NATURAL NUMBERS

• The set of natural numbers is denoted by N. Therefore N=[1,2,3…]

• The least number is 1.

• The greatest number is Infinite.

• Natural numbers are also called counting numbers.

WHOLE NUMBERS

• The set of whole numbers is denoted by W. Therefore W=[0,1,2,3…]

• The least number is 0.

• The greatest number is Infinite.

INTEGERS

• The set of integers is denoted by I or Z.

• Z=[…-3,-2,-1,01,2,3...]

• It is the collection of positive and negative numbers.

RATIONAL NUMBERS

The numbers which are in the form of p / q.

• (q is not equal to 0) were p , q are integers are called rational numbers.

• It is denoted by Q. therefore Q=[p / q (p not equal to 0)].EX:3/4,1/2,-3/2.

IRRATIONAL NUMBERS• A number S is called Irrational , if it cannot

be written in the form of p / q (q is not equal to 0,were p and q are integers .

• There are Infinite many Irrational numbers.

• Examples-(0.1011011101111…)

REAL NUMBERS

• It is the collection of rational and irrational numbers.

• Every real number is represented by a unique point on the number line.

• Also every point on the number line represents a unique real number.

NUMBER LINE-5 -4 -3 -2 -1 0 1 2 3 4

5

Number line is the collection of positive and negative numbers.

PYTHAGORAS(569BC-479BC)

The Pythagoras in Greece, followers of the famous mathematician and philosopher Pythagoras, were the first to discover the numbers which were not rational, around 400BC these numbers are called irrational numbers, because they cannot be written in the form of a ratio of integers

In the 1870s two German mathematicians , Cantor and Dedekind , showed that corresponding to every real number , there is a point on the real number line, and corresponding to every point on the number line , there exists a unique real number

DEDEKIND.R CANTOR.G

REAL NUMBERS AND THEIR DECIMAL EXPANSIONS

Decimal expansions of 10/3

3.333…10

9

10

9

10

9

10

9

1

3 In the division 10/3 after a certain stage we get a repeating digits.

REMAINDER:- 1,1,1,1,1…

Divisor:- 3

DECIMAL EXPANSIONS OF 7/8

0.875

7.0

64

60

56

40

40

0

8In the division of 7/8 after some steps the remainder becomes 0.

We call the decimal expansion of such numbers terminating.

REMAINDER:-6,4,0Divisor:- 8

DECIMAL EXPANSIONS OF1/70.142857

1.0

7

30

28

20

14

60

56

40

35

50

49

1

In the division of 1/7 if the remainders repeat then we get a repeating block of digits in the quotient in the case of 1/7 we get repeating block 142857 and in the case of 10/3 repeating digit is 3 we call the decimal expansion of such numbers non terminating and recurring decimal expansion

REMAINDER:-3,2,6,4,5,1,3,2,6,4,5,1,…Divisor:-7

7

REPRESENTING THE REAL NUMBERS ON THE NUMBER LINE

OPERATIONS ON REAL NUMBERS

• The sum or difference of a rational number and an irrational number is a irrational .

• The product or quotient of a non-zero rational number with an irrational number is irrational

• If we add or subtract, multiply or divide two irrationals, the result may be irrational or rational.

ARCHIMEDESThe Greek genius Archimedes was the first to compute digits in the decimal expansion of 22/7 he showed 3.140845 <22/7<30142857.aryabhatta , the great mathematician, found the value of 22/7 correct to four decimal places (3.1416).