ofwhich is itself a vector product. That is:

A × (B × C) = (A.C)B – (A.B)CSimilarly

(A × B) × C = (A.C)B – (B.C)AThese are equal only when A, B, and C

are mutually perpendicular.

227

triple vector product

trirectangular /trÿ-rek-tang-gyŭ-ler/Having three right angles. See spherical tri-angle.

trisection /trÿ-sek-shŏn/ Division intothree equal parts.

trivial solution A solution to an equationor set of equations that is obvious and givesno useful information about the relation-ships between the variables involved. Forexample, x2 + y2 = 2x + 4y has a trivial so-lution x = 0; y = 0.

trochoid /troh-koid/ The curve describedby a fixed point on the radius or extendedradius of a circle as the circle rolls (in aplane) along a straight line. Let r be the ra-dius of the circle and a the distance of thefixed point from the center of the circle. Ifa > r the curve is called a prolate cycloid, ifa < r the curve is called a curtate cycloid,and if a = r the curve is a CYCLOID.

truncated Describing a solid generatedfrom a given solid by two non-parallelplanes cutting the given solid.

truncation The dropping of digits from anumber, used as an approximation whenthat number has more digits than it is con-venient or possible to deal with. This is dif-ferent to ROUNDING since in rounding thenumber with the required digits is the near-est number with the required digits, whichis not necessarily the number obtained bytruncation. For example, if truncated toone decimal place 2.791 and 2.736 bothgive 2.7 whereas rounding of the first num-ber gives 2.8 and rounding of the secondnumber gives 2.7.

truth table In logic, a mechanical proce-dure (sometimes called a matrix) that canbe used to define certain logical operations,and to find the truth value of complexpropositions or statements containingcombinations of simpler ones. A truthtable lists, in rows, all the possible combi-nations of truth values (T = ‘true’, F =‘false’) of a PROPOSITION or statement, andgiven an initial assignment of truth or fal-sity to the constituent parts, mechanically

assigns a value to the whole. The truth-table definitions for conjunction, disjunc-tion, negation, and implication are shownat those headwords.

An example of a truth table for a com-plex, or compound proposition is shown inthe illustration. The assigning of valuesproceeds in this way: on the basis of thetruth values of P and Q, the simple propo-sitions are given truth values, writtenunder the sign ∧ (in P∧Q) and under ∼P.Using these, truth values can then be as-signed to the whole; in the example this isin effect a complex disjunction and the val-ues are written under the sign ∨.

Thus in the case where P is true and Qis false, P∧Q will be false, ∼P will be false,and therefore the whole will be false. Seealso symbolic logic.

truth value The truth or falsity of aproposition in logic. A true statement orproposition is indicated by T and a falseone by F. In computer logic, the digits 1and 0 are often used to denote truth values.See also truth table.

Turing machine /tewr-ing/ An abstractmodel of a computer that consists of a con-trol or processing unit and an infinitelylong tape divided into single squares alongits length. At any given time each square ofthe tape is either blank or contains a singlesymbol from a fixed finite list of symbolss1, s2, …, sn. The Turing machine movesalong the tape square by square and reads,erases, and prints symbols. At any giventime the Turing machine is in one of a fixedfinite number of states represented by q1,q2, …, qn. The ‘program’ for the machine ismade up of a finite set of instructions of theform qisjskXqj, where X is either R (moveto the right), L (move to the left), or N(stay in the same position). Here, qi is the

trirectangular

228

P Q (P∧Q) ~P

T TT FF TF F

TFFF

TFTT

FFTT

∧

Truth table

state of a machine reading sj, which itchanges to sk, then moves left, right, orstays and completes the operation by goinginto state qj. A Turing machine can be usedto define computability. The machine isnamed for the British mathematician AlanMathison Turing (1912–54). See com-putability.

turning effect The ability of a force tocause a body to rotate about an axis of ro-tation. It is described quantitatively by thetorque, sometimes called the moment ofthe force, which is the product of the mag-nitude of the force, denoted F, and the per-pendicular distance d of the force from theaxis of rotation. The unit of torque is thenewton meter (Nm).

turning point A point on the graph of a

function at which the slope (gradient) ofthe tangent to a continuous curve changesits sign. If the slope changes from positiveto negative, that is, the y-coordinate stopsincreasing and starts decreasing, it is amaximum point. If the slope changes fromnegative to positive it is a minimum point.Turning points may be local maxima andminima or absolute maxima and minima.All turning points are STATIONARY POINTS.At a turning point the derivative, dy/dx, ofthe curve y = f(x) is zero.

two-dimensional Having length andbreadth but not depth. Flat shapes, such ascircles, squares, and ellipses, can be de-scribed in a coordinate system using only two variables, for example, two-dimensional Cartesian coordinates with anx-axis and a y-axis. See also plane.

229

two-dimensional

unary operation /yoo-nă-ree/ A mathe-matical procedure that changes one num-ber into another. For example, taking thesquare root of a number is a unary opera-tion. Compare binary operation.

unbounded An unbounded function is afunction that is not bounded – it has nobounds or limits. Intuitively this meansthat for any number N there is a value ofthe function numerically greater than N,i.e. there is a point x such that |f(x)| > N.For example, the function f(x) = x is un-bounded on the real axis, and f(x) = 1/x isunbounded on the interval 0 < x ≤ 1.

undecidability /un-di-sÿ-dă-bil-ă-tee/ Inlogic, if a formula or sentence can beproved within a given system the formulaor sentence is said to be decidable. If everyformula in a system can be proved then thewhole system is decidable. Otherwise thesystem is undecidable. The process of de-termining which systems are decidable andwhich are not is an important branch oflogic.

underdamping /un-der-damp-ing/ Seedamping.

unicursal /yoo-nă-ker-săl/ Describing aclosed curve that, starting and finishing atthe same point, can be traced in one sweep,without having any part retraced.

uniform acceleration Constant accelera-tion.

uniform distribution See distributionfunction.

uniform motion A vague phrase, usuallytaken to mean motion at constant velocity(constant speed in a straight line).

uniform speed Constant speed.

uniform velocity Constant velocity, de-scribing motion in a straight line with zeroacceleration.

union Symbol: ∪ The combined set of allthe elements of two or more sets. If A = 2,

U

AAA BBB

E

A ∪ B

Union: the shaded area in the Venn diagram is the union of sets A and B.

4, 6 and B = 3, 6, 9, then A ∪ B = 2, 3,4, 6, 9. See also Venn diagram.

unique factorization theorem A resultin number theory which states that a posi-tive integer can be expressed as a productof prime numbers in a way which isunique, except for the order in which theprime factors are written. This theorem fol-lows from fundamental axioms for primenumbers. The factorization of a numberinto prime numbers (and powers of primenumbers) is called the prime decomposi-tion of that number.

uniqueness theorem /yoo-neek-nis/ Atheorem which shows that there can onlybe a single entity satisfying a given condi-tion, for example only one solution to agiven equation. The proof of such a theo-rem usually proceeds by assuming thatthere exist two distinct entities satisfyingthe condition and showing that this leadsto a contradiction.

unique solution The only possible valueof a variable that can satisfy an equation.For example, x + 2 = 4 has the unique so-lution x = 2, but x2 = 4 has no unique solu-tion because x = +2 and x = –2 both satisfythe equation.

unit A reference value of a quantity usedto express other values of the same quan-tity. See also SI units

unitary matrix A matrix in which the in-verse of the matrix is the complex conju-gate of the transpose of the matrix. If thedeterminant of a unitary matrix has a value1 then the matrix is said to be a special uni-tary matrix. The set of all unitary N × Nmatrices form a group called the unitarygroup, denoted U(N), with the group of allspecial unitary N × N matrices forming agroup called the special unitary group, de-noted SU(N). There are many importantphysical applications of unitary groups andmatrices.

unit circle A circle in which the radius isone unit and the origin of the coordinatesystem is the center of the circle. The equa-

tion for a unit circle in Cartesian coordi-nates is x2 + y2 = 1. A unit circle in the com-plex plane gives the set of complexnumbers z for which the modulus |z | hasthe value 1.

unit fraction See fraction.

unit matrix (identity matrix) Symbol: I Asquare MATRIX in which the elements in theleading diagonal are all equal to one, andthe other elements are zero. If a maxtrix Awith m rows and n columns is multipliedby an n × n unit matrix, I, it remains un-changed, that is IA = A. The unit matrix isthe identity matrix for matrix multiplica-tion.

unit vector A vector with a magnitude ofone unit. Any vector r can be expressed interms of its magnitude, the scalar quantityr, and the unit vector r′′ which has the samedirection as r. The vector r = rr′′, where r isthe magnitude of r′′. In three-dimensionalCartesian coordinates with origin O unitvectors i, j, and k are used in the x-, y-, andz-directions respectively.

universal quantifier In logic, a symbolmeaning ‘for all’ and usually written ∀ .For example (∀ x)Fx means ‘for all x, theproperty F is true’.

universal set Symbol: E The set that con-tains all possible elements. In a particularproblem, E will be defined according to thescope of the problem. For example, in acalculation involving only positive num-bers, the universal set, E, is the set of allpositive numbers. See also Venn diagram.

unstable equilibrium Equilibrium suchthat if the system is disturbed a little, thereis a tendency for it to move further from itsoriginal position rather than to return. Seestability.

upper bound See bound.

upthrust An upward force on an object ina fluid. In a fluid in a gravitational field thepressure increases with depth. The pres-sures at different points on the object will

231

upthrust

therefore differ and the resultant is verti-cally upward. See also Archimedes’ princi-ple.

utility programs PROGRAMS that help inthe general running of a computer system.

Utility programs have a number of uses.They can be used, for example, to makecopies of files (organized collections ofdata) and to transfer data from one storagedevice to another, as from a magnetic tapeunit to a disk store. See also program.

utility programs

232

validity In logic, a property of arguments,inferences, or deductions. An argument isvalid if it is impossible that the conclusionbe false while the premisses are true. Thatis, to assert the premisses and deny the con-clusion would be a contradiction.

variable A changing quantity, usually de-noted by a letter in algebraic equations,that might have any one of a range of pos-sible values. Calculations can be carriedout on variables because certain rulesapply to all the possible values. For exam-ple, to carry out the operation of squaringall the integers between 0 and 10, andequation can be written in terms of an in-teger variable n, : y = n2, with the conditionthat n is between 0 and 10 (0<n<10). y iscalled a dependent variable because itsvalue depends on the value chosen for n,i.e. it can only have the values 1, 4, 9, …etc. An independent variable has no suchrelationship with another variable. For ex-ample, if one variable, x, denotes the num-ber of students in a school and another, ydenotes the proportion of the total numberof students who want to take schoollunches, then x and y are independent vari-ables and a change in either of them willnot affect the other. However, their prod-uct, xy, will affect a third quantity – thenumber of lunches ordered. Variables mayalso denote quantities other than ordinaryarithmetic numbers, for example, vectorvariables and matrix variables.

variance A measure of the dispersion of astatistical sample. In a sample of n obser-vations x1, x2, x3, … xn with a samplemean -x, the sample variance is

s2 = [(x1 – -x)2 + (x2 – -x)2 +(x3 – -x)2 + …(xn – -x)2]/(n – 1)

See also standard deviation.

variational principle A mathematicalprinciple stating that the value of somequantity either has to be a minimum or(more rarely) a maximum out of all possi-ble values. Many physical laws can bestated as variational principles, with FER-MAT’S PRINCIPLE of geometrical optics beingan example of this. Variational principlesare also used to calculate quantities of in-terest in physical science and engineeringapproximately.

VDU See visual display unit.

vector /vek-ter/ A measure in which direc-tion is important and must usually be spec-ified. For instance, displacement is a vectorquantity, whereas distance is a scalar.Weight, velocity, and magnetic fieldstrength are other examples of vectors –they are each quoted as a number with aunit and a direction. Vectors are often de-noted by printing the symbol in bold italictype F. Vector algebra treats vectors sym-bolically in a similar way to the algebra ofscalar quantities but with different rulesfor addition, subtraction, multiplication,etc.

Any vector can be represented in termsof component vectors. In particular, a vec-tor in three-dimensional Cartesian coordi-nates can be represented in terms of threeunit vector components i, j, and k directedalong the x-, y-, and z-axes respectively. IfP is a point with coordinates (x1,y,z1), thenthe vector OP is equivalent to ix1 + jy1 +kz1.

See also vector difference; vector sum;vector multiplication.

vector calculus The branch of mathe-matics that combines differential and inte-gral calculus with vectors. There are three

233

V

types of operator associated with differen-tiation in vector calculus: CURL, DIV, GRAD.Some of the key theorems for integrals invector calculus are GAUSS’S THEOREM,GREEN’S THEOREM, and STOKES’ THEOREM.Vector calculus was originally formulatedfor vectors in two and three dimensions inEuclidean space, but can be generalized tohigher dimensions and curved manifolds.There are many physical applications ofvector calculus, particularly in the theoryof electricity and magnetism.

vector difference The result of subtract-ing two VECTORS. On a vector diagram,two vectors, A and B, are subtracted bydrawing them tail-to-tail. The difference, A– B is the vector represented by the linefrom the head of B to the head of A. If Aand B are parallel, the magnitude of the dif-ference is the difference of the individualmagnitudes. If they are antiparallel, it is thesum of the individual magnitudes.

The vector difference may also be cal-culated by taking the difference in the mag-nitudes of the corresponding componentsof each vector. For example, for two planevectors in a Cartesian coordinate system.

A = 4i + 2jB = 2i + j

where i and j are the unit vectors parallel tothe x- and y-axes respectively,

A – B = 2i + jSee also vector sum.

vector equation of a line An equationfor a straight line in space when it is ex-pressed in terms of vectors. If a is the posi-tion vector of a point A on the line and v isany vector which has its direction along theline, then the line is the set of points Pwhich have their position vector p given byp = a + cv, for some value of c. This type ofequation is a vector equation of a line. Theform of the equation means that the pointP can only be on the line if p–a is propor-

vector difference

234

b r

a

The parallelogram law: r is the resultant ofa and b.

The polygon of vectors: r is the resultant.

e

d

c

b

a

r

Resolution of the vector r into different pairs of components

r r r

Vectors: addition and resolution of vectors

tional to v, i.e. the two vectors are in thesame direction (or in opposite directions).

If a line is specified by two points A andB with position vectors a and b respectivelythen defining v = b–a produces a vectorequation of the line in the form p = (1– c) a + cb.

vector equation of a plane An equationfor a plane when it is expressed in terms ofvectors. If a point A in the plane has a po-sition vector a and n is a normal vector, i.e.a vector which is perpendicular to theplane, then the plane is the set of all pointsP with position vector p which satisfies theequation (p–a).n = 0. This type of equationis a vector equation of a plane. This equa-tion can be written in the form p.n = c,where c is a constant. The vector forms ofequations of a plane can be converted intoan equation for a plane in terms of theCartesian coordinates x, y, and z by takingcomponents of the vectors.

vector graphics See computer graphics.

vector multiplication Multiplication oftwo or more vectors. This can be defined intwo ways according to whether the result isa vector or a scalar. See scalar product; vec-tor product; triple scalar product; triplevector product.

vector product A multiplication of twovectors to give a vector. The vector productof A and B is written A ×× B. It is a vector ofmagnitude ABsinθ, where A and B are themagnitudes of A and B and θ is the anglebetween A and B. The direction of the vec-tor product is at right angles to A and B. Itpoints in the direction in which a right-hand screw would move turning from A to-ward B. An example of a vector product isthe force F on a moving charge Q in a fieldB with velocity v (as in the motor effect).Here

F = QB ×× v

235

vector product

Basic vectors: the vector OP can berepresented as ix + jy + kz .

y

x

i

z

k

jO

P(x, y, z)

Vector product c = a x b.

a

bc

Vectors: base vectors and vector product.

Another example is the product of aforce and a distance to give a moment(turning effect), which can be representedby a vector at right angles to the plane inwhich the turning effect acts. The vectorproduct is sometimes called the cross prod-uct. The vector product is not commutativebecause

A ×× B = –(B ×× A)It is distributive with respect to vector

addition:C ×× (A + B) = (C ×× A) + (C ×× B)

The magnitude of A ×× B is equal to thearea of the parallelogram of which A and Bform two non-parallel sides. In a three-di-mensional Cartesian coordinate systemwith unit vectors i, j, and k in the x, y, andz directions respectively,

A ×× B = (a1i + a2j + a3k) ××(b1i + b2j + b3k)

This can also be written as a determi-nant. See also scalar product.

vector projection The vector resultingfrom an orthogonal projection of one vec-tor on another. For example, the vectorprojection of A on B is bAcosθ where θ isthe smaller angle between A and B, and b isthe unit vector in the direction of B. Com-pare scalar projection.

vectors, parallelogram of See parallelo-gram of vectors.

vectors, triangle of See triangle of vec-tors.

vector space A vector space V over ascalar field F is a set of elements x, y, …,called vectors, together with two algebraicoperations called vector addition and mul-tiplication of vectors by scalars, i.e. by ele-ments of F, such that: 1. the sum of two elements is written x + y,and V is an Abelian group with respect toaddition;2. the product of a vector, x, and a scalar,a, is written ax and a(bx) = (ab)x, 1x = xfor all a and b in F and all x in V;3. the distributive laws hold, i.e. a(x + y) =ax + ay, (a + b)x = ax + bx for all a and bin F and all x and y in V.

The scalars may be real numbers, com-plex numbers, or elements of some otherfield.

vector sum The result of adding two VEC-TORS. On a vector diagram, vectors areadded by drawing them head to tail. Thesum is the vector represented by thestraight line from the tail of the first to thehead of the last. If they are parallel, themagnitude of the sum is the sum of themagnitudes of the individual vectors. Iftwo vectors are anti-parallel, then the mag-nitude of the sum is the difference of the in-dividual magnitudes.

The vector sum may also be calculatedby summing the magnitudes of the corre-sponding components of each individualvector. For example, for two plane vectors,A = 2i + 3j and B = 6i + 4j, in a Cartesiancoordinate system with unit vectors i and jparallel to the x- and y-axes respectively,the vector sum, A + B, equals 8i + 7j.

See also vector difference.

velocities, parallelogram of See paral-lelogram of vectors.

velocities, triangle of See triangle of vec-tors.

velocity Symbol: v Displacement per unittime. The unit is the meter per second(m s–1). Velocity is a vector quantity, speedbeing the scalar form. If velocity is con-stant, it is given by the slope of a posi-tion/time graph, and by the displacementdivided by the time taken. If it is not con-stant, the mean value is obtained. If x is thedisplacement, the instantaneous velocity isgiven by

v = dx/dtSee also equations of motion.

velocity ratio See distance ratio.

Venn diagram /ven/ A diagram used toshow the relationships between SETS. Theuniversal set, E, is shown as a rectangle. In-side this, other sets are shown as circles. In-tersecting or overlapping circles areintersecting sets. Separate circles are setsthat have no intersection. A circle inside

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236

another is a subset. A group of elements, ora subset, defined by any of these relation-ships may be indicated by a shaded area inthe diagram. The diagram was devised bythe British mathematician John Venn(1834–1923).

vertex (pl. vertexes or vertices) 1. A pointat which lines or planes meet in a figure;for example, the top point of a cone orpyramid, or a corner of a polygon or poly-hedron.2. One of the two points at which an axisof a conic cuts the conic. See ellipse; hyper-bola; parabola.

vertical A direction that is at right anglesto the horizontal (see horizontal); in every-day terms, upright. In geometry, measure-ments PERPENDICULAR to the base line areoften described as being vertical.

vertically opposite angles One of thetwo pairs of equal angles formed when twostraight lines cross each other.

vibration (oscillation) Any regularly re-peated to-and-fro motion or change. Ex-amples are the swing of a pendulum, thevibration of a sound source, and thechange with time of the electric and mag-netic fields in an electromagnetic wave.

virtual work The work done if a systemis displaced infinitesimally from its posi-tion. The virtual work is zero if the systemis in equilibrium.

visual display unit (VDU) A computerterminal with which the user can commu-nicate with the computer by means of akeyboard; this input and also the outputfrom the computer appears on a cathode-ray screen. A VDU can operate both as ininput and output device.

volt Symbol: V The SI unit of electricalpotential, potential difference, and e.m.f.,defined as the potential difference betweentwo points in a circuit between which aconstant current of one ampere flows whenthe power dissipated is one watt. One voltis one joule per coulomb (1 V = 1 J C–1).The unit is named for the Italian physicistCount Alessandro Volta (1745–1827).

volume Symbol: V The extent of the spaceoccupied by a solid or bounded by a closedsurface, measured in units of length cubed.The volume of a box is the product of itslength, its breadth, and its height. The SIunit of volume is the cubic meter (m3).

volume integral The integral of a func-tion over a volume. To emphasize thethree-dimensional nature of the integral, avolume integral is sometimes written withthree integral signs and dxdydz. There aremany physical applications of physical in-tegrals, particularly to the theory of elec-tricity and magnetism.

volume of revolution The volume V ob-tained by rotating the region bounded by y= f(x), the x-axis and the lines x = a and x= b through one complete revolution, i.e.360%, about the x-axis (with it being as-sumed that f(x) is continuous in this inter-val). This volume is given by

∫ baabπy2dx.

The expression can be derived by divid-ing the volume of revolution into very thinslabs. The volume of each slab is πy2δx,where δx is the difference between x valuesin a slab. Summing over all the slabs be-tween x = a and x = b gives the desired ex-pression for V.

vulgar fraction See fraction.

237

vulgar fraction

X

X• •

Vertically opposite angles formed at the inter-section of two straight lines.

Wallis’s formula an expression for π/2 interms of an infinite product:

π = 2.2.4.4.6.6.8.8… = Π∞ 4n2

2 = 1 3 3 5 5 7 7 9 n = 1 (4n2–1)This product is sometimes known as Wal-lis’s product. The formula is named for theEnglish mathematician and theologianJohn Wallis (1616–1703).

wallpaper symmetry A name given tothe types of regular repeating patterns thatare possible in two dimensions, i.e. thetwo-dimensional analog of SPACE GROUPS.There are 17 different types of possiblewallpaper symmetry.

watt /wot/ Symbol: W The SI unit ofpower, defined as a power of one joule persecond. 1 W = 1 J s–1. The unit is named forthe British instrument maker and inventorJames Watt (1736–1819).

wave A method of energy transfer involv-ing some form of vibration. For instance,waves on the surface of a liquid or along astretched string involve regular to-and-fromotion of particles about a mean position.Sound waves carry energy by alternatecompressions and rarefactions of air (orother media). In electromagnetic waves,electric and magnetic fields vary at rightangles to the direction of propagation ofthe wave. At any particular instance, agraph of displacement against distance is aregular repeating curve – the waveform orwave profile of the wave. In a traveling (orprogressive) wave the whole periodic dis-placement moves through the medium. Atany point in the medium the disturbance ischanging with time. Under certain condi-tions a STATIONARY (or standing) WAVE canbe produced in which the disturbance doesnot change with time.

For the simple case of a plane progres-sive wave the displacement at a point canbe represented by an equation:

y = asin2π(ft – x/λ)where a is the amplitude, f the frequency, xthe distance from the origin, and λ thewavelength. Other relationships are:

y = asin2π(vt – x)/λwhere v is the speed, and

y = asin2π(t/T – x/λ)where T is the period. Note that if theminus sign is replaced by a plus sign in theabove equations it implies a similar wavemoving in the opposite direction. For a sta-tionary wave resulting from two waves inopposite directions, the displacement isgiven by:

Y = 2acos2πx/λSee also longitudinal wave; transverse

wave; phase.

wave equation A second-order partialdifferential equation that describes wavemotion. The equation

∂2u/∂x2 = (1/c2)∂2u/∂t2

might represent, for example, the verticaldisplacement u of the water surface as aplane wave of velocity c passes along thesurface, the horizontal position and thetime being given by x and t respectively.The general solution of this one-dimen-sional wave equation is a periodic functionof x and t. There exist analogous waveequations for two and three dimensions.

waveform See wave.

wavefront A continuous surface associ-ated with a wave radiation, in which all thevibrations concerned are in phase. A paral-lel beam has plane wavefronts; the outputof a point source has spherical wavefronts.

238

W

wavelength Symbol: λ The distance be-tween the ends of one complete cycle of awave. Wavelength relates to the wavespeed (c) and its frequency (v) thus:

c = vλ

wave motion Any form of energy trans-fer that may be described as a wave ratherthan as a stream of particles. The term isalso sometimes used to mean any harmonicmotion.

wave number Symbol: σ The reciprocalof the wavelength of a wave. It is the num-ber of wave cycles in unit distance, and isoften used in spectroscopy. The unit is themeter–1 (m–1). The circular wave number(Symbol: k) is given by

k = 2πσ

weber /vay-ber/ Symbol: Wb The SI unitof magnetic flux, equal to the magneticflux that, linking a circuit of one turn, pro-duces an e.m.f. of one volt when reduced tozero at a uniform rate in one second. 1 Wb= 1 V s. The unit is named for the Germanphysicist Wilhelm Eduard Weber (1804–91).

weight Symbol: W The force by which amass is attracted to another, such as theEarth. It is proportional to the body’s mass(m), the constant of proportionality beingthe gravitational field strength (i.e. the ac-celeration of free fall). Thus W = mg, whereg is the acceleration of free fall. The massof a body is normally constant, but itsweight varies with position (because it de-pends on g).

Although mass and weight are oftenused interchangeably in everyday lan-guage, they are different in scientific lan-guage and must not be confused.

weighted mean See mean.

weightlessness An apparent loss ofweight experienced by an object in free fall.Thus for a person in an orbiting spacecraft,the weight in the Earth’s frame of referenceis the centripetal force necessary to main-tain the circular orbit. In the frame of ref-

erence of the spacecraft the person feelsthat he or she has no weight.

wheel and axle A simple MACHINE con-sisting of a wheel on an axle that has a ropearound it. An effort applied to the wheel istransmitted to a load exerted at the axlerope. The force ratio (mechanical advan-tage) is equal to rW/rA where rW is the ra-dius of the wheel and rA that of the axle.

whole numbers Symbol: W The set of in-tegers 1,2,3,…, excluding zero.

word The basic unit in which informationis stored and manipulated in a computer.Each word consists usually of a fixed num-ber of BITS. This number, known as theword length, varies according to the type of computer and may be as few as eight or as many as 60. Each word is given aunique address in store. A word may rep-resent an instruction to the computer or apiece of data. An instruction word is codedto give the operation to be performed andthe address or addresses of the data onwhich the operation is to be performed. Seealso byte.

word processor A microcomputer that isprogrammed to help in preparing text forprinting or data transmission. A general-purpose computer can be used as a wordprocessor by means of a suitable applica-tions program.

work Symbol: W The work done by aforce is the product of the force and the dis-tance moved in the same direction:

work = force × displacementWork is in fact a process of energy

transfer and, like energy, is measured injoules. If the directions of force (F) and mo-tion are not the same the component of theforce in the direction of the motion is used.

W = Fscosθwhere s is displacement and θ the angle be-tween the directions of force and motion.Work is the scalar product of force and dis-placement.

world curve See space–time.

239

world curve

write head A device, part of a computersystem, that records data onto a magneticstorage medium such as tape or disk. Seealso input device.

Wronskian /vrons-kee-ăn/ A determinantwhich can be used to examine whether thefunctions f1(x), f2(x), fn(x) of x, each ofwhich has the non-vanishing derivativesare linearly independent. The WronskianW is defined to be the determinant shown.If W ≠ 0 then the n functions are linearly in-

dependent but if W ≠ 0 the functions arelinearly dependent.

The concept of the Wronskian is used inthe theory of differential equations. TheWronskian is named for the Polish mathe-matician J. M. H. Wronski (1778–1853).

write head

240

f1(x) f2(x) . . . fn(x)

f ‘1(x) f ‘

2(x) . . . f ‘1(x)

f (1n−1)(x) f (

2n−1)(x) f(

nn−1)(x)

. . .

. . .

. . .W =

yard A unit of length now defined as0.9144 meter. y-coordinate See ordinate.

yield The income produced by a stock orshare expressed as a percentage of its mar-ket price.

yocto- Symbol: y A prefix denoting 10–24.For example, 1 yoctometer (ym) = 10–24

meter (m).

yotta- Symbol: Y A prefix denoting 1024.For example, 1 yottameter (Ym) = 1024

meter (m).

Young modulus The ratio of the stress tothe strain in a deformed body in the case ofelongation or compression of the body.

There are several other types of elasticmodulus i.e. the ratio of stress to strain, de-pending on the nature of the deformationof the body. The bulk modulus is the ratioof the pressure applied to a body to thefraction by which the volume of the bodyhas decreased. The shear modulus, alsoknown as the rigidy is the ratio of the shearforce per unit area divided by the deforma-tion of the body, measured in radians.Since these elastic moduli all have ratios ofstress to strain they have the dimensions offorce per unit area since stresses have thedimensions of force per unit area andstrains are dimensionless numbers. Themodulus is named for the British physicistand physician Thomas Young (1773–1829).

241

Y

zepto- Symbol: z A prefix denoting 10–21.For example, 1 zeptometer (zm) = 10–21

meter (m).

zero (0) The number that when added toanother number gives a sum equal to thatother number. It is included in the set of in-tegers but not in the set of whole numbers.The product of any number and zero iszero. Zero is the identity element for addi-tion.

zero function A function f(x), for whichf(x) = 0 for all values of x, where x belongsto the set of all real numbers.

zero matrix See null matrix.

zeta function See Riemann zeta function.

zetta- Symbol: Z A prefix denoting 1021.For example, 1 zettameter (Zm) = 1021

meter (m).

zone A part of a sphere produced by twoparallel planes cutting the sphere.

Zorn’s lemma /zornz/ (Kuratowski–Zornlemma) If a set S is partially ordered andeach linearly ordered subset has an upperbound in S, then S contains at least onemaximal element, i.e. an element x suchthat there is no y in S for x < y. The lemmawas first discovered in 1922 by the Polishmathematician Kazimierz Kuratowski(1896–1980) and independently in 1935by the German-born American mathemati-cian Max Zorn (1906–93).

242

Z

APPENDIXES

Arithmetic and algebra

equal to =not equal to ≠identity ≡approximately equal to

approaches →proportional to ∝less than <greater than >less than or equal to ≤greater than or equal to ≥much less than <<much greater than >>plus, positive +minus, negative –plus or minus ±multiplication a × b

a.bdivision a ÷ b

a/bmagnitude of a |a|factorial a a!logarithm (to base b) logbacommon logarithm log10anatural logarithm logea or lnasummation ∑continued product Π

Geometry and trigonometry

angle

triangle square

circle

parallel to ||perpendicular to

congruent to ≡similar to ~sine sin

245

Symbols and Notation

Appendix I

cosine costangent tancotangent cot, ctnsecant seccosecant cosec, cscinverse sine sin–1, arc sininverse cosine cos–1, arc cosinverse tangent tan–1, arc tanCartesian coordinates (x, y, z)spherical coordinates (r, θ, φ)cylindrical coordinates (r, θ, z)direction numbers or cosines (l, m, n)

Sets and logic

implies that ⇒is implied by ⇐implies and is implied

by (if and only if) ⇔set a, b, c,... a, b, c, ...is an element of ∈is not an element of ∉such that :number of elements in set S n(S)universal set E or empty set ∅complement of S S′union ∪intersection ∩is a subset of ⊂corresponds one-to-one with ↔x is mapped onto y x→yconjunction ∧disjunction ∨negation (of p) ~p or ¬pimplication → or ⊃biconditional (equivalence) ≡ or ↔the set of natural numbers

the set of integers

the set of rational numbers

the set of real numbers

the set of complex numbers

Appendix I

246

Symbols and Notation

Calculus

increment of x x or δx

limit of function of x as x approaches a Lim f(x)

x→a

derivative of f(x) df(x)/dx or f′(x)

second derivative of f(x) d2f(x)/dx2 or f″(x)etc. etc.

indefinite integral of f(x)with respect to x f(x)dx

definite integral withlimits a and b

a

bf(x)dx

partial derivative of function f(x,y) with respect to x ∂f(x,y)/∂x

247

Appendix I

Symbols and Notation

acceleration aangle θ, φ, α,

β, etc.angular acceleration αangular frequency, 2πf ωangular momentum Langular velocity ωarea Abreadth bcircular wavenumber kdensity ρdiameter ddistance s, Lenergy W, Eforce Ffrequency f, νheight hkinetic energy Ek, Tlength lmass mmoment of force M

moment of inertia Imomentum pperiod τpotential energy Ep, Vpower Ppressure pradius rreduced mass µrelative density dsolid angle Ω, ωthickness dtime ttorque Tvelocity vviscosity ηvolume Vwavelength λwavenumber σweight Wwork W, E

248

Symbols for Physical Quantities

Appendix II

Plane figures

Figure Dimensions Perimeter Area

triangle sides a, b, and c, a + b + c ½bc.sinAangle A

square side a 4a a2

rectangle sides a and b 2(a + b) a × b

kite diagonals c and d ½c × d

parallelogram sides a and b distances 2(a + b) a.c or b.dc and d apart

circle radius r 2πr πr2

ellipse axes a and b 2π√[(a2 + b2)/2] πab

Solid figures

Figure Dimensions Area Volume

cylinder radius r, height h 2πr(h + r) πr2h

cone base radius r, πrl πr2h/3slant height l,height h

sphere radius r 4πr2 4πr3/3

249

Areas and Volumes

Appendix III

sin x x/1! – x3/3! + x5/5! – x7/7! + x9/9! – ...

cos x 1 – x2/2! + x4/4! – x6/6! + x8/8! – ...

ex 1 + x/1! + x2/2! + x3/3! + x4/4! + x5/5! + ...

sinh x x + x3/3! + x5/5! + x7/7! + x9/9! + ...

cosh x 1 + x2/2! + x4/4! + x6/6! + x8/8! + ...

loge(1+x) x – x2/2 + x3/3 – x4/4 + x5/5 – ...

loge(1–x) –x – x2/2 – x3/3 – x4/4 – x5/5 – ...

(1 + x)n 1 + nx + n(n – 1)x2/2! + n(n – 1)(n – 2)x3/3! + ...for |x| < 1

f(a + x) f(a) + xf′(a) + (x2/2!)f′′(a) + (x3/3!)f′′′(a) + (x4/4!)f′′′′(a) + ...where f′(a) denotes the first derivative, f′′(a) thesecond derivative, etc.

f(x) f(0) + xf′(0) + (x2/2!)f′′(0) + (x3/3!)f′′′(0) + (x4/4!)f′′′′(0) + ...

250

Expansions

Appendix IV

x is a variable, u is a function of x, and a and n are constants

Function f(x) Derivative df(x)/dx

x 1

ax a

axn naxn–1

eax aeax

loge x 1/x

loga x (1/x)loge a

cos x –sin x

sin x cos x

tan x sec2 x

cot x –cosec2 x

sec x tan x.sec x

cosec x –cot x.cosec x

cos u –sin u.(du/dx)

sin u cos u.(du/dx)

tan u sec2 u.(du/dx)

loge u (1/u)(du/dx)

sin–1 (x/a) 1/√(a2 – x2)

cos–1 (x/a) –1/√(a2 – x2)

tan–1 (x/a) a/(a2 + x2)

251

Derivatives

Appendix V

x is a variable and a and n are constants. Note that a constant of integra-tion C should be added to each integral.

Function f(x) Integral f(x)dx

a ax

x x2/2

xn xn+1/(n + 1)

1/x loge x

eax eax/a

loge ax xloge ax – x

cos x sin x

sin x –cos x

tan x loge (cos x)

cot x loge (sin x)

sec x loge (sec x + tan x)

cosec x loge (cosec x – cot x)

1/√(a2 – x2) sin–1 (x/a)

–1/√(a2 – x2) cos–1 (x/a)

a/(a2 + x2) tan–1 (x/a)

252

Integrals

Appendix VI

Addition formulae:sin(x + y) = sinx cosy + cosx sinysin(x – y) = sinx cosy – cosx sinycos(x + y) = cosx cosy – sinx sinycos(x – y) = cosx cosy + sinx sinytan(x + y) = (tanx + tany)/(1 – tanx tany)tan(x – y) = (tanx – tany)/(1 + tanx tany)

Double-angle formulae:sin(2x) = 2 sinx cosxcos(2x) = cos2x – sin2xtan(2x) = 2tanx/(1 – tan2x)

Half-angle formulae:sin(x/2) = ±√[(1 – cosx)/2]cos(x/2) = ±√[(1 + cosx)/2]tan(x/2) = sinx/(1 + cosx)

= (1 – cosx)/sinx

Poduct formulae:sinx cosy = ½[sin(x + y) + sin(x – y)]cosx siny = ½[sin(x + y) – sin(x – y)]cosx cosy = ½[cos(x + y) + cos(x – y)]sinx siny = ½[cos(x – y) – cos(x + y)]

253

Trigonometric Formulae

Appendix VII

Length

To convert into multiply by

inches meters 0.0254feet meters 0.3048yards meters 0.9144miles kilometers 1.60934nautical miles kilometers 1.85200nautical miles miles 1.15078kilometers miles 0.621371kilometers nautical miles 0.539957meters inches 39.3701meters feet 3.28084meters yards 1.09361

Area

To convert into multiply by

square inches square centimeters 6.4516square inches square meters 6.4516 × 10–4

square feet square meters 9.2903 × 10–2

square yards square meters 0.836127square miles square kilometers 2.58999square miles acres 640acres square meters 4046.86acres square miles 1.5625 × 10–3

square centimeters square inches 0.155square meters square feet 10.7639square meters square yards 1.19599square meters acres 2.47105 × 10–4

square meters square miles 3.86019 × 10–7

square kilometers square miles 0.386019

Volume

To convert into multiply by

cubic inches liters 1.63871 × 10–2

cubic inches cubic meters 1.63871 × 10–5

cubic feet liters 28.3168cubic feet cubic meters 0.0283168cubic yard cubic meters 0.764555gallon (US) liters 3.785438gallon (US) cubic meters 3.785438 × 10–3

gallon (US) gallon (UK) 0.83268

254

Conversion Factors

Appendix VIII

Mass

To convert into multiply by

pounds kilograms 0.453592pounds tonnes 4.53592 × 10–4

hundredweight (short) kilograms 45.3592hundredweight (short) tonnes 0.0453592tons (short) kilograms 907.18tons (short) tonnes 0.90718kilograms pounds 2.204623kilograms hundredweights (short) 0.022046kilograms tons (short) 1.1023 × 10–3

tonnes pounds 2204.623tonnes hundredweights (short) 22.0462tonnes tons (short) 0.90718

The short ton is used in the USA and is equal to 2000 pounds. The shorthundredweight (also known as the cental) is 100 pounds.

The long ton, which is used in the UK, is equal to 2240 pounds (1016.047kg). The long hundredweight is 112 pounds (50.802 kg). 1 long ton equals20 long hundredweights.

Force

To convert into multiply by

pounds force newtons 4.44822pounds force kilograms force 0.453592pounds force dynes 444822pounds force poundals 32.174poundals newtons 0.138255poundals kilograms force 0.031081poundals dynes 13825.5poundals pounds force 0.031081dynes newtons 10–5

dynes kilograms force 1.01972 × 10–6

dynes pounds force 2.24809 × 10–6

dynes poundals 7.2330 × 10–5

kilograms force newtons 9.80665kilograms force dynes 980665kilograms force pounds force 2.20462kilograms force poundals 70.9316newtons kilograms 0.101972newtons dynes 100000newtons pounds force 0.224809newtons poundals 7.2330

255

Appendix VIII

Conversion Factors

Work and energy

To convert into multiply by

British Thermal Units joules 1055.06British Thermal Units calories 251.997British Thermal Units kilowatt-hours 2.93071 × 10–4

kilowatt-hours joules 3600000kilowatt-hours calories 859845kilowatt-hours British Thermal Units 3412.14calories joules 4.1868calories kilowatt-hours 1.16300 × 10–6

calories British Thermal Units 3.96831 × 10–3

joules calories 0.238846joules kilowatt hours 2.7777 × 10–7

joules British Thermal Units 9.47813 × 10–4

joules electron volts 6.2418 × 1018

joules ergs 107

electronvolts joules 1.6021 × 10–19

ergs joules 10–7

Pressure

To convert into multiply by

atmospheres pascals* 101325bars pascals 100000pounds per square pascals 68894.76inch

pounds per square kilograms per square 703.068inch meter

pounds per square atmospheres 0.068046inch

kilograms per square pascals 9.80661meter

kilograms per square pounds per square 1.42234 × 10–3

meter inchkilograms per square atmospheres 9.67841 × 10–5

meterpascals kilograms per square 0.101972

meterpascals pounds per square 1.45038 × 10–4

inchpascals atmospheres 9.86923 × 10–6

*1 pascal = 1 newton per square meter

Appendix VIII

256

Conversion Factors

1 1 1 1.000 1.0002 4 8 1.414 1.2603 9 27 1.732 1.4424 16 64 2.000 1.5875 25 125 2.236 1.710

6 36 216 2.449 1.8177 49 343 2.646 1.9138 64 512 2.828 2.0009 81 729 3.000 2.08010 100 1 000 3.162 2.154

11 121 1 33` 3.317 2.22412 144 1 728 3.464 2.28913 169 2 197 3.606 2.35114 196 2 744 3.742 2.41015 225 3 375 3.873 2.466

16 256 4 096 4.000 2.52017 289 4 913 4.123 2.57118 324 5 832 4.243 2.62119 361 6 859 4.359 2.66820 400 8 000 4.472 2.714

21 441 9 261 4.583 2.75922 484 10 648 4.690 2.80223 529 12 167 4.796 2.84424 576 13 824 4.899 2.84425 625 15 625 5.000 2.924

26 676 17 576 5.099 2.96227 729 19 683 5.196 3.00028 784 21 952 5.292 3.03729 841 24 389 5.385 3.07230 900 27 000 5.477 3.107

31 961 29 791 5.568 3.14132 1 024 32 768 5.657 3.175

257

n n2 n3 √n3√n

Powers and Roots

Appendix IX

33 1 089 35 937 5.745 3.20834 1 156 39 304 5.831 3.24035 1 225 42 875 5.916 3.271

36 1 296 46 656 6.000 3.30237 1 369 50 653 6.083 3.33238 1 444 54 872 6.164 3.36239 1 521 59 319 6.245 3.39140 1 600 64 000 6.325 3.420

41 1 681 68 921 6.403 3.44842 1 764 74 088 6.481 3.47643 1 849 79 507 6.557 3.50344 1 936 85 184 6.633 3.53045 2 025 91 125 6.708 3.557

46 2 116 97 336 6.782 3.58347 2 209 103 823 6.856 3.60948 2 304 110 592 6.928 3.63449 2 401 117 649 7.000 3.65950 2 500 125 000 7.071 3.684

51 2 601 132 651 7.141 3.70852 2 704 140 608 7.211 3.73353 2 809 148 877 7.280 3.75654 2 916 157 464 7.348 3.78055 3 025 166 375 7.416 3.803

56 3 136 175 616 7.483 3.82657 3 249 185 193 7.550 3.84958 3 364 195 112 7.616 3.87159 3 481 205 379 7.681 3.89360 3 600 216 000 7.746 3.915

61 3 721 226 981 7.810 3.93662 3 844 238 328 7.874 3.95863 3 969 250 047 7.937 3.97964 4 096 262 144 8.000 4.00065 4 225 274 625 8.062 4.021

Appendix IX

258

n n2 n3 √n3√n

Powers and Roots

66 4 356 287 496 8.124 4.04167 4 489 300 763 8.185 4.06268 4 624 314 432 8.246 4.08269 4 761 328 509 8.307 4.10270 4 900 343 000 8.367 4.121

71 5 041 357 911 8.426 4.14172 5 184 373 248 8.485 4.16073 5 329 389 017 8.544 4.17974 5 476 405 224 8.602 4.19875 5.625 421 875 8.660 4.217

76 5 776 438 976 8.718 4.23677 5 929 456 533 8.775 4.25478 6 084 474 552 8.832 4.27379 6 241 493 039 8.888 4.29180 6 400 512 000 8.944 4.309

81 6 561 531 441 9.000 4.32782 6 724 551 368 9.055 4.34483 6 889 571 787 9.110 4.36284 7 056 592 704 9.165 4.38085 7 225 614 125 9.220 4.397

86 7 396 636 056 9.274 4.41487 7 569 658 503 9.327 4.43188 7 744 681 472 9.381 4.44889 7 921 704 969 9.434 4.46590 8 100 729 000 9.487 4.481

91 8 281 753 571 9.539 4.49892 8 464 778 688 9.592 4.51493 8 649 804 357 9 644 4.53194 8 836 830 584 9.695 4.54795 9 025 857 375 9.747 4.563

96 9 216 884 736 9.798 4.57997 9 409 912 673 9.849 4.59598 9 604 941 192 9.899 4.61099 9 801 970 299 9.950 4.626100 10 000 1 000 000 10.000 4.642

259

Appendix IX

n n2 n3 √n3√n

Powers and Roots

260

The Greek Alphabet

A α alphaB β betaΓ γ gamma∆ δ deltaE ε epsilonZ ζ zetaH η etaΘ θ thetaI ι iotaK κ kappaΛ λ lambdaM µ mu

N ν nuΞ ξ xiO ο omikronΠ π piP ρ rhoΣ σ sigmaT τ tauΥ υ upsilonΦ φ phiX χ chiΨ ψ psiΩ ω omega

Appendix X

Organizations:

American Mathematical Society www.ams.org

Mathematical Association of America www.maa.org

International Mathematical Union www.mathunion.org

London Mathematical Society www.lms.ac.uk

Mathematics Foundation of America www.mfoa.org

General resources:

Mathematics on the Web www.ams.org/mathweb

Mathematics WWW Virtual Library euclid.math.fsu.edu/Science/math.html

MathGate Homepage www.mathgate.ac.uk

Mathematical Resources www.ama.caltech.edu/resources.html

Math-Net Links www.math-net.de/links/show?collection=math

Mathematics Resources on the WWW mthwww.uwc.edu/wwwmahes/homepage.htm

History and biography:

History of Mathematics Archive www-history.mcs.st-and.ac.uk/history

261

Web Sites

Appendix XI

Bittinger, Marvin L. Basic Mathematics. 9th ed. New York: Addison Wesley, 2002.

Boyer, Carl B. and Isaac Asimov. A History of Mathematics. 2nd ed. New York: Wiley,

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