245705033-103340355-Real-Number-System-ppt

8/21/2019 245705033-103340355-Real-Number-System-ppt

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Maths project.Maths project.

topic:topic: real number system.real number system.

produced by,produced by,

s.vishnu vardan.s.vishnu vardan.

Contents:Contents:1. Introduction,1. Introduction,

2. Realnumber system,2. Realnumber system,

4. Rational numbers,4. Rational numbers,6. Integers,6. Integers,

9. ecimal numbers,9. ecimal numbers,

12.natural numbers,12.natural numbers,1!."hole numbers,1!."hole numbers,

16.Comparision,16.Comparision,

Conclusion.Conclusion.


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#$ #$

%%sspro&ect.pro&ect.


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Introduction.Introduction.

Real number systemReal number system is denoted by theis denoted by the

symbol'''''''''''''''''''''''symbol''''''''''''''''''''''' In mathematics, aIn mathematics, a real numberreal number is a value thatis a value that

represents a )uantity along a continuous line.represents a )uantity along a continuous line.

Real numbers can be thought o* as points on anReal numbers can be thought o* as points on an

in+nitely long line called the number line or realin+nitely long line called the number line or realline, "here the points corresponding to integersline, "here the points corresponding to integersare e)ually spaced.are e)ually spaced.

http://en.wikipedia.org/wiki/File:Latex_real_numbers.svghttp://en.wikipedia.org/wiki/File:Latex_real_numbers.svghttp://en.wikipedia.org/wiki/File:Latex_real_numbers.svg


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Real number system.Real number system. #he real numbers include #he real numbers include

all the rational numbers,all the rational numbers,such as the integer ! .such as the integer ! .

the *raction4-.the *raction4-.

all the irrationalall the irrational

numbers such asnumbers such as/2/2 ii..ee.,1.41421!6....,1.41421!6...

the s)uare root o* 2, anthe s)uare root o* 2, an

irrational algebraicirrational algebraicnumber.number. 00 i.e.,i.e.,.141!926!..., .141!926!...,

#RC33# #RC33#

573R.573R.

#he #he real linereal line can becan be

thought o* as a partthought o* as a parto* theo* thecomple8 planecomple8 plane, and, andcorrespondingly,correspondingly,

comple8 numberscomple8 numbers include realinclude realnumbers as anumbers as aspecial case.special case.

http://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Real_linehttp://en.wikipedia.org/wiki/Real_linehttp://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/Real_linehttp://en.wikipedia.org/wiki/Pi


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Rational numbers.Rational numbers.

InIn mathematicsmathematics,,

aa rational numberrational number isisanyany numbernumber that canthat canbe e8pressed as thebe e8pressed as the)uotient)uotient oror*raction*raction p p--qq o* t"oo* t"ointegersintegers, "ith the, "ith thedenominatordenominator qq notnot

e)ual to ero.e)ual to ero.inceince qq may be e)ualmay be e)ualto 1, every integer is ato 1, every integer is arational number.rational number.

#he #he setset o* allo* all

rational numbers isrational numbers isusually denoted byusually denoted bya bold*acea bold*ace QQ ;or;orblac


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Integers.Integers. #he #he integersintegers are *ormed by theare *ormed by the natural numbersnatural numbers

including ?including ? ??,, 11,, 22,, , ... together "ith the, ... together "ith thenegativesnegatives o* the non'ero natural numberso* the non'ero natural numbers 11,,2, , ...2, , ...

on negative integerson negative integers ;purple>;purple> and negativeand negativeintegersintegers ;red>;red>..

n integer isn integer is positive positive i* it is greater than eroi* it is greater than eroandand negativenegative i* it is less than ero. @ero is de+nedi* it is less than ero. @ero is de+nedas neither negative nor positive.as neither negative nor positive.

http://en.wikipedia.org/wiki/Natural_numbershttp://en.wikipedia.org/wiki/Natural_numbershttp://en.wikipedia.org/wiki/0_(number)http://en.wikipedia.org/wiki/0_(number)http://en.wikipedia.org/wiki/1_(number)http://en.wikipedia.org/wiki/1_(number)http://en.wikipedia.org/wiki/2_(number)http://en.wikipedia.org/wiki/2_(number)http://en.wikipedia.org/wiki/3_(number)http://en.wikipedia.org/wiki/3_(number)http://en.wikipedia.org/wiki/Negative_numberhttp://en.wikipedia.org/wiki/Negative_numberhttp://en.wikipedia.org/wiki/%E2%88%921_(number)http://en.wikipedia.org/wiki/%E2%88%921_(number)http://en.wikipedia.org/wiki/File:Number-line.svghttp://en.wikipedia.org/wiki/%E2%88%921_(number)http://en.wikipedia.org/wiki/Negative_numberhttp://en.wikipedia.org/wiki/3_(number)http://en.wikipedia.org/wiki/2_(number)http://en.wikipedia.org/wiki/1_(number)http://en.wikipedia.org/wiki/0_(number)http://en.wikipedia.org/wiki/Natural_numbers


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Properties of addition and multiplication on integers

Addition Multiplication

Closure: a + b is an integer a × b is an integer

Associativity: a + (b + c) = (a + b)

+ c a × (b × c) = (a × b) × c

Commutativity: a + b = b + a a × b = b × a

Existence of anidentity element:

a + 0 = a a × 1 = a

Existence of

inverse elements: a + (−a) = 0

An inverse element usually does not

exist at all.

Distributivity: a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) +

(b × c)

No zero divisors:

If a × b = 0 t!en a = 0 or b = 0 (or

bot!)

http://en.wikipedia.org/wiki/Closure_(mathematics)http://en.wikipedia.org/wiki/Associativityhttp://en.wikipedia.org/wiki/Commutativityhttp://en.wikipedia.org/wiki/Identity_elementhttp://en.wikipedia.org/wiki/Inverse_elementhttp://en.wikipedia.org/wiki/Distributivityhttp://en.wikipedia.org/wiki/Zero_divisorhttp://en.wikipedia.org/wiki/Zero_divisorhttp://en.wikipedia.org/wiki/Distributivityhttp://en.wikipedia.org/wiki/Inverse_elementhttp://en.wikipedia.org/wiki/Identity_elementhttp://en.wikipedia.org/wiki/Commutativityhttp://en.wikipedia.org/wiki/Associativityhttp://en.wikipedia.org/wiki/Closure_(mathematics)


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ecimal numbers.

#he #he decimaldecimal numeral systemnumeral system hashas tenten as itsas its

basebase. It is the numerical base most "idely used. It is the numerical base most "idely usedby modern civiliations.by modern civiliations.

Decimal notationDecimal notation o*ten re*ers to a base'1?o*ten re*ers to a base'1?positional notationpositional notation such as the Aindu'rabicsuch as the Aindu'rabicnumeral system.numeral system.

ho"ever, it can also be used more generally toho"ever, it can also be used more generally to

re*er to non'positional systems suchre*er to non'positional systems suchas Roman or Chinese numerals "hich are alsoas Roman or Chinese numerals "hich are alsobased on po"ers o* ten.based on po"ers o* ten.

http://en.wikipedia.org/wiki/Numeral_systemhttp://en.wikipedia.org/wiki/Numeral_systemhttp://en.wikipedia.org/wiki/10_(number)http://en.wikipedia.org/wiki/10_(number)http://en.wikipedia.org/wiki/Base_(exponentiation)http://en.wikipedia.org/wiki/Base_(exponentiation)http://en.wikipedia.org/wiki/Positional_notationhttp://en.wikipedia.org/wiki/Positional_notationhttp://en.wikipedia.org/wiki/Positional_notationhttp://en.wikipedia.org/wiki/Base_(exponentiation)http://en.wikipedia.org/wiki/10_(number)http://en.wikipedia.org/wiki/Numeral_system


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1/2 = 0.51/2 = 0.5 1/20 = 0.051/20 = 0.05 1/5 = 0.21/5 = 0.2 1/50 = 0.021/50 = 0.02

1/4 = 0.251/4 = 0.25 1/40 = 0.0251/40 = 0.025 1/25 = 0.041/25 = 0.04

1/8 = 0.1251/8 = 0.125 1/125= 0.0081/125= 0.008 1/10 = 0.11/10 = 0.1

1/3 = 0.333333…1/3 = 0.333333… 1/9 = 0.111111…1/9 = 0.111111…

100-1=99=9×11100-1=99=9×11 1/11 = 0.090909…1/11 = 0.090909…

1000-1=9×111=27×371000-1=9×111=27×37

1/27 = 0.037037037…1/27 = 0.037037037… 1/37 = 0.027027027…1/37 = 0.027027027…

1/111 = 0 .009009009…1/111 = 0 .009009009…

1/81= 0.012345679012…1/81= 0.012345679012…


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Natural numbers.Natural numbers. In mathematics, theIn mathematics, the natural numbersnatural numbers are theare the

ordinary "hole numbers usedordinary "hole numbers used

*or counting and ordering .*or counting and ordering . #hese purposes are related to the linguistic #hese purposes are related to the linguistic

notions o* cardinal and ordinal numbers,notions o* cardinal and ordinal numbers,

respectively .respectively .

later notion is that o* a nominal number, later notion is that o* a nominal number,"hich is used only *or naming."hich is used only *or naming.

Broperties o* the natural numbers relatedBroperties o* the natural numbers relatedto divisibility, such as the distribution o* primeto divisibility, such as the distribution o* primenumbers, are studied in number theory.numbers, are studied in number theory.


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Broblems concerning counting andBroblems concerning counting andordering, such as partition enumeration,ordering, such as partition enumeration,are studied in combinatorics.are studied in combinatorics.

ome authors use the term naturalome authors use the term naturalnumber to e8clude ero and "holenumber to e8clude ero and "holenumber to include it.number to include it.

others use "hole number in a "ay thatothers use "hole number in a "ay thate8cludes ero, or in a "ay that includese8cludes ero, or in a "ay that includesboth ero and the negative integers.both ero and the negative integers.


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whole numbers.whole numbers.

DholeDhole

numbers arenumbers arenaturalnatural

numbersnumbersincluding E?F.including E?F.


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Comparision betweenComparision between

andand


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hat Is A Number?

hat Is A Number?

What is a number?What is a number?

re these numbers? re these numbers? !s 11 a number?!s 11 a number?

33?33? What ab"ut 0#$%&? !s this a number?What ab"ut 0#$%&? !s this a number?

'es it is an an(ient number -0945732'es it is an an(ient number -0945732


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Some ancient numbers

ome ancient numbers


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Messages

essages

The number system we have today haveThe number system we have today have

come through a long route, and mostlycome through a long route, and mostly

from some far away lands, outside offrom some far away lands, outside ofEurope.Europe.

They came about because human beingsThey came about because human beingswanted to solve problems and createdwanted to solve problems and created

numbers to solve these problems.numbers to solve these problems.


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Limit of Four

imit of Four

Take a look at the next picture, and try toTake a look at the next picture, and try toestimate the quantity of each set of objects in aestimate the quantity of each set of objects in asinge visual glance, without countingsinge visual glance, without counting..

Take a look again.Take a look again. More difficult to see the objects more than four.More difficult to see the objects more than four.

Everyone can see the sets of one, two, and ofEveryone can see the sets of one, two, and ofthree objects in the figure, and most people canthree objects in the figure, and most people cansee the set of four.see the set of four.

ut that!s about the limit of our natural ability tout that!s about the limit of our natural ability tonumerate. eyond ", quantities are vague, andnumerate. eyond ", quantities are vague, and

our eyes alone cannot tell us how many thingsour eyes alone cannot tell us how many thingsthere are.there are.


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Limits Of Four

imits Of Four


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Egyptian 3

gyptian 3

rd

d

entury !

entury !


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retan "#$$%"&$$!

retan "#$$%"&$$!


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Eng'and(s )fi*e%barred

ng'and(s )fi*e%barred

gate+

ate+


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,o- to ount -ith )'imit

o- to ount -ith )'imit

of four+

f four+

)ere is a *i+ure t" sh" "u hat e"e)ere is a *i+ure t" sh" "u hat e"e

hae use.hae use.

he &ema "* e uineahe &ema "* e uinea


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.he

he

E'ema'ema

of

f

Ne-

e-

/uine

uine

a


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.he /ree0 Numera'

he /ree0 Numera'

System

ystem


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Arithmetic -ith /ree0 Numera' System

rithmetic -ith /ree0 Numera' System


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1oman Numera's

oman Numera's

11 !! 2020

22 !!!! 2525

33 !!!!!! 2929 !!

44 !! 5050

55 7575

66 !! 100100

1010 500500 1111!! 10001000::

1616!!

" tr these;" tr these;

1.1. !!2.2.

3.3. !!!!

4.4. !!5.5. ::!::!


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1oman Numera's 2 .as0

oman Numera's 2 .as0

"

!!


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"

:::!:::!

!!


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"

:::!!!:::!!!

-- :!!:!!

-- :!:!-- !!!!!!

''((())

37283728

-- 18521852

-- 12311231

-- 413413*#*


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"

##

MMM+''

7575

## 5050

#-%


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ra-bac0s of positiona'

ra-bac0s of positiona'

numera' system

umera' system

)ar t" reresent ar+er)ar t" reresent ar+er

numbersnumbers

)ar t" " arithmeti( ith ar+er)ar t" " arithmeti( ith ar+er

numbers, trin+ " 23456 #numbers, trin+ " 23456 #

987654987654


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he sear(h as "n *"r "rtabe reresentati"nhe sear(h as "n *"r "rtabe reresentati"n"* numbers"* numbers

" mae r"+ress, humans ha t" s"e a" mae r"+ress, humans ha t" s"e atri( r"bem;tri( r"bem;

What is the smaest set "* smb"s in hi(hWhat is the smaest set "* smb"s in hi(hthe ar+est numbers (an in the"r bethe ar+est numbers (an in the"r bereresente?reresente?


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South American Maths

outh American Maths

The Maya

The )ncas


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tentiestenties unitsunits

Mayan Maths

ayan Maths

tentiestenties unitsunits * x *% / - 0 "-

&$ x *% / 0 #1


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!aby'onian Maths

aby'onian Maths

The abylonians


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3600s3600s 60s60s 1s1s

!

a

b

y

'

o

n

I

a

n

si#tiessi#ties unitsunits01" 0 #1%"


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4ero and the Indian Sub%

ero and the Indian Sub%

ontinent Numera' System


'"u n" the'"u n" the "ri+in "* the "siti"na number, an its"ri+in "* the "siti"na number, an itsraba(s.raba(s.

>ne "* its imits is h" " "u reresent tens, hunres,>ne "* its imits is h" " "u reresent tens, hunres,et(.et(.

number sstem t" be as e**e(tie as "urs, it must "ssess number sstem t" be as e**e(tie as "urs, it must "ssessa er".a er". !n the be+innin+, the ("n(et "* er" as sn"nm"us ith!n the be+innin+, the ("n(et "* er" as sn"nm"us ith

emt sa(e.emt sa(e. @"me s"(ieties (ame u ith s"uti"ns t" reresent@"me s"(ieties (ame u ith s"uti"ns t" reresent

An"thin+B.An"thin+B. he $ab"nians e*t bans in a(es here er"es sh"uhe $ab"nians e*t bans in a(es here er"es sh"u

be.be. he ("n(et "* AemtB an An"thin+B starte be("min+he ("n(et "* AemtB an An"thin+B starte be("min+

sn"nm"us.sn"nm"us. !t as a "n+ time be*"re er" as is("ere.!t as a "n+ time be*"re er" as is("ere.


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4ero and the Indian Sub%

ero and the Indian Sub%



We hae t" than the :athemati(ians *"rWe hae t" than the :athemati(ians *"r

"ur m"ern number sstem."ur m"ern number sstem.

@imiarit beteen the !nian numera@imiarit beteen the !nian numerasstem an "ur m"ern "nesstem an "ur m"ern "ne


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Indian Numbers

ndian Numbers


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!inary Numbers

inary Numbers


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ifferent !ases

ifferent !ases

hunreshunres tenstens unitsunits11 22 55

&*&%

0 & x &%% / * x &% /

ase &% 2+ecimal34

ei+htsei+hts *"urs*"urs t"st"s unitsunits

11 11 11 00

&&&%* 0 & x $ / & x " / & x * / %

0 &" 2base &%3

ase * 2inary34


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5ythagoras .heorem

ythagoras .heorem

b

c

a

a* 0 b

* / c

2


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5ythagoras! Theorem

&

&

a

a* 0 &

* / &

*

6o

a* 0 *a 0 7


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Messages

essages

The number system we have today have comeThe number system we have today have come

through a long route, and mostly from some farthrough a long route, and mostly from some far

away lands, outside of Europe.away lands, outside of Europe.

They came about because human beings wantedThey came about because human beings wantedto solve problems and created numbers to solveto solve problems and created numbers to solve

these problems.these problems.

8umbers belong to human culture, and not8umbers belong to human culture, and not

nature, and therefore have their own long history.nature, and therefore have their own long history.


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6uestions to As0

uestions to As0

Ourse'*es7

urse'*es7

!s this the en "* "ur number sstem?!s this the en "* "ur number sstem?

re there +"in+ t" be an m"re (han+es re there +"in+ t" be an m"re (han+es

in "ur resent numbers?in "ur resent numbers? !n 300 ears *r"m n", i the numbers!n 300 ears *r"m n", i the numbers

hae (han+e a+ain t" be s"methin+hae (han+e a+ain t" be s"methin+

ese?ese?


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3 great ideas made our

great ideas made our

modern number system

odern number system

>ur m"ern number sstem as a resut "* a>ur m"ern number sstem as a resut "* a

("nCun(ti"n "* 3 +reat ieas;("nCun(ti"n "* 3 +reat ieas;

the iea "* atta(hin+ t" ea(h basi( *i+urethe iea "* atta(hin+ t" ea(h basi( *i+ure+rahi(a si+ns hi(h ere rem"e *r"m a+rahi(a si+ns hi(h ere rem"e *r"m aintuitie ass"(iati"ns, an i n"t isua e"eintuitie ass"(iati"ns, an i n"t isua e"ethe units the reresentethe units the reresente

the rin(ie "* "siti"nthe rin(ie "* "siti"n

the iea "* a *u "erati"na er", *iin+ thethe iea "* a *u "erati"na er", *iin+ theemt sa(es "* missin+ units an at the sameemt sa(es "* missin+ units an at the sametime hain+ the meanin+ "* a nu numbertime hain+ the meanin+ "* a nu number


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