1. 1
NUMBER
SYSTEM
TYPES
OF
NUMBERS

2. Number Systems
Lets study about various types
of numbers and number line
and how to represent these
numbers on it…..
2

3. Number Systems
NUMBERS
0 8 9
Hi… All these
are
Numerals….
Denoted by Group of digits called Numerals…
3

4. Number Systems
NATURAL NUMBERS “N”
INTEGERS “Z”
WHOLE NUMBERS “W”
THE NUMBER LINE
4

5. Number Systems
NATURAL NUMBERS
8 9
Counting numbers are Natural Numbers.
5

6. Number Systems
ODD
NUMBERS
Counting
numbers not
divisible by
two “2”
EVEN & ODD NUMBERS
6

7. Number Systems
WHOLE NUMBERS
0 8 9
Counting numbers along with zero are Whole
Numbers.
7

8. Number Systems
INTEGERS
3 4...
Counting numbers, zero and negatives of
counting are Integers.
8

9. Number Systems
INTEGERS can be…
Negative Integers e.g. -1, -2, -3, -4,…
Non Negative Integers e.g. 0, 1, 2, 3, 4,…
Positive Integers e.g. 1, 2, 3, 4, 5, 6, 7, 8,…
9

10. Number Systems
Denoted by ‘r’ are numbers which
can be written in p/q form, where
p and q are integers and q≠ 0.
Rational Numbers
Example
Collection of rational number is
denoted by “Q”
10
RationalNumbers

11. 11
Number Systems
Some Important Points:-
 Every integer, whole and natural number is a rational number.
 Number of rational numbers between two rational numbers is
infinite.
 Suppose a and b are two rational numbers, then
a x b
a + b = Rational Number
a + b
a ÷ b (where b is non-zero i.e. b ≠ 0)
Closure property under addition, subtraction, multiplication and
division is satisfied by rational numbers.
RationalNumbers

12. Number Systems
Irrational Numbers
All numbers which cannot be written
in p/q form, where p and q are
integers and q≠ 0
IrrationalNumbers
Example
12
Square roots of all positive integers are not irrational e.g.
9 = 3 (rational number)

13. 13
Number Systems
Some Important Points:-
 Irrational numbers have non-terminating and non-repeating
decimal expression.
 Irrational numbers can be easily represented on number line by
using Pythagoras Theorem where
In right angled ∆[Hypotenuse]2 = [Base]2 + [Perpendicular]2
 Suppose a and b are two irrational numbers, then
a x b
a + b = Not always an Irrational Number
a + b
a ÷ b
Closure property under addition, subtraction, multiplication and
division is not satisfied by irrational numbers.
Irrational Numbers

14. 14
RATIONAL NUMBER
TERMINATING or NON-
TERMINATING RECURRING
DECIMAL EXPANSION DECIMAL EXPANSION
IRRATIONAL NUMBER
NON-TERMINATING or
NON-RECURRING
DECIMAL EXPANSION DECIMAL EXPANSION
Number Systems
Decimal Expansion

15. 15
Number Systems
Real Numbers
Real numbers include all rational and all
irrational numbers. Denoted by ‘R’
RealNumbers
THE REAL NUMBER LINE

16. 16
IRRATIONAL
NUMBERS
RATIONAL
NUMBERS
REAL NUMBERS
NATURAL NUMBERS
INTEGERS
WHOLE NUMBERS
So…What we learned today
Number Systems

17. 17
Number Systems
RealNumbers
Real Numbers :-
 Both rational and irrational numbers together makes a collection
of real numbers.
 On a number line, there is a unique real number corresponding to
every point and also corresponding to each real number there is a
unique point.
 Suppose we have one rational and one irrational number, then:-
Rational Number + Irrational Number = Irrational Number
Rational Number - Irrational Number = Irrational Number
Rational Number x Irrational Number = Irrational Number, Rational
Number ≠ 0
Rational Number / Irrational Number = Irrational Number, Rational
Number ≠ 0
Real numbers also satisfy the various laws i.e. commutative,
associative and distributive laws etc.

18. 18
Number Systems
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