4. 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.
5. 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.
6. 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.
7. 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.
8. 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…)
9. 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.
10. NUMBER LINE
-5 -4 -3 -2 -1 0 1 2 3 4 5
Number line is the collection of
positive and negative numbers.
11. 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
12. 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
DEDEKIND.R CANTOR.G
number line , there
exists a unique real
number
13. REAL NUMBERS AND THEIR DECIMAL EXPANSIONS
Decimal expansions of 10/3
3 3.333… In the division 10/3 after a certain
stage we get a repeating digits.
10
9
10
9
10
9
REMAINDER:- 1,1,1,1,1…
10
9 Divisor:- 3
1
14. DECIMAL EXPANSIONS OF 7/8
In the division of 7/8 after some steps
8 0.875
the remainder becomes 0.
7.0
64 We call the decimal expansion of such
numbers terminating.
60
56
40
40
REMAINDER:-6,4,0
0 Divisor:- 8
15. DECIMAL EXPANSIONS OF1/7
7 0.142857
1.0 In the division of 1/7 if the remainders
7 repeat then we get a repeating block of
30
digits in the quotient in the case of 1/7 we
28
get repeating block 142857 and in the case
20
14
of 10/3 repeating digit is 3 we call the
60 decimal expansion of such numbers non
56 terminating and recurring decimal
40 expansion
35
50
REMAINDER:-3,2,6,4,5,1,3,2,6,4,5,1,…
49
1
Divisor:-7
17. 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.
18. ARCHIMEDES
The Greek genius
Archimedes was the first
to compute digits in the
decimal expansion of 22/7
he showed 3.140845
<22/7<30142857.aryabhat
ta , the great
mathematician, found the
value of 22/7 correct to
four decimal places
(3.1416).
