NCV2 Mathematics Hands-On Training 2010 Syllabus - Sample chapter

Mathematics Hands-On TrainingSample Chapter

Dear lecturer

We have completely revised our Mathematics NCV 2 textbook according to the new curriculum to be implemented in 2011. The completed learner book and lecturer guide will be submitted to the DOE for screening on 15 November 2010. We are confident that the book will be approved. The book will be completed and ready to order by early December 2010.

This is a sample chapter to give you an idea of the look and feel of the revised book. As you will notice, we have made the learner book more user-friendly by including the following features:

• Pre-knowledgesections• Interesting‘Didyouknow’facts• Hintstoassistwithunderstandingcalculations• Relevantdefinitions• Manyworked-outexamples• Avarietyofformativeandsummativeassessments• Chaptersummary• Answerstoactivities

The learner book will be accompanied by a comprehensive lecturer guide that will include:

• Newrevisedcurriculum• Pacesetter• Worked-outanswerstoallactivitiesinlearnerbook• Additionalactivities(formativeassessments)withsolutions• Examplesofexaminationtypequestions

Youwillfindacatalogueandorderformincludedinthepack.Pleasecontactusatinfo@futuremanagers.netor021-4623572foranyqueries.

© Future Managers 2010

All rights reserved. No part of this book may be reproduced in any form, electronic, mechanical, photocopying, or otherwise, without prior permission of the copyright owner.

ISBN 9781920364274

First published 2010

Published by Future Managers (Pty) LtdPO Box 13194, Mowbray, 7705Tel (021) 462 3572Fax (021) 462 3681E-mail: [email protected]: www.futuremanagers.net

FutureManagers

1

Chapter 1

Numbers

After completing this chapter, you will be able to:

1. Usecomputationaltoolsandstrategiesandmakeestimatesandapproximations.1.1 Useascientificcalculatorcorrectlytosolveexpressionsinvolvingaddition,

subtraction,multiplication,division,squares,cubes,squarerootsandcuberoots.

1.2 Estimateandapproximatephysicalquantitiestosolveproblemsinpracticalsituations.Quantitiesincludelength,time,massandtemperature.

2. Demonstrateanunderstandingofnumbers,relationshipsamongnumbersandnumbersystemsandrepresentnumbersindifferentways.2.1 Identifyrationalandirrationalnumbers.2.2 Round off rational and irrational numbers to an appropriate degree of

accuracy.2.3 Convert rational numbers between terminating and recurring decimals to

theforma__b

;a,b∈Z;b≠02.4 Applythefollowinglawsofexponents. am×an=am+n am÷an=am–n

(am)n=am×n (ab)m=ambm

(ambn)p=ampbnp (a__b

)m

=am___

bm

(am___

bn)p=a

mp____

bnp a-n=1___an

1___a-n=an a0=1

n√___

am=am__n

2.5 Rationalise fractions with surd denominators (binomial and monomialdenominators)withoutusingacalculator.

2.6 Add,subtract,multiplyanddividesimplesurds.2.7 Manipulatesimpletechnicalandnon-technicalformulae.2.8 Solveanunknownvariableinsimpletechnicalandnon-technicalformulae.2.9 Identifyandworkwitharithmeticsequencesandseries.

2

Chapter 1

1.1 Use computational tools and strategies and make estimates and approximations

1.1.1 How to use a scientific calculator

IntroductionLearninghowtouseascientificcalculatorwill requiretimeandpatienceas

no one can teach you how to do it.You should be quite familiar with your

calculatorbynow.Yourlecturermaybeabletoassist,butyoushouldalsorefer

totheinstructionbookletprovidedwithyourspecificcalculator.

Pre-knowledgeIf a calculation has more than one operation we use BODMAS/BIDMAS to

performthecalculationinafixedorder.Thescientific calculatorisprogrammed

tofollowthisorderautomatically.Rememberthatyouhavetopresstheright

keystoget to thecorrectanswers!Calculatorshavedifferentways inwhich

thesequenceofnumbers ispunched in.This isknownas thealgorithms of arithmeticorthepriority of calculations.

BODMAS or BIDMAS B–Brackets B–Brackets

O–Off I–Indices

D–Division D–Division

M–Multiplication M–Multiplication

A–Addition A–Addition

S–Subtraction S–Subtraction

RememberTheoperationsxand÷shouldbedonebefore+and–• 3+4×5=3+20

=23Iftheoperationshaveequalpriority,youshouldworkfromlefttoright• 8+4–5=12–5

=7Iftherearebrackets,youshouldworkoutthepartinthebracketsfirst• 10×(7–5)÷4=10×2÷4

=20÷4 =5

Work with someone in your class who has the same calculator as you to calculate the following:

Calculate: Solution:

102–34×5+96 28

102–34×(5+96) -3332

(102–34)×5+96 436

(102–34)×(5+96) 6868

DefinitionAnalgorithmisasetofstep-by-step

procedurethatproducesananswertoaparticularproblem

DefinitionBODMASisan

acronymthatservesasareminderof

theorderinwhichcertainoperationshavetobecarriedoutwhenworking

withequationsandformulas.

3

Numbers

Rounding off to THREE decimal placesLet us briefly revise the method for rounding 514,342615 off to THREE decimal places:

If the fourth decimal place is 5 or higher, the third decimal is increased by 1. "the third decimal will be 3If the fourth decimal place is 4 or lower, the third decimal remains the same. "the third decimal will remain 2∴514,342615 ≈ 514,343

Using the different keys on your calculatorThe diagram illustrates an electronic scientific calculator. Many of the keys have meanings that are the same as the corresponding symbols in arithmetic and algebra.

The display is at the top of the calculator. As you type the numbers, they will appear in the display. The results of the calculations will also appear in the display.

Below is a table with useful keys of some of the functions:

Keys Function Example

+/− Changes the sign of a number from positive to negative

Press 3 +/− ; Answer = −3 or

Press −3; Answer = −3

1 __ x ; x−1 Inverts the value of the display

Press 4 1 _ x = ; Answer = 1 __ 4 = 0,25

π Enters the value of π to the full accuracy the calculator is capable of

≈ 3,141592654

x2 Calculates the square of a number

Press 3 x2 = ; Answer = 9∴ 32 = 9

x3 Calculates the cube of a number

Press 3 x3 = ; Answer = 27∴ 33 = 27

yx

x

xy

^Calculates all other powers

Press 2 yx 4 = ; Answer = 16

∴ 24 = 16

sincostan

Calculates the corresponding trigonometric function of the value in the display

sin 30° = 0,5; tan 45° = 1; cos 30° = 0,866These functions will be described in the next chapter.

∙ Enters the decimal point into the display. When using the calculator you must work with the decimal point, not the decimal comma

Press 3.4This is the same as 3,4A calculator uses a decimal point, not decimal comma

NoteYou will learn more about rounding off when you deal with

irrational numbers on page 13.

Note≈ is the symbol

for ‘approximately equal to’.

Did you know?It is acceptable to use a comma (,) or point (.) as a decimal separator

in South Africa. For example we can write either 2,346 or

2.346.We will use the comma (,) as a

separator to avoid confusion.

HintSome of the keys of different calculators

have different notations, but have

meanings that are the same.

4

Chapter 1

ab__c __  Fractionkey Tofind1__2;press1 ab__

c 2 =

ortofind12__3press1 ab__

c 2 ab__

c 3 =

2ndForSHIFT

Secondfunctionkeyisusedtodothefunctionsthatarewrittenabovethekeys.

Press SHIFT 3√__

R 8 = ;Answer=2

∴3√__

8=2

EXPor×10x

Changesanumberfromscientificnotationtoanordinarynumber

Towrite4,2×10–3asanordinarynumberpress:4,2 ×10x –3 = ;Answer=0,0042

√___

√__

R Calculatesthevalueofasquareroot

Press √__

R25 = ;Answer=5

3√__

R 3√___

Calculatesthevalueofacuberoot

Press 3√__

R27 = ;Answer=3

√__

 

y1_xx√__

R Calculatesthevalueofallotherroots

Press9 x√__

R512 = ;Answer=2

∴9√____

512=2

n!X!

Calculatesthefactorialofanumber.

Ifthedisplaycontains4andyoupressthiskey,thecalculatorwillworkoutthevalueof4×3×2×1=24

% Calculatespercentages(peronehundred)

Towrite12___50

asapercentage:

Press12 ÷ 50 SHIFT ( = ;

Answer=24%

Otherhelpfulkeysthatyoumayneed:

M+ Storesthevalueinthedisplayinthememoryofthecalculator.

RCLorMR Insertsthecontentsofthecalculator’smemoryintothedisplay.

ACorCE Clearsthecontentsofthedisplay.

Addition, subtraction, multiplication and division by using a calculatorYou can practice the examples below to familiarise yourself with a specificcalculator.

Examples Key Sequence Answer

1. 9–3_____2 9 – 3 = ÷ 2 = 3

2. (9−3)÷2 ( 9 – 3 ) ÷ 2 = 3

3. 9−3÷2 9 – 3 ÷ 2 = 7,5

4. 9−3__2 9 – ( 3 ÷ 2 ) = 7,5

5. 16+4[8×(3−4)] 16 + 4 ( 8 × ( 3 – 4 ) = −16

6. 1_______7,5−3,6

1 ÷ ( 7,5 – 3,6 ) = 0,256

Explanations:

• For no. 1:– Theentirenumeratorisdividedby2,therefore = mustbeenteredafter

9−3beforedividingby2.

Did you know?e1=2,718

Press:1 ex = ;

Answer=2,718

Proof:ex=1+x+ x2

_____1×2

+

x3_________

1×2×3

+ x4____________

1×2×3×4 + …

e1=1+1+12__

2+1

3__

6

+14___

24 + …

=2,718

Hint( "opensa

bracket

) "closesabracket

5

Numbers

• For no. 2:–Bracketsareusedinthecalculation–Question1and2areidentical

• For no. 3 and 4:–Thecalculatordivides3by2beforesubtractingfrom9.(BODMAS).–Thereforetheanswerisnot3asthecaseinno.1and2.

• For no. 5 and 6:–Makesurethatyouenterthebracketsasexplainedintheexample.

Calculating squares, cubes and higher powers by using a calculatorThefollowingkeysondifferentcalculatorsperformthesamefunction:

yx=xy=x=^

Examples Key Sequence Answer

1. 2462246 x2 = 60516

2. 13313 x3 = 2197

3. 272 x 7 = 128

4. 20×12220 × 12 x2 = 2880

5. 232−16323 x2 – 16 x3 = −3567

6. (26+15)3 ( 26 + 15 ) x3 = 68921

7. (152−92)4 ( 15 x2 – 9 x2 ) x 4 = 429981696

8. (−1,4)4 ( – 1,4 ) x 4 = 3,842

Calculating square roots, cube roots and higher roots by using a calculatorThefollowingkeysondifferentcalculatorsperformthesamefunction:

√___

= x1__y = y1__x = x√__

y

Examples Key Sequence Answer (rounded off to 3 decimal places)

1. √____

144 √___

144 = 12

2. 3√_____

4216 SHIFT 3√___

4216 = 16,155

3. 5√_____

1883 5 SHIFT √___

1883 = 4,518

4. √______

9+16 √___

( 9 + 16 ) = 5

5. 3√____

612−√___

42 3√___

612 = – √___

42 = 2,009

6. 3,81_2 3,8 x 1 ÷ 2 = 1,949

NoteTheexamplesare

explainedaccordingtothekeysandfunctionsona“Casiofx−82ES”scientificcalculator..

6

Chapter 1

ExAMplESSolveforxwiththeaidofacalculatortothreedecimalplaces:

Example 1

x=3√_____________

(3,2)4−(4,1___

2π)

3

Solution:

SHIFT 3√___

( 3,2 ) x 4 ) – ( 4,1 ÷ ( 2 × π ) ) x3 =

Answer:4,711

Example 2

x=4,2(2+6,1)3,2−9,4_____

(1,3)2

Solution:4,2 ( 2 + 6,1 ) x 3,2− ( 9,4 ÷ 1,3 x2 ) =

Answer: 3386,012

Assessment activity 1.1

Workwithsomeoneinyourclasswhohasthesamecalculatorasyourstodo

thefollowingactivity.

1. Useyourcalculatortocalculatethefollowing:(Roundyouranswersoffto

THREEdecimalplaces)

(a) (100−84)÷3 (b) 840÷21+3 (c) 522−143

(d) (43,6−19,2)4 (e) 1_____2,8π2 (f)

3√_____

1314

(g) (5√_____

1256+√____

315)2

2. SolveforxwiththeaidofacalculatortoTHREEdecimalplaces:

(a) x= 1____________(2,36−1,2)2

(b) x=(16,42)2,1+√_________

13,4−π2__

4

(c) x=3√____

436+(13,2)3

______12

− 1____3π2 (d) x=6,4(3−1,2)2,1−

3,4_____

(4,2)2

(e) x=√

_____0,36−(3,27)2

______________√

_____1,57 (f) x=

3√_______________

(5,99)4+(3,91_____

2,6)3

1.1.2 Estimate and approximate physical quantities for example length, time, mass and temperature

IntroductionAphysical quantityisaphysicalpropertythatcanbequantified.Thismeans

thatitcanbemeasured and/or calculated.Itcanbeexpressedinnumbers,for

example:

• ‘weight’isaphysicalquantitythatcanbeexpressedbyabasicmeasurement

suchaskilograms.

• ‘length’isaphysicalquantitythatcanbeexpressedbyabasicmeasurement

suchasmeter.

NoteThesolutions

totheexercisesandsummative

assessmentsofeachmodulearegivenin

thebackofthisbook.Usethistocheckyour

answers.

DefinitionAnapproximation(≈)isastatedvalueofanumberthatiscloseto(butnotequalto)

thetruevalueofthatnumber.

For example: A decimal fraction like 145,65 can be approximated to 146 or to one decimal place

145,7 etc.

7

Numbers

ExAMplES1. Youcanestimate theamountofmoneyyouwouldneed toattend the

soccercupfinals:• Transportbytaxi: R55• Ticket: R60• Food: R35• Supporter’scap: R45• Approximateamountofmoneyneeded:≈R195

2. Estimatethefollowinganswer:• 3,9×6,2≈4x6≈24Theactualansweronthecalculatoris:24,18

Measuring instrumentsScientists, engineers and other humans use a vast range of instruments toperform their measurements. These instruments may range from simpleobjectssuchasrulerstoelectronicmicroscopes.

Youneedtoknowsomeofthemeasuringinstruments,aswellastheunitsofmeasurement.Thetablebelow isasummaryof thephysicalquantities thatyouneedtoknow.Otherquantitiesthatarenotinthecurriculumforlevel2studentsarevolume,capacity,force,energy,speed,area,etc.

All measuring instruments are subjected to varying degrees of instrumenterror and measurement uncertainty. Therefore all measurements done areapproximate.

Quantity Units Symbols Measuring instrument Conversions

Length(distance)

millimetrescentimetresmetreskilometres

mmcmmkm

rulersverniercallipersmeasuringtapeOdometer

1cm=10mm1m=100cm1m=1000 mm1km=1000m

Time secondsminuteshoursdayyear

sminhdy

AnaloguewatchesDigitalwatchesStopwatchesClocksCalender(bycountingdays)EggtimerSundial

1min=60s1h=60min1day=24h1year=365/366days

Mass milligramsgramskilogramstons

mggkgt

KitchenscalesBathroomscalesBalancesMassspectrometer

1g=10mg1kg=1000g1t=1000kg

Temperature degreesCelsiusKelvin

°C

K

Thermometer 0°C≈273K100°C≈373K0K≈-273°C

DefinitionAnestimationis

anapproximationofaquantitywhich

hasbeendecidedbyjudgementratherthancarryingout

theprocessneededtoproduceamoreaccurateanswer.

For example: You can estimate the number of people in the room as 30,

when the actual count shows it is 26.

8

Chapter 1

Measuring lengthRulersareusedtomeasuresmallerdistancesinmmorcm.Vernier clippers can be used to measure very small lengths, for example afractionofamm.Electronic devicesareusedtomeasurelongerdistances,forexampletheodometer.

Measuring timeAnalogue clocksaremarkedin12-hourintervals.• 11a.m.means11o’clockinthemorning• 11p.m.means11o’clockintheeveningDigital clocksmeasure24-hourtime.• 11:00means11o’clockinthemorning• 23:00means11o’clockintheevening

Measuring massKitchen scales are used to measure quantities in grams or kilograms, forexample,250gofflour.

Measuring temperatureThermometersareusedtomeasuretemperatureindegreesCelsius.Thehumanbodyhasanormaltemperatureof37°C.TochangetemperaturesbetweentheCelsiusandKelvinscalesuse:• Temperaturein°C=temperatureinKelvinminus273• TemperatureinK=temperaturein°Cplus273

ExAMplESExamplesofconversionsbetweendifferentmetricunits:

Convert: Method:

1. 2cmtomm 1 cm = 10 mm∴2cm=2×10=20mm

2. 24,3cmtomm 1 cm = 10 mm∴ 24,3cm=24,3×10=243mm

3. 2kgtogram 1 kg = 1000 g∴2kg=2×1000=2000g

4. 0,32kgtog 1 kg = 1000 g0,32kg=0,32×1000=320g

5. 1260gtokg 1 kg = 1000 g

∴1260g=1260_____1000

=1,26kg

6. 3htomin 1 h = 60 min∴3h=60×3=180min

7. 4mintos 1 min = 60 s∴4min=4×60=240s

8. 8pmtodigitaltime 20:00

9. 15:45toanaloguetime 3:45pm

10 20°CtoK 20°C+273=293KRemember:0°C≈273K

9

Numbers

Assessment activity 1.2Convertthefollowingmeasurementstothegivenunits:1. 0,65kgtog 2. 1548gtokg3. 11amtodigitaltime 4. 13:15toanaloguetime5. 10p.m.todigitaltime 6. 18,5cmtomm7. 50,54kmtom 8. 2mtomm9. 12,5gtomg 10. 3,2ttokg11. 0,008ttog 12. 200mgtog13. 65000kgtot 14. 200mmtocm15. 14,5gtokg 16. 4425,63mtokm17. 37°CtoK 18. -1°CtoK19. 273Kto°C 20. 301Kto°C

1.2 Demonstrate an understanding of numbers and relationships among numbers and number systems and represent numbers in different waysIntroductionWeworkwithnumbersonadailybasis,oftenwithoutreally thinkingaboutthem. Numbers are used in newspapers (prices, dates), on television or theradio,time,soccerresults,labelsonfood,barcodes,addresses,cellularphones,money,etc.Thisiswhyitisimportantforyoutobeabletoworkwithnumbersandtohaveanunderstandingofthenumbersystem.

Below is a summary of the relationships among several types of numbersillustratedwithaflattwodimensionalVenn-diagram.

Complex numbers

REAl NUMBERS (R)

IMAGINARY NUMBERS

2+3i3–2i√

___–2

√_____

–513

RATIONAl NUMBERS (Q)

IRRATIONAl NUMBERS

(Q1)√

__2

1,414213π

√__

5(Decimaldoesnotrepeat)

INTEGERS (Z)

WHOlE NUMBERS (No)

NATURAl NUMBERS

(N)1;2;3;4…

0;1;2;3;4…

…-3;-2;-1;0;1;2;3…

…3√___

–8;–2__3;0;11__

2;

3√___

–8;4;317;…

Did you know?Zerowasonlyaccepted

bypeoplefromthe18thcenturyonwards.Thenaturalnumbersetwasextendedto

include0.Thisnumbersetiscalledthewhole

numbers:N0={0;1;2;3;4;…}

10

Chapter 1

Belowisadetailedexplanationofthedifferentnumbersystems:

Classification of numbers

Symbol Definition Example

Naturalnumbers

N Numbersusedforcounting;startsat1 N={1;2;3;4;5;…}

Wholenumbers

No Naturalnumbersplusthenumber0 No={0;1;2;3;4;5;…}

Integers Z Positiveandnegativewholenumbers Z={...−3;−2;−1;0;1;2;3;…}

Rationalnumbers

Q Anynumberthatcanbewrittenintheforma__

b;aandbareintegers;b≠0

Q={…−2__3;−4__7;−8__

9;4;1__4;619…}

Irrationalnumbers

Q1 Anynumberthatcannotbeexpressedasanintegerdividedbyaninteger;a__

b.

Thedecimalsneverterminateandneverrepeatwithapattern

Squarerootsofprimenumbersandπareincluded.Q1={…√

__2;√

__3;√

__5…π…}

Realnumbers R Rationalandirrationalnumbers QandQ1

Imaginarynumbers

i=imaginaryunit

Squarerootsofnegativenumbers {√___

−2;√___

−3;√___

−4;√___

−5…}

Complexnumbers

C Combinationofarealnumberandanimaginarynumber;a+biora+bj

3+2ior3+2j(ThiswillbecoveredinLevel3and4)

Primenumbers

Numbersthatareonlydivisibleby1orthenumberitselfwithoutaremainder

{2;3;4;7;11;13…}(1isnotaprimenumber)

Evennumbers

Arewholenumberswhich,whendividedby2havenoremainder

{2;4;6;8…}

Unevennumbers

Arewholenumberswhich,whendividedby2havearemainderof1

{1;3;5;7;…}

1.2.1 Rational and irrational numbers

The nature of rational and irrational numbers

Introduction

Numberssuchas√__

4and√___

16canbewrittenasnumbersthatwearefamiliar

with,forexample:

• √__

4=2

• √___

16=4

These numbers are called rational numbers

However,whenyoutry towritedownthevalueof √__

2,youneedtouseyour

calculator.Theanswerfor√__

2onyourcalculatorwillbeanapproximationas

thecalculatorcanonlydisplayalimitednumberofdecimalplaces.

• √__

2≈1,414213562

• √______

112,2≈10,59245014

These numbers are called irrational numbers

Did you know?Negativenumberslike–3wasonlyaccepted

inthe17thcentury.Peoplethoughtthat4–7hadnoanswer!

11

Numbers

Rational numbersA rational number can be written in the form a __

b , where a and b are integers and

b≠0.

For example:• 3 = 3 __

1

• – 3 __ 4

• 0,1•4• (This number is called a terminating decimal and can be written as a

fraction. This will be explained later in this chapter.)

ExamplEsThe following numbers are all rational since they can be written in the form a __

b :

Number −3,5 1 1 __ 3 0,09090909… 3,1

•42857

•8

Written in the form a __

b − 7 __

2 4 __

3 1 ___ 11 22 ___ 7 8 __ 1

Rational numbers (Q) include integers, whole numbers and natural numbers as well as fractions.

The table below provide you with examples to work through in groups/pairs in the classroom.

Number Write in form a __ b a, b E Z b≠0;

Yes/NoRational: Yes/No

1. 1 2 __ 3 8 __

3 Yes Yes

2. 2, 5 2 5 ___ 10

= 25 ___ 10

= 5 __ 2 Yes Yes

3. − 10 ___ 3 − 10 ___

3 or − 10 _____

3 Yes Yes

4. √___

16 √___

16 = 4 or 4 __ 1 Yes Yes

5. 3 √____

−27 −3 = −3 ___ 1 or 3 ___

−1 Yes Yes

6. −5 − 5 __ 1 or − 5 ____

1 or 5 ___

−1 Yes Yes

7. √__

7 √__

7 ___ 1

√__

7 is not an integer

No, irrational

8. π 3,141592654 (cannot be written as a __

b )

No, irrational

Irrational numbersIrrational numbers can only be written in number form (using no symbols) as a never-ending, non-repeating decimal fraction. Irrational numbers cannot be expressed as an integer divided by an integer. These numbers cannot be expressed as fractions in the form a __

b .

For example:• 1,234567891011.... • √

___ 10 = 3,162277...

• π = 3,1415927....

Hint Division of a number by zero is undefined in any number system,

e.g. a __ 0 undefined,

but 0 __ a = 0.

A whole number multiplied by 0

equals 0. You can check all of the above with your

calculator.

12

Chapter 1

Irrationalnumbersdorepresentrealquantities.

You may be tempted to think that irrational numbers do not represent realquantitiesinoureverydayworld.Youwouldnotthinkofgoingintoashopandaskingfor√

__2kgofsugaroraskingfor√

__2slicesofcake.

Let'slookatthesimpleright-angledtriangleshowninthediagrambelow.Eachofthetwoshortersidesis1mlong.

InthissectionwewillmakeuseofageometricfactknownastheTheorem ofpythagoras.Thistheoremtellsustherelationshipthatexistsbetweenthethreesidesofanyright-angledtriangle.Thelongestsideofaright-angledtriangle,thesideoppositetherightangle,isknownasthehypotenuse.Ifwedenotethelengthofthehypotenusebycandthelengthsoftheothertwosidesbyaandb,thenwewillalwaysfindthat:

c2 = a2 + b2

Thefollowingdiagramshowsarightangledtriangleinwhichaandbareboth1unitlong.

c2 =a2+b2

=12+12 =1+1 =2∴c =√

__2

Whatthisshowsusisthat√__

2canrepresentanactualphysicalquantity.Itisthelengthofthelineinthediagramabove.

This leads us to the very interesting conclusion that there are some actualmeasurementsthatcannotbewrittendownexactly,eventhoughtheyexist!

Anumberthatcanonlybeexpressedexactlybyusingarootsigniscalledasurd.•

√__

4,forexampleisnotasurd.Itcanbeexpressedas2whichisarationalnumber.

• 3√__

2,forexampleisasurd.Itcannotbeexpressedasanexactvalue.


Arethefollowingnumbersrationalorirrational?Drawandcompletethetablebelow:

Number Rational or Irrational

Number Rational or Irrational

1. 7,8• 6. 22___

7

2. 5√__

4 7. 3√___

64

3. 2√__

3 8. 4,12

4. 4,128128… 9. −1___12

5. π 10. 2,13612143

A

B

Cb=1metre

a=1metrec=?metre

Note√

__2,forexample,iscalledasurd.

Wewillbedoingcalculationswithsurds

laterinthischapter.

13

Numbers

1.2.2 Rounding off rational and irrational numbers

IntroductionIneverydaymeasurementsandamountsofmoneyweoftenneedtoworkwithapproximatenumbersornumbersthathavebeenroundedoff.InMathematicsyouwillbeaskedtoroundofftoagivennumberofdecimalplaces.Forexample,theinstructioninthefinalexaminationpapercanbeasfollows:

‘Round your answers off to THREE decimal places, unless stated otherwise.’

Pre-knowledgeUsethefollowingrulestoassistyouwhenroundingoff:1. Lookatthevalueofthedigit to rightofthespecifieddigit.2. Round upifthevalueis5,6,7,8or9. Add 1tothespecifieddigit. Forexample:

– Round17,5867offto1decimalplace ∴ 17,5867≈17,6Round17,5867offto2decimalplaces ∴ 17,5867≈17,59Round17,5867offto3decimalplaces ∴ 17,5867≈17,587Ifthevalueis0,1,2,3,or4thedigitmustbeleftunchanged.

ExAMplEπcanbeshowntobeanirrationalnumber.Ithasbeencalculatedtooveramilliondecimalplacesandnorecurringpatterninthedigitshasbeenfound.Herearethefirstdigits:3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128

Expressπasanapproximaterationalnumberto4decimalplaces.

Solutionπ≈3,1416.The4thdecimalplaceisa5,butthe5thisa9.Thereforechangethe5toa6.

Fractional approximations to πThefirst few fractionalapproximationsofπareas follows: 3, 22___

7, 333____

106, 355____

113,

103993_______33102

,104348_______33215

.


1. CalculateandgivetheanswerscorrecttoTHREEdecimalplaces:(a) 3,864×0,236 (b) 0,006749÷0,000382 (c) 0,00745÷1,7

2. CalculateandgivetheanswerscorrecttoTWOdecimalplaces:(a) 2π(1,2)2 (b) π(3,05)2×0,006

3. Round98,784035to:(a) 2decimalplaces (b) 1decimalplace (c) 5decimalplaces

Did you know?Irrationalnumberscan

beapproximatedtorationalnumbersbyroundingthemoffto

anyrequireddegreeofaccuracy.

14

Chapter 1

1.2.3 Convert rational numbers between terminating and recurring decimals

IntroductionDecimalsareused forexample tomeasuredistance, lengthandamountsofmoney.YouwillnotexpressapriceasR201__

2orR16211___

12.

ThiswillbeexpressedasR20,50orR162,92respectively.

A fraction can be converted to a decimal bydividing thenumeratorby thedenominator.Theanswerwilleitherbeaterminating decimalorarecurring decimal.

Pre-knowledge

3__4"numerator=topnumber"denominator=bottomnumber

∴3__4=0,75

Powersoftenareasfollows:• 101=10• 102=100• 103=1000• 104=10 000,etc.

Terminating decimalsA terminating decimal can be described in everyday language as a decimalfractionthatdoesnotgoonforever.Ithasafixed number of decimal places.Forexample:• 0,4;0,123;1,61;0,747474

Example 1Howtoconvertaterminatingdecimaltoafraction:

• 0,5: whichcanbewrittenas 5___10

=1__2 !use10asthedenominator

• 0,42: whichcanbewrittenas 42____100

!use100asthedenominator

=21___50

!simplify

• 4,543:whichcanbewrittenas4543_____1000

!use1000asthedenominator

or4543_____1000


1. Convert each rational number to a decimal. Round your answers off toTHREEdecimals:(a) 3__

4 (b) 12___

17 (c) −4__

9 (d) 34__

5

2. Convertthefollowingdecimalstofractionsinthesimplestform:(a) 0,3 (b) 0,54 (c) 0,613(d) 0,0035 (e) 3,34 (f) 0,001

NoteAllfractionscanbewrittenasdecimals.

HintThenumberofthe

decimalsisthesameasthenumber

ofthezerosinthedenominator.

DefinitionAterminating decimalisarational number

thatcanbewrittenasafractionwithapowerof10asadenominator.

Thereforedecimalscanbeconverted

tofractionsbyrepresentingthemastenths,hundredths,

thousandths,etc.

15

Numbers

Recurring decimalsRecurringdecimalsgoonforeverandaremuchtoolongtoworkwith.Youhavetoroundthemoff.Recurringdecimalscannot be written as fractionswithadenominatorthatisapowerof10.

ExAMplESExamplesofrecurringdecimalsare:0,666666666...=0,6

• !thedotshowsthatitisonlythe6thatrepeats

2,141414...=2.1•4• !thedotshowsthatboththe1andthe4repeat

3,5222222...=3,52• !thedotshowsthatthe2repeats,butnotthe5

16,315315...=16,3•15

• !the dot shows that all the numbers between the

dotsrepeat

It is important for you to know how your calculator deals with recurring decimals.

ExAMplES

Example 1

Enter1÷6onyourcalculator:• Ifyouget0,166666 6667thenyourcalculatorroundsoffthelastdecimal

place• Ifyouget0,166666 6666thenyourcalculatortruncates(cutsoff)thelast

recurringdecimal

∴ 1__6=0,16

•

Example 2

Enter18___7onyourcalculator:

• Answer=2,571428571 =2,5

•71428

•

Conclusion:Somecalculatorswillroundthemoffandotherswillcutthemoffattheendofthedisplayarea.


Workingroups:

Writethefollowingasrecurringdecimals:1. (a) 0,8888…… (b) 13,626262……

(c) 3,6777…… (d) 5__6

(e) 32,435435435…… (f) 67,08910891……(g) 12__

3 (h) 2,1732732732……

2. Statewhetherthefollowingareterminatingorrecurringdecimals:(a) 2,333…… (b) 0,600000……(c) 4,333 (d) 2,463297(e) 6,352631927…… (f) 3,261261……(g) 6,060606……

DefinitionArecurring decimalisadecimalfraction

whichgoesonREPEATINGitself

withoutend.

Remember22___7isanapproximation

forπ

Noteπ=3,141592654.....:

Irrational;non-terminating;non-recurringdecimal

22___7=3,142857142...... :Rational;recurring

decimal

NoteRemembertosimplify

allanswerstothesimplestform!

16

Chapter 1

Converting recurring decimals to fractions [ a __ b ; where a, b∈Z ; b ≠ 0 ].

ExamplEs

Write each recurring decimal as a fraction in the simplest form:

Example 1 0,5•

solution Let x = 0,555… (1) Then 10x = 5,555… (2)

10x − x = 5,555 − 0,555 ∴ 9x = 5

x = 5 __ 9

∴ 0,5• = 5 __

9

• Let the decimal be equal to x• Multiply by 10 to move the comma

after the FIRST recurring digit.• Subtract equation (1) from (2)

• Divide both sides by 9• Check the answer on your calculator

Example 2 0,1•8•


100x – x =18,181818 – 0,181818 99x = 18

x = 18 ___ 99

∴ x = 2 ___ 11

∴ 0,1•8• = 2 ___

11

• Let the decimal be equal to x• Multiply by 100 to move the comma

after the SECOND repeating digit• Subtract equation (1) from (2)

• Divide by 99 both sides • Write in the simplest form• Check the answer on your calculator

Example 3 0,5•14

•


1000x − x = 514,514514 − 0,514514 ∴ 999x = 514

x = 514 ____ 999

∴ 0,5•14

• = 514 ____

999

• Let the decimal be equal to x• Multiply by 1000 to move the

comma after the THIRD repeating digit.

• Subtract equation (1) from (2)• Divide both sides by 999• Check the answer on your

calculator

Example 4 0,02•8•

solution Let x = 0,0282828 10x = 0,282828… (1) 1000x = 28,282828… (2) 1000x − 10x = 28,282828 − 0,282828 990x = 28

x = 28 ____ 990

∴ x = 14 ____ 495

∴ x = 0,02•8• = 14 ____

495

• Let the decimal be equal to x• Multiply by 10• Multiply by 1000• Subtract: Equation (2) – equation (1)• Divide both sides by 990• Simplify the fraction

• Check the answer on your calculator

HintWe use algebra to convert recurring

decimals to fractions.

Did you know?A South African mathematician,

Stanley Skewes, claims that he has created one of the largest numbers used in

Mathematics:

• 101010

34

NoteRemember that you

can use your calculator to check your answers.

Example: Press 14 ÷ 495 =

Answer: 0,0282828……

17

Numbers


1. Expresseachrationalnumberasarecurringorterminatingdecimal:(a) 5__

8 (b) 11___

6 (c) 16___

37

(d) 8__7 (e) 423___

99

2. Converting the following recurring decimals to common fractions in thesimplestform:(a) 0,8

• (b) 0,6

•3• (c) 0,3

•12

•

(d) 0,36• (e) 0,01

•6• (f) 1,2

•4•

(g) 3,34•5•

1.2.4 Laws of exponents

Exponents (Indices)

IntroductionWe use scientific notation when we work with many digits (in very large orverysmallnumbers).Scientificnotationisusedtoexpressnumbersinamoremanageableform.For example: 0,00011123 is written in scientific notation as 1,1 x 10 –4.Therefore we can write very large numbers and very small numbers in a more manageable form.

Youneedtounderstandexponentialnotationeffectivelyinordertounderstandscientificnotation.

Youhavealreadyworkedwith the lawsof exponents.Youmustknow theselawsandalsohowtousethem.Youmustalsounderstandnegativeandpositiveexponents.Allofthiswillassistyouwhenyouhavetosimplifyorsolvealgebraicexpressionsandsolveequations.

Pre-knowledgeYou have already learnt how to apply the following laws of exponents inpreviousgrades:1. am×an=am+n 2. am÷an=am−n(m>nfornaturalnumbers)

3. (ab)n=anbn 4. (am)n=amn

Wheren,m∈N

Rememberthat:• a+a+a+a+a=5a• a×a×a×a×a=a5

−3xmcoefficient=−3

exponent(index)=mbase=xpower=xm

DefinitionAnexponent (index)

isamathematicalwayofrepresentinga

verylargenumberoraverysmallnumberinaformatthatisuser

friendly.

Remembera0=1(a∈R;a≠0)and

00=meaningless

Definitionan=a.a.a.a.a......a(ntimes)where

n∈Nanda>0Forexample:

28=2×2×2×2×2×2×2×2

andx5=x×x×x×x ×x

HintLawsofexponentsareonly valid for × and ÷Theyarenotvalidfor

+and–

18

Chapter 1

Example Base Exponent Coefficient of x:

1. x3 x 3 1; Remember that x3 = 1.x3

2. 4x2 x 2 4

3. −2x 1 _ 2 x 1 __

2 −2

4. − 1 __ 4 ax x 1 − 1 __

4 a; Remember that x = x1

• + × + = +• + × − = −• − × + = −• − × − = +

If the signs are the same, the answer is +If the signs are different, the answer is −

Also, when dividing:• + ÷ + = +• + ÷ − = −• − ÷ + = −• − ÷ − = +

Below is a table with a summary of the laws of exponents and their applications:

law Example Explanation

1. am × an = am + n 1.

2.

3.

4.

5.

6.

a4 × a2 = a4 + 2 = a6

x3 × x2 = x3 + 2 = x5

23.24 = 2 3 + 4 = 27

2x2y3 × 4xy3 = 8x3y6

3 × 32x × 33 = 32x + 4

x3 × x−2 = x3 + (−2) = x3 – 2 = x

When you multiply powers with the same base, you add the exponents.The base stays the same.

Remember:(+) × (−) = (−)

2. am ___ an = am − n 1.

2.

3.

4.

5.

a6 __

a3 = a6−3 =a3

24 ÷ 2 = 24−1 = 23 = 8

32 __

35 = 32 – 5 = 3–3 = 1 __ 33 = 1 ___

27

6x6 ____

2x2 = 3x4

a6x ___

a3 = a6x−3

When you divide powers with the same base, you subtract the exponents.For example no.3: Refer to deduction no. 1 Divide the coefficients and subtract exponents (indices)

3. (am)n = amn 1.

2.

3.

4.

5.

6.

(a4)2 = a4 × 2 = a8

(23)2 = 26= 64

(x−3 . y 1 _ 3 .z)3 = x−9 yz3

= yz3

___ x9

(27) 2 _ 3 = (33)

2 _ 3

= 32 = 9

(2x−3)2 = 22(x−3) = 22x−6

[(32)4]2 = [38]2 = 316

= 43046721

When you raise a power to a power, you must multiply exponents.

See deduction no. 2

Remember: 3 __ 1 × 2 __

3 = 2

DefinitionCoefficients are the

numerical parts of the expression

A power comprises of the following two

components: the base and the exponent

19

Numbers

4. (a.b)m = ambm 1.

2.

3.

4.

(a b)2 = a2b2

(2 a)3 = 23a3

= 8a3

6x = (2.3)x = 2x.3x

(2xn−3)2 = 22x2(n−3) = 4x2n−6

When you raise a product in a bracket to a power, each factor will be raised to that power.

5. ( a __ b

) m = am ___

bm

or

( a __ b

) −m

= ( b __ a ) m

= bm ___ am

1.

2.

3.

4.

5.

( a __ b

) 2 = a

2 ___

b2

( 3a ____ 2b2 )

3 = 3

3a3 ____

23b6

= 27a3 _____

8b6

( 12 ___ x ) a+3 = ( 22.3 ____ x ) a+3

= 22a+6.3a+3

________ xa+3

( 3 __ x2 ) −3

= ( x2 __

3 ) 3

= x6 _____

27

( 3−1 ___

2−1 ) −2 = ( 2 __

3 ) −2

= ( 3 __ 2

) 2

= 9 __ 4

= 2 1 __ 4

When you raise a fraction in a bracket to a power, the numerator and the denominator will both be raised to that power.

6. n √___

am = a m __ n

or

a m __ n = n √___

am

1.

2.

3.

4.

5.

6.

3 √__

a3 = a 3 _ 3 = a1 = a

√__

53 = 5 3 _ 2

4 1 _ 2 = √

__ 4 = 2

4 √__

x3 = x 3 _ 4

1 __ 3 × 9 = 1 __

3 × 9 __

1 = 3

2 × 1 __ 2 = 1

4 × 1 __ 2 = 2

Expressions that can be written in the form

n √___

am for a>0 and n∈N are called surds. The left- hand side of this equation is therefore the surd form of the right-hand side.

Deductions Applications

1. a0 = 1 (a ≠ 0) 1.

2.

3.

4.

5.

6.

x0 = 1

40 = 1

(a + b)0 = 1

2 (a + b)0 = 2(1) = 2

(−2)0 = 1

−(20) = − (1) = − 1

HintAn exponent is an

index. The plural of index is indices.

Laws of exponents = Laws of indices.

20

Chapter 1

2. a−m = 1 ___ am

or

1 ____ a−m = am

1.

2.

3.

a−4 = 1 __ a4

3−2 = 1 __ 32

= 1 __ 9

2−2 ___

3 = 1 ____

223

= 1 ___ 4.3

= 1 ___ 12

4.

5.

6.

( 3 __ 2 ) −2

= ( 2 __ 3 ) 2

= 22 __

32

= 4 __ 9

x ___ y−4 = x y4

a−2 b2 ______

a3 b−1 = b. b2 _____

a3.a2

= b3 ___

a5


1. Use appropriate index laws (laws of exponents)to simplify the following:(a) 2x2y × 3xy2 (b) x4 . x−2 (c) 2x+y . 2x−y

(d) a3x ÷ a2x (e) 2x+y ____

2x−y (f) 15h6 d4 _______

3d3h2

(g) 16 k3 t7 ÷ 48 kt5 (h) (3x3)3 (i) ( 5p2q3

______ 3r

) 4

(j) (3x+1)2

2. Evaluate by using the first deduction law: a0 = 1(a) 30 (b) 2xy0 (c) (3xy7)0

(d) 2 (8x)0 (e) 4a0 − 2 (f) 3x0 − (2y)0

(g) ( a + 3b ___ 4 ) 0 (h) x3. 2x2. 3x. x0. (2x)0

3. Express the following expressions in the simplest form and with positive exponents:(a) x−5 (b) 3x−6 (c) 3 ___

x−4

(d) (4x−3)4 (e) 3 p−4 q−2 z3 (f) a−4 . a3

(g) x−6 y4 × x3 y−2 (h) 6 x4 b3 × 3x−5 b (i) x3y5 ÷ x2 y7

(j) 3x2y

_____ 9xy2

HintsThe reciprocal of

• 2 is 1 __ 2

• − 1 __ 3 = –3

This is known as the multiplicative inverse.

21

Numbers

Simplify the following by using exponential laws. leave your answers with positive exponents.

Example 1

2a0×b÷1__b5

=2(1)×b÷1__b5

=2(1)×b×b5

=2(1)×b6

=2×b6

=2b6

• Applydeductionno.1

• Applylawno.2

Example 2

3a−5a2_______

3−1a−2b−4

=3.3.a2.a2b4_________

a5

=9a4b4_____

a5

=9b4___a


• Applylawno.1

• Applylawno.2

Example 3

(xy2)3

_____x−2y

×(x3y−2)3

=x3y6

____x−2y

×x9y−6 • Applylawno.3 Raisethepowersfirst

=x3.x2y6

_______y ×x9

__y6

• Makeexponentspositive Applydeductionno.2

=x14y6

_____y7 • Applylawno.1

=x14___y • Applylawno.2or

y6

__y7=y6−7=y−1=1__y

Example 4

[(−2x2)3]−2

=[−23x6]−2 • Applylawno.3 Simplifytheinnerbracketsfirst:()

=[−8x6]−2 • 23=2×2×2=8

= 1_______[−8x6]2 • Apply deduction no. 2 (Work with positive

exponentswherepossible)

= 1_____82x12

= 1_____64x12

• (−8)(−8)=+64

RememberAnegativeexponent

meansinvert,itdoesNOTresultina

negativeanswer.Anegativeexponent

indicatesdivision.Example 1

2–1≠–2but2–1=1__

2

Example 2

10–1=1___10

and10–2= 1_______

10×10= 1____

102

Remember• 1n=1

Therefore15=5

• (–1)n=1ifniseven

Example:(-1)6=1

• (–1)n=–1ifnisuneven

Example:(–1)5=–1

22

Chapter 1

Example 5

6(a−3b5)2

________a3b−4 × a2b5

________(3a−3b2)2

=6a−6b10_______

a3b−4 × a2b5_______

32a−6b4 • Applylawno.3

=6b10.b4_______

a6.a3 ×a2b5a6

______9b4 • Applydeductionno.2

=6b10.b4.a2.b5.a6______________

9.a6.a3.b4 • a__b

×c__d

=a×c______b×d

=6a8b19______

9a9b4 • Applylawno.1

=2b15____

3a • Simplify:6__

9=2__

3

Example 6

x4y+6z.x3y−3z

___________x4y+z

=x4y+6z+3y−3z

___________x4y+z • Applylawno.1

=x7y+3z______

x4y+z • Addliketerms(innumerator)

=x7y+3z−(4y+z) • Applylawno.2 RemembertheBRACKET!

=x7y+3z−4y−z • Removebrackets:multiply

=x3y+2z • Addliketerms

Pre-knowledge: Remember your laws of fractions• a__

b×c__

d=ac___

bd

• a__b

÷c__d

=a__b

×d__c=ad___bc

• a__b

+c__d

=ad+bc________bd


Simplify the following expressions by using exponential laws. Leave youranswerswithpositiveexponents.1. (x3y4)4×(3x2y4) 2. (

−3x2y______

2xy3)2

3. (x3y−2)3×(xy2)2

_____x5y−6 4. m

3n−6______

m−4n7×m−2n4______

m5n−9

5. 2(x−1)3y4

________(xy2)3

×(x−4y2

_____x3y−3)

3

6. [(−3x3)4]−3×(2x)0

7. (−3x3)(−2x3y4z)(−3z2)

___________________(4x3z)(−3yz)(−3xyz)

8. 3(x2y−4)2

________2(xy2)2

÷(xy)−3

________(3x−2y4)2

9. (3a−2b3c)2×2a3b−4

_________________(6a−1b2c3)3

10. 4a+1×36−a−1___________________

45−a+1×5a−1×81−1

11. (x15y10

______x−6y−14)

1_3×x

−1____

x2y3 12. a3x−2y.a4x−3y

____________a2x+6y

23

Numbers

Forthefollowingexamplesweuselawno.6:

n√___

am=(am)1_n=am__n

Example 73√__

x3

=x3_3

=x

or3√__

x3

=(x3)1_3

=x

Example 8√

_____81x6

=(92x6)1_2

=92×1_2x6×

1_2

=9x3

Example 9√

____9x4

=(9x4)1_2

=(32x4)1_2

=32×1_2.x4×

1_2

=3x2

Example 103√

______27x6y9

=3√______

33x6y9

=33_3x

6_3y9_3

=3x2y3

or3√

______27x6y9

=(27x6y9)1_3

=(33x6y9)1_3

=3x2y3

Example 114√

__a.(bc)−1_

2.√

__c3.√

__b.(a3)

1_4

=a1_4.b−1_

2.c−1_

2.c

3_2.b

1_2.a

3_4

=a1_4+3_

4.b−1_

2+1_

2.c−1_

2+3_

2

=a.b0.c2_2

=ac1

=ac

• Applydeductionno.3andlawno.3

• 1__4+3__

4=4__

4=1and

–1__2+3__

2=2__

2=1

• b0=1

Example 12

3√______

27x7y13

______xy

=3√_______

27x6y12

=(27x6y12)1_3

=(33x6y12)1_3

=3x2y4

• Applylawno.2


• Applylawno.3:3×3×3=27

Assessment activity 1.10Simplify the following expressions by using exponential laws. Leave youranswerswithpositiveexponents.1. 272_

3

2. (−81)1_4

3. √_____

81x2

24

Chapter 1

4. (x3y3_4)

2_3

5. (a−2____

2b0)−3×

3√__

a6

6. 3√_______

64a11b5_______

a2b2

7. 3√__

ab–2_______

√__

a3b−4

1.2.5 Rationalise fractions with surd denominators

IntroductionThereareanumberoftimeswhenitismoreappropriatetoleavetheresultofanequationinsurdformratherthantowritedownanapproximatedecimalorfractionalform.Thesewouldinclude:

• Iftheresultisanintermediatestageofacalculation.• Iftheresultisoftheoreticalratherthanpracticalimportance.• Anexactratherthananapproximatesolutionisrequired.

Asurdisasquarerootofawholenumberwhichproducesanirrational number.It can also be a cube (or other) root and is sometimes applied to an expression which contains a surd or surds.

The square root of any prime number is a surd.Forexample:• Theirrationalnumbers√

__2(≈1,414…)√

___11,

3√___

24,2√__

3,areallsurds.

Press √__

3onyourcalculator.Theansweronyourcalculator is≈1,732050808.Thedigitsneverendanddonotrecur.Thissurdisanirrational number.Inshort:surds cannot be written as fractions.

ExAMplES• √

__2≈1,4142135:surd ......can’tbesimplifiedfurther

• √__

4=2:notasurd ......canbesimplified:√__

4=2•

3√___

11=2,2239800:surd ......can’tbesimplifiedfurther•

3√___

27=3:notasurd ......canbesimplified:3√___

27=3

Calculations with surdsAllthecalculationsinthissectionoftheworkwillbedoneWITHOUTtheuseofacalculator.

Inordertosimplifysurdforms,youcanmakeuseofthefollowingtwobasicrules:

Rule 1:√___

xy=√__

x×√__

yorn√___

xy=n√__

x×n√__

y

ExAMplES

• √__

5×√__

3=√_____

3×5=√___

15• √

___xy=√

__x×√

__y

Rule 2: √__

x__y=√__

x___√__

yorn√__

x__y=n√__

x___n√

__y

HintIfitisarootand

irrational,itisasurdAlsotakenote:not all

rootsaresurds

NoteThesetworules

ONLYapplyifthesurdsarethesame

25

Numbers

ExamplEs

• √___

15 __ 4 = √___

15 ____ √__

4

• 3 √__

x ___ 3 √

__ y = 3 √

__ x __ y

ExamplEsExample 1Simplify √

___ 75

solution √

___ 75

= √______

25 × 3 (25 is the largest perfect square that is a factor of 75)

= √___

25 × √__

3 = 5 × √

__ 3

= 5 √__

3

Example 2Simplify √

____ 108

solution √

____ 108

= √______

36 × 3 (36 is the largest perfect square that is a factor of 108)

= √___

36 × √__

3 = 6 √

__ 3

Example 3

Simplify √___

27 ____ √

__ 3

solution √

___ 27 ____

√__

3

= √_____

9 × 3 ______ √

__ 3

= √__

9 × √__

3 _______ √__

3

= √__

9 = 3

Example 45 √

__ 6 × 4 √

__ 3

= 5 × 4 √_____

6 × 3

= 20 × √___

18

= 20 × √_____

9 × 2

= 20 × √__

9 × √__

2 = 20 × 3 ×

√__

2 = 60 √

__ 2

• Multiply the whole numbers first: 5 × 4

Then multiply the surds• Simplify: 9 is the largest perfect square that is a

factor of 18 (Remember √__

9 = 3)• √

_____ 9 × 2 = √

__ 9 × √

__ 2 : Rule no. 1

Example 5

4 √__

8 ____ √

__ 4

= 4 √_____

4 × 2 _____ 4

= 4 √__

2

• Rule no. 2

• Simplify: √______

4 × 2 _____ 4 = √__

8 __ 4 = √__

2

Rationalise the denominatorWhen you simplify surd forms, you must always ensure that the surds are in the numerator and not in the denominator. We can make the denominator of a fraction a rational number if the fraction is a surd.

Remember• √

__ 4 = √

_____ 2 × 2

= 2 Perfect square

• 3 √___

27 = 3 √_________

3 × 3 × 3 = 3

• √___

18 = √_____

2.3.3 = 3 √

__ 2

Not a perfect square

Remember also that n √___

xn = x

Hintperfect squares are 12, 22, 32, 42, 52,etc

Remember √

__ 3 ___

√__

3 = 1

26

Chapter 1

Togetridofthesurdinthedenominatoryoucanmultiplythenumeratorandthedenominatorbythesamequantity.ThisprocessiscalledRATIONAlISATION.

ExAMplESExample 11___√

__5

=1___√

__5×√

__5___

√__

5

=1×√__

5_______√

__5×√

__5

=√__

5___5

• Rationalise:multiplythedenominatorandnumeratorbythesamesurd

• √__

5×√__

5=√___

25=5

Example 23___√

__2

=3___√

__2×√

__2___

√__

2

=3×√__

2_______√

__2×√

__2

=3√__

2____2

• Multiply the denominator and numeratorbythesamesurd

• √__

2×√__

2=√__

4=2

Inthenextexamplesyouwillneedtofindtheconjugate.

Forexampletheconjugateof:• 3+√

__2is3−√

__2

"

Thesignofthesquarerootchanges• 2−√

__7is2+√

__7

Pre-knowledgeWhenwemultiplytwobinomials,weuseFOIL.

Forexample:(a+b)(c+d)

F : first×first = a×c = acO : outer×outer = a×d = adI : inner×inner = b×c = bcL : last×last = b×d = bd

∴(a+b)(c+d)=ac+ad+bc+bd

Whenyoumultiply(2+√__

2)(2−√__

2)youwillbeusingFOIL.

∴(2+√__

2)(2−√__

2)

=4+2√__

2−2√__

2−2=4–2=2

DefinitionConjugate anglesarepairsofangleswhichareaddedtoget360°

27

Numbers

Example 3 3_______2+√

__3

= 3_______2+√

__3×2−√

__3_______

2−√__

3

=3(2−√

__3)_______________

(2+√__

3)(2−√__

3)

= 6−3√__

3_________________4+2√

__3−2√

__3−3

=6−3√__

3________1

=6−3√__

3

• Multiplybytheconjugate

• UseFOIL: 3(2−√

__3)=3(2)–3(√

__3)=6−3√

__3

• √__

3×√__

3=√__

9=3 √

__3×2=2√

__3

• 2√__

3−2√__

3=0

Example 4 3_______3−√

__3

= 3_______3−√

__3×3+√

__3_______

3+√__

3

=3(3+√

__3)_______________

(3−√__

3)(3+√__

3)

= 9+3√__

3_________________9−3√

__3+3√

__3−3

=9+3√__

3________6

=3(3+√

__3)_________

6

=3+√__

3_______2

• Multiplybytheconjugate

• UseFOIL

• Factorise(ThiswillberevisedinChapter2)

• 3isacommonfactor

• Simplify


1. Write the following surds in the simplest form without the use of yourcalculator.(a) √

___20 (b) √

___45 (c)

3√___

16

(d) 3√__

8 (e) 3√____

343_____5 (f) √

__2.√

___18

2. Rationalisethedenominators:(a) 5___

√__

5 (b) 7___

√__

2 (c) 1____

3√__

3

(d) 1_______7−√

__3 (e) 5_______

8−√__

6 (f) 2+√

__5_______

2−√__

5

(g) 4_______4+√

__2 (h) 10_______

√__

5+2 (i) 2_______

√__

6−2

1.2.6 Add, multiply and divide simple surds

Pre-knowledgeIn theprevious sectionweexplained themethodsof themultiplicationanddivisionofsurds.

Wewillfocusontheadditionandsubtractingofsurdsinthissection.

RememberRememberthat√

_____x+y≠√

__x+√

__y

• Forexample

√___

16+√__

9≠√___

25because

√___

16+√__

9=4+3=7

28

Chapter 1

We can add or subtract surds when the terms contain like terms or the same terms.

exampleS

example 1 √

__ 3 + √

__ 3

= 2 √__

3

example 22 √

__ 5 + 3 √

__ 5

= 5 √__

5

example 35 √

__ 2 − 3 √

__ 2

= 2 √__

2

example 46 √

__ 2 − 3 √

__ 2 + √

__ 2

= 4 √__

2

example 5 √

__ 3 + 2 √

__ 2

= √__

3 + 2 √__

2


Simplify without the use of a calculator:

1. −3 √__

2 + √__

2

2. 4 √__

2 − 5 √__

3 + √__

2 + 2 √__

3

3. √__

6 + √__

3

4. √___

18 ____ 6 √

__ 2

5. 2 √__

3 × 3 √__

2

6. √___

54 − 3 √___

24 + 2 √__

6 _________________ √

____ 150

1.2.7 Manipulation of simple technical and non-technical formulae

Introduction

Engineers, technicians, accountants and many other professions use the skill of manipulating formulae daily.

Formulae are written so that:• a single variable, the subject of the formula, is on the left -hand side of the

equation. • everything else goes on the right-hand side of the equation.

General rules that you should remember when manipulating formulae:

Rule examples: change the subject to ‘a’

1. Additive inverse a + b = c a + b − b = c − b ∴ a = c – b

• subtract ‘b’ from both sides

a − b = c a − b + b = c + b ∴ a = c + b

• add ‘b’ to both sides

2. Multiplicative inverse

ab = c

ab ___ b

= c __ b

∴ a = c __ b

• divide both sides by ‘b’

a __ b

= c

b __ 1 × a __

b = c × b __

1

∴ a = bc

• multiply both sides by ‘b’

3. Raising to a power a2 = b a = ± √

__ b

• raise both sides of the equation to the same power

DefinitionAn equation is a statement where

two expressions (one of which may be a constant) have the

same valueFor example:

3x + 2 = 12 and2(x + 4) = 3x – 16

Definitionmanipulation

or changing the subject of a formula

is rearranging (transposing) it so that the value of a different

quantity from that given can be worked out. Manipulation is nothing more than solving equations

which you have been doing in grade 9

For example:• The formula

A = π r2 is rearranged to give r = A __ π

• The equation 4x + y = 3 can be transposed to y = –4x + 3

29

Numbers

√__

a =b (√

__a)2 =(b)2

∴a =b2

Or a1_

2 =b

(a1_2)2 =b2

∴a =b2

• squarebothsidesoftheequation

3√__

a =b (3√

__a)3 =b3

∴a =b3

Or a1_

3 =b

(a1_3)3 =b3

∴ a =b3

4. Inversion 1__a =1__b

+1__c

1__a =c+b_____bc

∴a = bc_____c+b

• Youcanonlyinvertasingleterm.∴findtheLCMoftheright-handsidebeforeyouinvertbothsides

Pre-knowledge LCM The lowestcommondenominatorof two (ormore) fractions is thesmallestnumber,intowhichalloftheirdenominatorswilldivide.ItistheLCMofthedenominators.Forexample:• the LCM of2__

3;1__

8and1__

6is 24

ExAMplESSolveforx:

Example 1

x−6+4=2xx−x−6+4=2x−x −2=x x=−2

• Subtractxonbothsides• Addliketerms• Swop sides so that‘x’ is on the left -hand

side(LHS)

Example 2

3(x+4)−9=63x+12−9=6 3x+3=6 3x+3−3=6−3 3x=3

3x___3=3__

3

x=1

• Removebrackets:multiply• Addliketerms• Subtract3onbothsides

• Divideby3onbothsides

• Simplify

DefinitionAformulaisa

statement,usuallywrittenasanequation,

givingtheexactrelationshipbetweencertainquantitiesso

thatwhenoneormorevaluesareknown,thevalueofoneparticularquantitycanbefound.

For example:Foracylinderofradiusrandheighth,thevolumeVcanbefoundfromtheformulaV = π r2h

NoteNote to the lecturer:Solvingoflinear

equationswillbedealtwithagainin

Chapter 2.

30

Chapter 1

Solvefora:

Example 3

a__3+4 =6

3__1×a__

3+4×3__

1=6×3__

1

a+12 =18 a+12−12 =18−12 a =6

• Multiplybothsidesby3• Simplify• Subtract12onbothsides• Simplify

Example 4

a+2_____3

+a__4=6

12___1×a+2_____3 +a__4×12___1=6×12___1

4(a+2)+3a=72 4a+8+3a=72 7a=72−8 7a=64

a=64___7

∴a=91__7

• MultiplybytheLCMonbothsides:LCM=12

• Removebrackets:multiply

• Subtract8onbothsides

• Divideby7onbothsides

• Simplify

Example 5

2(1−3a)+4=3(a−2)+2 2−6a+4=3a−6+2 6−6a=3a−4 6+4=3a+6a 10=9a 9a=10 a=10___

9

∴a=11__9

• Removebrackets:multiply• Addliketerms• Add4tobothsidesandadd6atobothsides

• Swopsides• Dividebothsidesby9


Makethesymbolwhichappearsinbracketsaftereachequationthesubjectoftheformula:1. 3x+4=7 (x)2. 11+5x=13y+4x (x)3. p−2y=5−q−3y (y)4. 2__

3x=4 (x)

5. 2x−y=5 (y)6. 3ax=5by (x)7. 5a2=3bx____c (x)8.

p__

4=

y__

2 (y)

9. 3(x+4)−9=6 (x)10. V=l×b×h (b)11. I=V__

R (R)

12. P=VI (I)13. v=s__t (t)14. a__x+1=y (x)

HintWhenmanipulatingformulae:Work step by step and do ONE operation at a time.

31

Numbers

ExAMplES

Example 1Make ‘u’ the subject of the formula:

v=u+atv−at=u+at−atv−at=u ∴u=v−at

• ‘v’isthesubject• Subtract‘at’frombothsides• Simplify• Swopsidessothatthesubjectison

theLHS

Example 2Make ‘t’ the subject of the formula:

v =u+atv−u =u+at−uv−u =at

v−u_____a =at___a

v−u_____a =t

t =v−u_____a

• ‘v’isthesubject• Subtract‘u’frombothsides• Simplify

• Dividebothsidesby‘a’

• Simplify• Swopsidessothatthesubjectison

theLHS

Example 3Make ‘h’ the subject of the formula:

V =l×b×h l×b×h =V

l×b×h________l×b

= V_____l×b

h =V__lb

• SwopsidessothatthesubjectisontheLHS

• Dividebothsidesby‘l×b’

Example 4Make ‘a’ the subject of the formula:

b= ac_______ab+d

b(ab+d)= ac_______ab+d

×(ab+d)

________1

ab2+bd=ac

ab2+bd−bd=ac−bd

ab2−ac=ac−bd−ac

ab2−ac=−bd

a(b2−c)=−bd

a(b2−c)

________(b2−c)

= −bd______b2−c

a= −bd______b2−c

• Multiplybothsidesby(ab+d)

• Removebrackets:multiply

• Subtract‘bd’frombothsides

• Subtract‘ac’frombothsidestogetthetermswithsubject‘a’ontheleftside

• Takeout‘a’asacommonfactor

• Dividebothsidesby‘b2−c’

32

Chapter 1

Example 5make ‘F’ the subject of the formula:

C = 5(F − 32)

_________ 9

9 × C = 5(F − 32)

_________ 9 × 9 __

1

9C = 5(F − 32)

9C ___ 5

= 5(F − 32)

_________ 5

9C ___ 5 = F− 32

9C ___ 5 + 32 = F − 32 + 32

9C ___ 5 + 32 = F

∴ F = 9C ___ 5 + 32

• ‘C’ is the subject

• Multiply by 9 on both sides to undo the dividing by 9

• Simplify• Divide each side by 5 to undo the

multiplying by 5• Simplify

• Add 32 to both sides

• Simplify

• Swop sides

Example 6make ‘c’ the subject of the formula:

A = (3bc)2

(3bc)2 = A 3bc = ± √

__ A

3bc ____ 3b

= ± √__

A _____ 3b

c = ± √__

A _____ 3b

• ‘A’ is the subject• Swop sides• Raise both sides of the equation to

the same power:[(3bc)2] 1 __ 2 = (A)

1 _ 2

• Divide by 3b on both sides

• Simplify

Example 7make ‘r’ the subject of the formula:

V = 1 __ 3 πr3h

3 __ 1 × V = 1 __

3 πr3h × 3 __

1

3V = πr3h

3V ___ πh

= πr3h _____ πh

3V ___ πh

= r3

3 √___

3V ___ πh

= r

r = 3 √___

3V ___ πh

• ‘V’ is the subject

• Multiply by 3 on both sides to undo the dividing by 3

• Divide each side by ‘πh’ to undo the multiplying by πh

• Simplify

• Raise both sides of the equation to the same power

• Swop sides

Example 8make ‘t’ the subject of the formula:

1 __ s + 1 __ t = 1 __ w

1 __ s + 1 __ t − 1 __ s = 1 __ w − 1 __ s

1 __ t = 1 __ w − 1 __ s

1 __ t = s − w ______ ws

∴ t = ws ______ s − w

• Subtract 1 __ s from both sides

• Simplify

• Find the LCM of the right side to get the right side to a single term before you invert

• Invert both sides

33

Numbers

Example 9make ‘I’ the subject of the formula:

WL = √______

I2 − R2

(WL)2 = ( √______

I2 − R2 )2

(WL)2 = I2 − R2

(WL)2 + R2 = I2 − R2 + R2

(WL)2 + R2 = I2

± √_________

W2L2 + R2 = I

∴ I = ± √_________

W2L2 + R2

• Remember √______

I2 − R2 ≠ I − R

• Square both sides

• Add ‘R2’ to both sides

• Simplify

• (WL)2 = W2L2

• Swop sides

Example 10make ‘y’ the subject of the formula:

y + 4

_____ 3 +

y __

4 = 3

12 ___ 1 ×

y + 4 _____

3 +

y __

4 × 12 ___

1 = 3 × 12 ___

1

4(y + 4) + 3y = 36

4y + 16 + 3y = 36

7y + 16 = 36

7y + 16 − 16 = 36 − 16

7y = 20

7y

___ 7 = 20 ___

7

y = 20 ___ 7

y = 2 6 __ 7

• Multiply by the LCM = 12 on both sides; each term

• 12(y + 4)

_________ 3 = 4(y + 4)

• Remove brackets (Multiply)

• Add like terms

• Subtract 16 from both sides

• Simplify

• Divide both sides by 7

• Simplify 20 ___ 7 = 2 6 __

7

Remember these guidelines when changing the subject of the formula before you do the next exercise.

• Get rid of the root signs.• Get rid of the fractions by multiplying by the LCM of the denominators.• Get rid of all the brackets.• Change the equation so that all the terms with the variable you want as the

subject are on the left side of the equation and all the other terms on the right side.

• Collect and add or subtract all the like terms.• Divide both sides by the coefficient of the variable you want as the subject

of the formula.• Before you invert fractions, get both sides to one term first.


Make the variable which appears in brackets after each equation the subject of the formula:

1. x + 2 _____ 3 + x __

4 = 3 (x)

2. v = u + at (a)

34

Chapter 1

3. 1 __ R

= 1 __ r1 + 1 __ r2

(r2)

4. A = πr(1 + rh) (h)

5. 1 __ 2 mv2 = E (v)

6. V = 4 __ 3 πr3 (r)

7. A = πr2 (r)

8. A = P(1 + in) (P)

9. A = P(1 + in) (i)

10. x2 + y2 = r2 (y)

11. √__________

sw(T − w)

__________ 12a

= D (T)

12. T = 2π √___

L __ G

(G)

1.2.8 Solving an unknown variable by means of substitution

ExamplEs

Example 1Calculate the value of ‘I’ if P = I2R and P = 650 and R = 1,23

P = I2R I2R = P I

2R ___ R = P __ R

I2 = P __ R

∴ I = ± √__

P __ R

I = ± √____

650 ____ 1,23

I = ±22,988

• Swop sides• Divide by ‘R’ on both sides

• Remember: a2 = b ∴ a = ± √

__ b

• Substitute the given values for ‘P’ and ‘R’

Example 2Calculate the value of ‘v’ if T = mv2

____ g and T = 15,3, m = 0,4 and g = 9,8

solution T = mv2

____ g ………(v)

mv2 ____ g = T

g × mv2 ____ g = T × g

mv2 = Tg

mv2 ____ m =

Tg ___ m

v2 = Tg

___ m

v = ± √___

Tg

___ m

v = ± √__________

(15,3)(9,8)

__________ 0,4

v = ±19,361

• Swop sides so that ‘v’ is on the left- hand side

• Multiply by ‘g’ on both sides

• Simplify: g __ 1 × mv2

____ g

• Divide by ‘m’ on both sides

• Simplify

• Raise both sides of the equation to the same power

• Substitute: T = 15,3, m = 0.4 and g = 9,8

• Use your calculator

HintChanging the subject

of a formula is the same as solving the

equation.

35

Numbers

Example 3Given:A=π(R2−r2)Calculatethevalueof‘r’ifA=310andR=21

A =π(R2−r2)

A__π =(R2−r2)

A__π−R2 =−r2

−A__π+R2=r2

r2 =R2−A__π

r =±√_______

R2−A__π

r =±√___________

(21)2−310____π

∴r =±18,502

• Dividebothsidesbyπ

• SubtractR2frombothsides

• Multiplyby−1onbothsides

• Swopsides

• Substitutethegivenvalue


1. Calculatethevalueof“T”inPv=nRTifn=14,R=25,4,v=12andP=200.

2. Calculatethetotalresistance,“R”inΩifr1=3Ω,r2=4Ωandr3=2Ωand

1__R

=1__r1+1__r2

+1__r3.

3. Given:T=2π√__

ℓ__g.Calculatethevalueof“g”ifT=36and“ℓ”=54.

4. IfA=πr√______

h2+r2,calculatethevalueof“h”ifA=106andr=0,2.

5. (a) Make‘h’thesubjectoftheformulaifS=2πr(h+r).

(b) Determine‘h’ifr=5,5andS=1102.GiveyouranswercorrecttoTHREEdecimalplaces.

6. Calculate‘d’ifT2=−12,a=−24andn=2inT2=a+(n−1)d.

7. IfZ=√_________

R2−W2L2,calculatethevalueof“L”ifR=156,W=315andZ=124.

8. Calculatethevalueof“F”ifC=28inC__5=F−32___

9

9. Calculatethevalueof‘x’ifa=2,1,y=3,4,b=21andz=1,5ina=xyz______

xy+b.

10.Calculatethevalueof‘r’ifP=1400,n=3andAt=1895inAt=P(1+r_____100

)n.

1.2.9 Arithmetic sequences and series

Formulas used with arithmetic sequences and arithmetic series:

Tofindanytermofanarithmeticsequence:

Tn=a+(n−1)d

where‘a’ is the first term of thesequence, ‘d’ is the commondifference, ‘n’ is the number ofthetermtofind.

Tofindthesumofacertainnumberoftermsofanarithmeticsequence:

Sn=n___

2[2a+(n−1)d]

where Sn is the sum of n terms(nthpartialsum),a is thefirst term,Tnisthenthterm.

36

Chapter 1

Arithmetic sequencesA progressionisasetofnumbersorvariableshavingapatternwhichenablesthenextitemtobedeterminedintermsofthepreviousitems.Herearesomeexamplesofprogressions.

Trytodeterminethepatternineachofthefollowingexamples.

1: 3;5;7;9;11;…2: 10;5;0;−5;−10;…3: 2;4;8;16;32;…4: 27;9;3;1;1__

3;1__

9;…

ExAMplES

Example 1 • 3;5;7;9;11;13;...isanarithmeticsequencewithacommon difference

of2

Asequenceusuallyhasarule,whichisawaytofindthevalueofeachterm.Forexample:thesequence{3;5;7;9;…}startsat3andjumps2everytime:

0 1 2 3

+2 +2 +2 +2

4 5 6 7 8 9 10

Thecommon difference=dd=T2–T1andd=T3-T2.........d=Tn-Tn – 1

‘d’canbecalculatedasfollows:d=T2–T1ord=T3–T2 =5–3 =9–7 =2 =2

Example 2 • 5;2;-1;-4;-7; ... isanarithmeticsequencewithacommondifference

of -3

‘d’canbecalculatedasfollows:d=T2–T1ord=T3–T2 =2–5 =–1–2 =–3 =–3

Thebehaviourofthearithmeticsequencedependsonthecommondifference‘d’.Ifthecommondifferenceis:• positive:Thetermswillgrowtowardspositive infinity.(Refertoexample1

above)• Negative:Thetermswillgrowtowardsnegative infinity.(Refertoexample2

above)

Thefollowingvariables(letters)areusedtorepresentthetermsinasequence:• Thevalueofthefirsttermofanarithmeticsequence:a• Theconstantdifference:d

NoteThedifferencebetween

anytwosuccessivenumbersofthe

sequenceisaconstant.

DefinitionAnarithmetic sequenceor

arithmetic progressionisasequencewhereeachnewtermafterthefirstiscalculatedbyaddingaconstant

amounttothepreviousterm.

Itisanorderedsetofnumbersorvariables.

3; 5; 7; 9 …

T1 T2 T3Second

termThirdterm

dotsmeangoesonforever

(infinite)

Firstterm

37

Numbers

• Terms:T

• Firstterm:T1

• Tenthterm:T10(thesameforterm2,3etc)

• Anyunknownterm:Tn(thenthterm)

The following is an example to prove that the following sequence is an

arithmeticsequence.

ExAMplE

Showthat5;10;15;20;...isanarithmeticsequence• 10–5=15–10=20–15=5 Thereisacommondifferenceof5• a=T1=5and d=T2–T1=10–5=5

Tocalculateterm1,term2,term3,etc.,weusethefollowingequations:

T1=a T2=a+d

T3=a+2d T4=a+3d

Forexampleinthesequence:

• 3;5;7;9;11;13;...andwithacommondifferenced=2

T1=a=3 T2=a+d=3+2=5

T3=a+2d=3+2(2)=7 T4=a+3d=3+3(2)=9etc.

ExAMplES

Example 1Calculatethetwelfthtermofthearithmeticsequence:5;14;23;32;…

Solution

a=5d=T2−T1

=14−5=9

n=12T12=?

∴Tn=a+(n−1)d T12=5+(12−1)(5) =5+(11)(5) =5+55 ∴T12=60

• Firstterm• Commondifference

• Calculatethe12thterm

• Equation• Substitutevalues

• Simplifythebrackets• Thetwelfthtermis60

NoteTofindanyterminanarithmeticsequencetheformula/equation

is:Tn=a+(n-1)d

38

Chapter 1

Example 2

Determinethe30thtermof:65;60;55;…

Given:a=65d=60−65 =−5

n=30∴T30=?

Tn=a+(n−1)d T30=65+(30−1)(−5) =65+(29)(−5) =65−145 ∴T30=−80

• Firstterm• Commondifference

• Numberofterms• Findthe30thterm

• Equation• Substitutevalues• Simplify• (+)×(−)=(−)• 30thtermis−80

Example 3

Calculatethecommondifferenceofanarithmeticprogressioniftheeighthtermis122andthefirsttermis−4.

Solution

Given:T8=122T1=−4=a

Tn=a+(n−1)d T8=−4+(8−1)d 122=−4+7d122+4=7d 7d=126 d=18

∴Commondifferenceisd=18

• Equation• Substitutevalues• T8=122• Add4tobothsides

• Dividebothsidesby7

Example 4

Thevalueofaterm,Tn,inanarithmeticsequenceis3.Calculatethenumberofthetermifthefirsttermis18andtheconstantdifferenceis−3.

Solution

Given:a=18d=−3Tn=3

39

Numbers

Tn=a+(n−1)d 3=18+(n−1)(−3) 3=18−3n+3 3=21−3n 3n=21−3

3n=18 n=6 ∴T6=3

• Equation• Substitutevalues• −3(n−1)=−3n+3• Addliketerms• Add −3n on both sides and

subtract3onbothsides• Divideby3onbothsides

• Thesixthtermis3

Example 5

Calculatethefirst3termsofanarithmeticsequenceifthefourthtermis25andtheeighteenthtermis123.

Solution

Given:T4=25T18=123

Calculate:T1;T2andT3

Tn=a+(n−1)d T4=a+(n−1)d 25=a+(4−1)d ∴25=a+3d………(1)

Tn=a+(n−1)d T18=a+(n−1)d 123=a+(18−1)d 123=a+17d………(2)

• Equation• T4isgiven:fourthterm• Substitute:T4=25• Simplify

• T18isgiven:eighteenthterm• Substitute:T18=123• Simplify

Tofindthefirstthreetermswehavetocalculatethevaluesof‘a’and‘d’firstbysolvingthetwoequations,no.(1)and(2)simultaneously.

∴25=a+3d………(1)123=a+17d………(2)

Equations(2)−(1):

123–25 =17d–3d 98 =14d ∴d =7

Substituted=7inequation(1)

∴25=a+3d 25=a+3(7) 25=a+2125−21=a a=4

• Commondifference

• Youcansubstituted=7inequation(1)or(2)

• Substituteinequation(1)• Simplify

40

Chapter 1

∴Tn=a+(n−1)d• T1=4• T2=4+(2−1)(7) =4+(1)(7) =11• T3=4+(3−1)(7) =4+(2)(7) =18

∴Thesequenceis4;11;18;25;…

• T1=a• Substitute• Simplify

• Substitute• Simplify


1. Whichofthefollowingarearithmeticsequencesandwhicharenot?

(a) −5;2;9;16;… (b) 20;14;8;2;−4;…(c) 3;6;12;24;… (d) 100;50;25;…(e) 9;9;9;9;9;… (f) 8;−8;8;−8;…(g) 12;11,5;11;10,5;… (h) 3;9;27;81;243;…(i) 1;4;9;16;25;… (j) 3__

4;1__

2;1__

4;0;−1__

4;−1__

2;…

2. Find the next two terms in the following arithmetic sequences. Use theformula:Tn=a+(n−1)d.(a) 5;11;17;23;… (b) 7;16;25;34;…(c) 20;13;6;−1;… (d) −8;−15;−22;−29;36;…(e) 1__

4;1__

2;3__

4;1;…

3. Calculatetheninthtermofthefollowingarithmeticsequences:

(a) 6;15;24;33;… (b) 6,8;14,1;21,4;28,7;… (c) 9;6;3;0;−3;…

4. Calculatethefirstfourtermsinthesequenceifthefirsttermis7andthecommondifference=−4.

5. Whichterminthesequence15;12;9;…isequalto−39?

6. Whichterminthefollowingarithmeticsequenceisequalto5,02?0,97;1,00;1,03;1,06;…

7. Findthetwenty-firsttermofanarithmeticsequenceofwhichthe6thtermis3andthe14thtermis19.

Arithmetic series

The following is not for examination purposes:

Formula for the sum of an arithmetic series

Supposewewishtosumtheseries3+7+11+15+19.Forasmallnumberofterms,youcouldjustaddthemup,butwearelookingforageneralmethod.Writetheseriesdown,thenunderneathitwriteitinreverseorder.

S5=3+7+11+15+19........... (1)S5=19+15+11+7+3........... (1)

41

Numbers

Nowaddtheequations:(1)+(2)2S5=22+22+22+22+22=5×22=110

HenceS5=110____

2 =55

Noticethatwhenyouaddthepairsofterms,eachpairaddsuptothesametotal.This method can be used to find a general sum to n terms for anyarithmeticseries.

Sn=a+(a+d)+(a+2d)+(a+3d)+…+[a+(n−1)d]............(3)Sn=[a+(n−1)d]+[a+(n−2)d]+[a+(n−3)d]+…+a...........(4)

Addingbothequations:(3)+(4)2Sn=[2a+(n−1)d]+[2a+(n−1)d]+[2a+(n−1)d]+…+[2a+(n−1)d]…nterms=n[2a+(n−1)d]

HenceSn=n__

2[2a+(n−1)d]

Thesum(ortotal)ofallthecomponentsofanarithmeticsequenceiscalledanarithmetic series.Forexample:• Ifthearithmeticsequenceisgivenas:3;7;11;15;19;...• theseriesis3+7+11+15+19+...Thefollowingexamplesareforforexaminationpurposes.

ExAMplES

Example 1Determinethesumofthefirst20termsofthearithmeticsequence:3+8+13+…

Solution

a=3d=T2−T1=8−3=5n=20 ∴S20=n__

2[2a+(n−1)d]

=20___2[2(3)+(20−1)(5)]

=10[6+95] ∴S20=1010

• Firstterm

• 20terms• Usetheequation:Sn

• Substitutevalues

• Simplify• Thesumis1010

Example 2Determine‘d’ifthefirstterminanarithmeticseriesis3andthesumofthefirst15termsis−165.

SolutionT1=a=3S15=−165d=?n=15

DefinitionAnarithmetic

seriesisanarithmeticsequence

orprogressionwthaddition(or

subtraction)signsinsertedbetweenthe

variousterms

42

Chapter 1

Sn=n__2[2a+(n−1)d]

−165=15___2[2(3)+(15−1)d]

−165=15___2[6+(15−1)d]

−330=15[6+14d]

−330____15

=6+14d

−22−6=14d −28=14d 14d=−28 d=−2

• Substitute

• Simplifybrackets

• Multiplybothsidesby2

• Dividebothsidesby15

• Subtract6onbothsides

• Swopsides• Divide14onbothsides

Example 3Calculatethefirsttermofanarithmeticsequenceifthesumof10termsis230andtheconstantdifferenceis4.

SolutionS10=230n=10d=4

Sn =n__2[2a+(n−1)d]

230 =10___2[2a+(10−1)(4)]

230 =5[2a+9(4)] 230 =5[2a+36] 230 =10a+180230−180=10a 50 =10a a =5

• Substitute

• Removebrackets:Multiply

• Subtract180onbothsides• Divideby10onbothsides

Example 4Atheatrehas60seatsinthefirstrow,68seatsinthesecondrow,76seatsinthethirdrowandsoforthinthesameincreasingpattern.Ifthetheatrehas20rowsofseats,howmanyseatsareinthetheatre?

SolutionTheseatingpatternisforminganarithmeticsequence.

60,68,76,…

Wewishtofind‘thesum’ofalltheseats.n=20,a=60,d=8andweneedS20forthesum.

Sn=n__2[2a+(n−1)d]

=20___2[2(60)+(20−1)(8)]

=10[120+19(8)] =2720

∴Thereare2720seatsinthetheatre.

43

Numbers

Example 5Calculatethesumofthefollowingseries:35+32+29+…+5

SolutionWeneedtoknowhowmanytermsareintheseries.Thereforeweneedtocalculatethevalueofnfirst:

Tn=a+(n−1)d 5=35+(n−1)(−3) 5=35−3n+3 5=38−3n3n=38−53n=33n=11

• a=35;d=32−35=−3• −3(n−1)=−3n+3

∴Sn=n__2[2a+(n−1)d]

=11___2[2(35)+(11−1)(−3)]

=11___2[70+10(−3)]

=11___2[70−30)

=11___2[40]

=220 ∴S11=35+32+29+…+5 =220

• n=11

• (+)×(−)=(−)

Example 6Calculatethesumofthefirst15termsofanarithmeticseriesifthefirsttermis12andthe20thtermis1380.

SolutionT1=a=12T20=1380n=15d=?

∴Tn=a+(n−1)d T20=12+(20−1)d1380=12+19d1368=19d d=72

Sn =n__2[2a+(n−1)d]

S15=15___2[2(12)+(15−1)(72)]

=15___2[24+14(72)]

=15___2[1032]

=7740

• Weneedthevaluefor‘d’• Substitute• Simplify• Dividebothsidesby19

NoteWhensolvingforn,besureyouranswerisapositiveinteger.Thereisnosuchthingasafractionalornegativenumberoftermsina

sequence!

44

Chapter 1


1. Foreachofthefollowingarithmeticprogressions,writedownthesumofthefirst18terms.(a) 6;9;12;15;… (b) −4;−2;0;2;… (c) 15,5;15;14,5;14;…

2. WritedownS4andS6forthefollowingarithmeticprogressionswhere:(a) a=5andd=−2 (b) a=−4andd=3

3. Determine‘a’if:(a) d=2andS8=16 (b) d=−5andS7=−126

4. Determine‘d’if:(a) a=7andS8=140 (b) a=7andS10=−110

5. (a) Showthat4;10;16;22;…isanarithmeticsequence.(b) Writedownaandd.(c) FindformulasforTnandSn

(d) HencedetermineT20andS20

6. (a) Showthat5__3;1;1__

3;−1__

3;−1;…isanarithmeticsequence.

(b) Writedownaandd.(c) FindformulasforTnandSn

(d) HencedetermineT10andS10

7. (a) Showthat20;13;6;−1;…isanarithmeticsequence.(b) Writedownaandd.(c) DetermineaformulaforTn

(d) HencedetermineT15

(e) Usingtheresultsin(a)and(d),determineS15

8. Inanarithmeticsequence,T3=7andT11=39.(a) Determineaandd. (b) FindTnandSn

9. Inanarithmeticsequence,T4=12andT7=−3.(a) DetermineT2 (b) DetermineS10

10.If 2; x; 11; y are four successive (follow one after the other) terms of anarithmeticsequence,determinexandy.

11.Afarmerexaminesasmallplantherecentlyplanted.Henoticesasingleshootonthestem.Aweeklater,therearefourshoots,aweeklatersevenshootsandbythefourthweektherearetenshoots.Ifthepatterncontinues,howmanyshootswouldthefarmerexpectaftertenweeks?

45

Numbers

SUMMARY OF CHAPTER 1

Before you do the summative assessments you should know the following:1. How your calculator works for:

• Addition,multiplication,division,squares,cubes,squareroots,cuberootsetc.

2. The number system

Complex numbers

REAl NUMBERS (R)

IMAGINARY NUMBERS

2+3i3–2i√

___–2

√_____

–513

RATIONAl NUMBERS (Q)

IRRATIONAl NUMBERS

(Q1)√

__2

1,414213π

√__

5(Decimaldoesnotrepeat)

INTEGERS (Z)

WHOlE NUMBERS (No)

NATURAl NUMBERS

(N)1;2;3;4…

0;1;2;3;4…

…-3;-2;-1;0;1;2;3…

…3√___

–8;–2__3;0;11__

2;

3√___

–8;4;317;…

• Identifyrationalnumbersasnumbersthatcanbewrittenasfractionsintheforma __

b .

Q={…−2__3;−4__

7;−8__

9;4;1__

4;619…}

• Irrational numbers cannot be expressed as fractions in the form a __b .

Irrationalnumbersarenon-recurring,non-terminatingnumbers Squarerootsofprimenumbersandπareincluded.

Q1={…√__

2;√__

3;√__

5…π…}

3. Remember the rules for rounding off.• Lookatthevalueofthedigit to rightofthespecifieddigit.• Round off by adding 1 to the digitifthevalueis5, 6, 7, 8 or 9.• Ifthevalueis0, 1, 2, 3, or 4thedigitmustbeleftunchanged.

4. How to convert rational numbers between terminating and recurring decimals.• A terminating decimal is a rational number that can be written as a

fractionwithapowerof10asadenominator.Aterminaldecimalhasafixed number of decimal places.0,4; 0,123; 1,61; 0,747474

46

Chapter 1

0,42 : which can be written as 42 ____ 100

! use 100 as the denominator

= 21 ___ 50

! simplify

• A recurring decimal is a decimal fraction which goes on REPEATING itself without end.

Recurring decimals go on forever and are much too long to work with. You have to round them off. Recurring decimals cannot be written as fractions with a denominator that is a power of 10.

0,666 666 666... = 0,6• ! the dot shows that it is only the 6 that repeats

5. laws of exponents Deductions

6. How to rationalise fractions with surd denominators without a calculator.• In order to simplify surd forms, you can make use of the following two

basic rules: Rule 1: √

___ xy = √

__ x × √

__ y or n √

___ xy = n √

__ x × n √

__ y

Rule 2: √__

x __ y = √__

x ____ √__

y or n √___

x __ y = n √__

x ___ n √

__ y

• also remember that: √__

a ____ √

__ a = 1 and √

__ a × √

__ a = a

1 ___ √

__ 5

= 1 ___ √

__ 5 × √

__ 5 ___

√__

5

= 1 × √__

5 ________ √

__ 5 × √

__ 5

= √__

5 ___ 5

3 _______ 2 + √

__ 3

= 3 _______ 2 +

√__

3 × 2 − √

__ 3 _______

2 − √__

3

= 3(2 − √

__ 3 ) _________________

4 – 2 √__

3 + 2 √__

3 – 3

= 3(2 – √

__ 3 ) _________

1

= 6 − 3 √__

3

am × an = am + n

am ___ an = am − n

(am)n = amn

(a.b)m = ambm

( a __ b

) m

= am ___

bm or ( a __ b

) −m

= ( b __ a ) m

= bm ___ am

n √___

am = am/n

oram/n =

n √___

am

a0 = 1 (a ≠ 0)

a−m = 1 ___ am or 1 ____ a−m = am

47

Numbers

7. How to add, subtract, multiply and divide surds.• Wecanonlyaddorsubtractsurdswhenthetermscontainliketermsor

thesameterms.√

__3+2√

__3=3√

__3and

4√__

2–6√__

2=–2√__

2

8. How to manipulate formula / changing the subject of a formula.• Remembertokeepthesubjectinmind.Itmayinfluenceyourmethod• Knowallthestepswhenmanipulating.Refertopage32.

9. How to work with arithmetic sequences and series.Knowtheformulasusedwitharithmeticsequencesandarithmeticseries:

• Tofindanytermofanarithmeticsequenceusethefollowingformula:

Tn=a+(n−1)d , wherea isthefirsttermofthesequence,d isthecommondifference,nisthenumberofthetermtofind.

• Tofindthesumofacertainnumberoftermsofanarithmeticsequenceusethefollowingformula:

Sn=n__2[2a+(n−1)d],whereSnisthesumofnterms(nthpartialsum),ais

thefirstterm,Tnisthenthterm.

SUMMATIvE ASSESSMENT

MARK ALLOCATION: 40 TIME: 1 h 15 min

Question 1Multiple choice questionsVariouspossibleoptionsareprovidedasanswerstothefollowingquestions.Choosethecorrectanswersandwriteonlytheletters(A-D)nexttothequestionnumbers(1.1–1.3)inyourworkbook.

1.1 Whichofthefollowingnumbersisanexampleofarationalnumber?A (1+π)3 B

3√___

−8 C 1___√

__3 D (1+√

__3)2 (1)

1.2 If−3y+x=6,then:A y=1__

3x−6 B y=−1__

3x+2 C y=−1__

3x+6 D y=1__

3x−2 (1)

1.3 Whichofthefollowingisafalsestatement?

A 1__2=3__x;∴x=6 B 5__

7=x___

28;∴ x=20

C x__9=3___

27;∴ x=1 D 15___

30=x__

5;∴ x=10 (1)

[3]

Question 22.1 Withtheaidofacalculator,solvethefollowingtoTHREEdecimalplaces:

2.1.1 2,4(2+6,1)3,1−8,4_____

(0,3)2 (1)

2.1.2 Solveforx:

x=3√__________________

3,6______

(15,4)4−π2+(3,4)6 (1)

48

Chapter 1

2.2 Convertthefollowingmeasurement: 1000000mmtocm (1)

2.3 Convertthefollowingdecimalfractionstotheforma__b

;a;b∈Z;b≠0.Expressyouranswerinthesimplestform.

2.3.1 0,425 (2) 2.3.2 0,4

•1• (3)

2.4 Rationalisethedenominator:

2.4.1 3√__

5____√

__6 (2)

2.4.2 8_______√

__5−1

(3)

2.5 Simplifythefollowingwithoutusingacalculator:

√___

54+5√__

6−3√___

24_________________√

____150 (3)

[16]Question 33.1 Simplify the following by using laws of exponents. Leave answers with

positiveexponents.

3.1.1 x4y−3×x5y2

____xy6×x3y−2 (2)

3.1.2 (3m4n3______

2mn)2×

9(2m−2n3)3

__________3mn

(3)

3.1.3 (27____125

)−2__3

(3)

3.2 Maketheletterinbracketsthesubjectoftheformula.

r=3√___

3v____4π

………(v) (3)

3.3 Solvefor‘b’in1__a=1__b

+1__cifa=2,5andc=3,71 (4)[15]

Question 4Susanbuildsthefollowingthreefiguresusingcubes.Shecontinuesusingthesamepattern.

Figure 1 Figure 2 Figure 3

4.1 Howmanycubeswillsheusetobuildthe18thfigure? (3)4.2 Ifsheonlybuilds35figures,howmanycubeswouldshehaveused

in total? (3)[6]

Total: 40

49

Numbers

ANSwERS

ASSESSMENT ACTIvITY 1.11a. 5,333 2a. 0,743

1b. 43 2b. 359,990

1c. -40 2c. 199,213

1d. 354453,530 2d. 21,799

1e. 0,036 2e. -8,055

1f. 10,953 2f. 10,888

1g. 480,267

ASSESSMENT ACTIvITY 1.21. 650g

2. 1,548kg

3. 11:00

4. 1:15pm

5. 22:00

6. 185mm

7. 50540m

8. 2000mm

9. 125mg

10. 3200kg

11. 8000g

12. 20g

13. 65t

14. 20cm

15. 0,0145kg

16. 4,426km

17. 310K

18. 272K

19. 0°C

20. 28°C

ASSESSMENT ACTIvITY 1.31. Rational

2. Rational

3. Irrational

4. Rational

5. Irrational

6. Rational

7. Rational

8. Rational

9. Rational

10. Rational

ASSESSMENT ACTIvITY 1.41a. 0,912

1b. 17,668

1c. 0,004

2a. 9,05

2b. 0,18

3a. 98,78

3b. 98,8

3c. 98,78454


1b. 0,706

1c. -0,444

1d. 3,800

2a. 3__10

2b. 27__50

2c. 613____1000

2d. 7____2000

2e. 317__50

2f. 1____1000


•

1b. 13,6•2•

1c. 3,67•

1d. 0,83•

1e. 32,4•35

•

1f. 67,0•891

•

1g. 1,6•

1h. 2,17•32

•

2a. recurring

2b. recurring

2c. terminating

2d. terminating

2e. non-recurring,non-terminating

2f. recurring

2g. recurring

50

Chapter 1


1b. 1,83•

1c. 0,4•32

•

1d. 1,1•42857

•

1e. 4,2•3•

2a. x=8__9

2b. x=7__11

2c. x=104___333

2d. x=11__30

2e. x= 8___495

2f x=41__33

2g. x=184___55

ASSESSMENT ACTIvITY 1.81a. 6x3y3

1b. x2

1c. 22x

1d. ax

1e. 22y

1f. 5h4d

1g. k2t2____

3

1h. 27x9

1i. 625p8q12

_______81r4

1j. 32x+2

2a. 1

2b. 2x

2c. 1

2d. 2

2e. 2

2f. 2

2g. 1

2h. 24x8

3a. 1__x5

3b. 3__x6

3c. 3x4

3d. 256___x12

3e. 3z3____

p4q2

3f. 1__a

3g. y2

__x3

3h. 18b4____x

3i. x__y2

3j. x__3y

ASSESSMENT ACTIvITY 1.91. 3x14y20

2. 9x2___

4y4

3. x6y4

4. 1

5. 2y13

___x27

6. 1________531441x36

7. –x2y2

____2

8. 27x___2y

9. a2_____

12b4c7

10. 1

11. x4y5

12. a5x–11y

ASSESSMENT ACTIvITY 1.101. 9

2. 3

3. 9x

4. x2y1__2

5. 8__a4

6. 4a3b

7. 3b6___a

ASSESSMENT ACTIvITY 1.111a. 2√

__5

1b. 3√__5

1c. 23√__3

1d. 2

1e. 7__5

1f. 6

2a. √__5

2b. 7√__2____

2

2c. √__3___

9

2d. 7+√__3______

46

2e. 40+5√__6_______

58

2f. –9–4√__5

2g. 8–2√__2______

7

2h. 10√__5–20

2i. √__6+2

51

Numbers

ASSESSMENT ACTIvITY 1.121. –2√

__2

2. 5√__2–3√

__3

3. √__6+√

__3

4. 1__2

5. 6√__6

6. –1__5

7. √__2___

2

ASSESSMENT ACTIvITY 1.131. x=1

2. x=13y–11

3. y=5–q–p

4. x=1__6

5. y=–5+2x

6. x=5yb___

3a

7. x=5a2c____3b

8. y=p__2

9. x=1

10. b=V__l h

11. R= V __I

12. I= P __V

13. t=s __v 14. x= a____

y–1

ASSESSMENT ACTIvITY 1.141. x=4

2. a=v–u____t

3. r2=Rr1_____

r1–R

4. h=A–πr_____πr2

5. V=±√___

2E__m

6. r=3√___

3V___4π

7. r=√__

A___π

8. P= A_____1+in

9. i=A−P_____Pn

10. y=±√_____

r2−x2

11. T=12aD2+SW2__________

SW

12. G=4π2L____T2

ASSESSMENT ACTIvITY 1.151. T=6,749

2. R=12__13

Ω

3. g=1,645

4. h=168,704

5a h=s−2πr2______

2πr

5b h=26,389

6 d=12

7 L=0,3

8 F=82,4

9 x=-21,618

10 r=10,618

ASSESSMENT ACTIvITY 1.161a. Yes

1b. Yes

1c. No

1d. No

1e. No

1f. No

1g. Yes

1h. No

1i. No

1j. Yes

2a. T5=29;T6=35

2b. T5=43;T6=50

2c. T5=-8;T6=-15

2d. T6=-43;T7=-15

2e. T5=5__4;T6=3__

2

3a. T9=78

3b. T9=65,2

3c. T9=-15

4. 7;3;-1;-5;….

5. T19=-39

6. T136=5,02

7. T21=33

ASSESSMENT ACTIvITY 1.171a. S18=567

1b. S18=234

1c. S18=202,5

2a. S4=8;S6=0

2b. S4=2;S6=21

3a. a=-5

3b. a=-3

4a. d=3

4b. d=-4

5a. constantdifferenceof6

5b. a=4 d=6

5c. Tn=6n-2 Sn=n+3n2

5d. S20=1220 T20=118

6a. constantdifference

6b. a=5__3 d=-2__

3

6c. Tn = 7__3 – 2__

3 n Sn = 2n – 1__

3 n2

6d. T10=-13__3 S10=-40__

3

7a. constantdifference

7b. a=20;d=-7

7c. Tn=27−7n

7d. T15=78

7e. S15=-435

8a. a=-1;d=4

8b. Tn = 4n − 5 Sn = −3n + 2n2

9a. T2=22

9b. S10=45

10. y=31__2;x=13__

2

11. T10=28

Date post:	22-Nov-2014
Category:	Documents
Upload:	future-managers-pty-ltd
View:	185 times
Download:	5 times

NCV2 Mathematics Hands-On Training 2010 Syllabus - Sample chapter

Documents