Real Number System

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. Decimal numbers,9. Decimal numbers,

12.natural numbers,12.natural numbers,

15.whole numbers,15.whole numbers,

16.Comparision,16.Comparision, Conclusion. Conclusion.

TO TO

‘‘s s

project.project.

Introduction.Introduction. Real number systemReal number system is denoted by the is denoted by the symbol-----------------------symbol----------------------- In mathematics, a In mathematics, a real numberreal number is a value that  is a value that

represents a quantity along a continuous line.represents a quantity along a continuous line.

Real numbers can be thought of as points on an Real numbers can be thought of as points on an infinitely long line called the number line or real infinitely long line called the number line or real line, where the points corresponding to integers line, where the points corresponding to integers are equally spaced.are equally spaced.

Real number system.Real number system. The real numbers include The real numbers include

all the rational numbers, all the rational numbers, such as the integer −5 .such as the integer −5 .

the fraction4/3.the fraction4/3. all the irrational all the irrational

numbers such as numbers such as √2 √2 ii..ee.,1.41421356....,1.41421356...

the square root of 2, an the square root of 2, an irrational algebraic irrational algebraic number. number. ππ  i.e.,i.e.,3.14159265..., A 3.14159265..., A TRANCENDENTAL TRANCENDENTAL NUMBER.NUMBER.

The The real linereal line can be  can be thought of as a part thought of as a part of the of the complex planecomplex plane, and , and correspondingly, correspondingly, complex numberscomplex numbers include real  include real numbers as a numbers as a special case.special case.

Rational numbers.Rational numbers.

In In mathematicsmathematics, , a a rational numberrational number is  is any any numbernumber that can  that can be expressed as the be expressed as the quotientquotient or  or fraction fraction pp//qq of two  of two integersintegers, with the , with the denominatordenominator  qq not  not equal to zero. equal to zero. Since Since qq may be equal  may be equal to 1, every integer is a to 1, every integer is a rational number. rational number.

The The setset of all  of all rational numbers is rational numbers is usually denoted by usually denoted by a boldface a boldface QQ (or  (or blackboard boldblackboard bold ,  , UnicodeUnicode ℚ), which  ℚ), which stands for stands for quotientquotient..

Integers.Integers. The The integersintegers are formed by the  are formed by the 

natural numbersnatural numbers including 0 including 0 00, , 11, , 22, , 33, ... , ... together with the together with the negativesnegatives of the non-zero  of the non-zero natural numbers natural numbers −1−1, −2, −3, ..., −2, −3, ...

Non negative integers Non negative integers (purple)(purple) and negative and negative integers integers (red)(red)..

An integer is An integer is positivepositive if it is greater than zero  if it is greater than zero and and negativenegative if it is less than zero. Zero is  if it is less than zero. Zero is defined as neither negative nor positive.defined as neither negative nor positive.

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 an identity element:

a + 0  =  a a × 1  =  a

Existence of inverse elements:

a + (−a)  =  0An 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, then a = 0 or b = 0 (or both)

Decimal numbers. The The decimaldecimal  numeral systemnumeral system has  has tenten as its  as its 

basebase. It is the numerical base most widely used . It is the numerical base most widely used by modern civilizations.by modern civilizations.

Decimal notationDecimal notation often refers to a base-10  often refers to a base-10 positional notationpositional notation such as the Hindu-Arabic  such as the Hindu-Arabic numeral system. numeral system.

however, it can also be used more generally to however, it can also be used more generally to refer to non-positional systems such refer to non-positional systems such as Roman or Chinese numerals which are also as Roman or Chinese numerals which are also based on powers of ten.based on powers of ten.

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…

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

ordinary whole numbers used ordinary whole numbers used for counting and ordering .for counting and ordering .

These purposes are related to the linguistic These purposes are related to the linguistic notions of cardinal and ordinal numbers, notions of cardinal and ordinal numbers, respectively . respectively .

A later notion is that of a nominal number, A later notion is that of a nominal number, which is used only for naming. which is used only for naming.

Properties of the natural numbers related Properties of the natural numbers related to divisibility, such as the distribution of prime to divisibility, such as the distribution of prime numbers, are studied in number theory. numbers, are studied in number theory.

Problems concerning counting and Problems concerning counting and ordering, such as partition enumeration, ordering, such as partition enumeration, are studied in combinatorics.are studied in combinatorics.

Some authors use the term "natural Some authors use the term "natural number" to exclude zero and "whole number" to exclude zero and "whole number" to include it.number" to include it.

others use "whole number" in a way that others use "whole number" in a way that excludes zero, or in a way that includes excludes zero, or in a way that includes both zero and the negative integers.both zero and the negative integers.

whole numbers.whole numbers.

Whole Whole numbers are numbers are natural natural numbers numbers including “0”.including “0”.

Comparision betweenComparision between

andand

What Is A Number?What Is A Number?

What is a number?What is a number?

Are these numbers? Are these numbers? Is 11 a number? Is 11 a number? 33?33? What about 0xABFE? Is this a number? What about 0xABFE? Is this a number?

Yes it is an ancient number -0945732 Yes it is an ancient number -0945732

Some ancient numbersSome ancient numbers

MessagesMessages

The number system we have today have The number system we have today have come through a long route, and mostly come through a long route, and mostly from some far away lands, outside of from some far away lands, outside of Europe. Europe.

They came about because human beings They came about because human beings wanted to solve problems and created wanted to solve problems and created numbers to solve these problems.numbers to solve these problems.

Limit of FourLimit of Four Take a look at the next picture, and try to Take a look at the next picture, and try to

estimate the quantity of each set of objects in a estimate the quantity of each set of objects in a singe 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 of Everyone can see the sets of one, two, and of

three objects in the figure, and most people can three objects in the figure, and most people can see the set of four.see the set of four.

But that’s about the limit of our natural ability But that’s about the limit of our natural ability to numerate. Beyond 4, quantities are vague, to numerate. Beyond 4, quantities are vague, and our eyes alone cannot tell us how many and our eyes alone cannot tell us how many things there are.things there are.

Limits Of FourLimits Of Four

Egyptian 3Egyptian 3rdrd Century BC Century BC

Cretan 1200-1700BCCretan 1200-1700BC

England’s “five-barred England’s “five-barred gate”gate”

How to Count with “limit How to Count with “limit of four”of four”

Here is a figure to show you what people Here is a figure to show you what people have used.have used.

The Elema of New GuineaThe Elema of New Guinea

The The Elema Elema of New of New GuineaGuinea

The Greek Numeral The Greek Numeral SystemSystem

Arithmetic with Greek Numeral SystemArithmetic with Greek Numeral System

Roman NumeralsRoman Numerals11 II 20 20 XXXX22 IIII 2525 XXVXXV33 IIIIII 2929 XIXXIX44 IVIV 5050 LL55 VV 7575 LXXVLXXV66 VIVI 100100 CC1010XX 500500 DD1111XIXI 10001000MM1616XVIXVI

Now try these:Now try these:

1.1. XXXVIXXXVI2.2. XLXL3.3. XVIIXVII4.4. DCCLVIDCCLVI5.5. MCMLXIXMCMLXIX

Roman Numerals – Task 1 Roman Numerals – Task 1 CCLXIVCCLXIV

++ DCLDCL

++ MLXXXMLXXX

++ MDCCCVIIMDCCCVII

MMMDCCXXVIIIMMMDCCXXVIII

-- MDCCCLIIMDCCCLII

-- MCCXXXIMCCXXXI

-- CCCCXIIICCCCXIII

LXXVLXXV

xx LL

Roman Numerals – Task 1Roman Numerals – Task 1

MMMDCCCIMMMDCCCI

CCLXIVCCLXIV

++ DCLDCL

++ MLXXXMLXXX

++ MDCCCVIIMDCCCVII

264264

++ 650650

++ 10801080

++ 18071807 3801

Roman Numerals – Task 1Roman Numerals – Task 1

MMMDCCXXVIIIMMMDCCXXVIII

-- MDCCCLIIMDCCCLII

-- MCCXXXIMCCXXXI

-- CCCCXIIICCCCXIIICCXXXII

37283728

-- 18521852

-- 12311231

-- 413413232

Roman Numerals – Task 1Roman Numerals – Task 1

LXXVLXXV

xx LL

MMMDCCL

7575

xx 5050

3750

Drawbacks of positional Drawbacks of positional numeral systemnumeral system

Hard to represent larger Hard to represent larger numbersnumbers

Hard to do arithmetic with larger Hard to do arithmetic with larger numbers, trying do 23456 x numbers, trying do 23456 x 987654987654

The search was on for portable The search was on for portable representation of numbersrepresentation of numbers

To make progress, humans had to solve a To make progress, humans had to solve a tricky problem:tricky problem:

What is the smallest set of symbols in What is the smallest set of symbols in which the largest numbers can in theory which the largest numbers can in theory be represented?be represented?

South American MathsSouth American Maths

The Maya

The Incas

twentiestwenties unitsunits

Mayan MathsMayan Maths

twentiestwenties unitsunits 2 x 20 + 7 = 47

18 x 20 + 5 = 365

Babylonian MathsBabylonian Maths

The Babylonians

3600s3600s 60s60s 1s1s

BBaabbyylloonnIIaan n

sixtiessixties unitsunits =64 = 3604

Zero and the Indian Sub-Zero and the Indian Sub-Continent Numeral SystemContinent Numeral System You know the You know the origin of the positional number, and its origin of the positional number, and its

drawbacks.drawbacks. One of its limits is how do you represent tens, hundreds, One of its limits is how do you represent tens, hundreds,

etc.etc. A number system to be as effective as ours, it must A number system to be as effective as ours, it must

possess a zero.possess a zero. In the beginning, the concept of zero was synonymous In the beginning, the concept of zero was synonymous

with empty space.with empty space. Some societies came up with solutions to represent Some societies came up with solutions to represent

“nothing”. “nothing”. The Babylonians left blanks in places where zeroes should The Babylonians left blanks in places where zeroes should

be.be. The concept of “empty” and “nothing” started becoming The concept of “empty” and “nothing” started becoming

synonymous.synonymous. It was a long time before zero was discovered.It was a long time before zero was discovered.

Zero and the Indian Sub-Zero and the Indian Sub-Continent Numeral SystemContinent Numeral System

We have to thank the Mathematicians for We have to thank the Mathematicians for our modern number system.our modern number system.

Similarity between the Indian numeral Similarity between the Indian numeral system and our modern onesystem and our modern one

Indian NumbersIndian Numbers

Binary NumbersBinary Numbers

Different BasesDifferent Baseshundredshundreds tenstens unitsunits

11 22 55

12510 = 1 x 100 + 2 x 10 + 5

Base 10 (Decimal):

eightseights foursfours twostwos unitsunits

11 11 11 00

11102 = 1 x 8 + 1 x 4 + 1 x 2 + 0 = 14 (base 10)

Base 2 (Binary):

Pythagoras TheoremPythagoras Theorem

b

c

a

a2 = b2 + c2

Pythagoras’ Theorem

1

1

a

a2 = 12 + 12

So a2 = 2a = ?

MessagesMessages The number system we have today have come The number system we have today have come

through a long route, and mostly from some far through a long route, and mostly from some far away lands, outside of Europe.away lands, outside of Europe.

They came about because human beings wanted They came about because human beings wanted to solve problems and created numbers to solve to solve problems and created numbers to solve these problems.these problems.

Numbers belong to human culture, and not Numbers belong to human culture, and not nature, and therefore have their own long history.nature, and therefore have their own long history.

Questions to Ask Questions to Ask Ourselves.Ourselves.

Is this the end of our number system?Is this the end of our number system? Are there going to be any more changes Are there going to be any more changes

in our present numbers?in our present numbers? In 300 years from now, will the numbers In 300 years from now, will the numbers

have changed again to be something have changed again to be something else?else?

3 great ideas made our 3 great ideas made our modern number systemmodern number system

Our modern number system was a result of aOur modern number system was a result of aconjunction of 3 great ideas:conjunction of 3 great ideas: the idea of attaching to each basic figure the idea of attaching to each basic figure

graphical signs which were removed from all graphical signs which were removed from all intuitive associations, and did not visually intuitive associations, and did not visually evoke the units they represented evoke the units they represented

the principle of position the principle of position the idea of a fully operational zero, filling the the idea of a fully operational zero, filling the

empty spaces of missing units and at the same empty spaces of missing units and at the same time having the meaning of a null number time having the meaning of a null number