1. Presented by Jahnvi
Tanwar
Number system

2. Introduction
This chapter is based on real numbers, on their
decimal representation and their representation
on number line. Real numbers are the collection
of all the rational and irrational numbers. It is
denoted by the symbol R. Every real number can
be represented on the number line with a
unique point on the number line. Also, every
point on number line represents a unique real
number. That is why we call the number, line a
real number line.

3. Classification Of Real Numbers
Real
Numbers
Irrational
Numbers
Rational
Numbers
Integers
Natural
numbers
Whole
numbers

4. Whole Numbers
Whole numbers include 0 and all the whole numbers. It
is denoted by W.
W={0,1,2….}
Natural numbers
The counting numbers are known as natural
numbers. The collection of whole numbers are
denoted by N.
N={1,2,3…..}
Integers
The collection of natural numbers, 0 and their
negatives are known as integers. It is denoted by Z.
Z={-1,-5,0,2,10,100,-1234 etc.}

5. Rational Numbers
Numbers in the form where p and q are integers
and q is not equal to 0 are known as rational
numbers. It is denoted by Q.
Q=
The decimal representation of rational numbers is
terminating or non-terminating and recurring.
2,0,-3, etc.

6. Irrational Numbers
Numbers which can’t be written in the form of in of
the form where p and q are integers and q is not
equal to are known as irrational numbers.
The decimal representation of irrational numbers is
non-terminating or non- recurring.

7. Decimal Expansion of Real Numbers
 The decimal expansion of a rational number is
either terminating or non terminating repeating.
Moreover, a number whose decimal expansion is
terminating or non terminating repeating is
rational. For e.g. 3/5, 1.25 etc.
 The decimal expansion of an irrational number is
non-terminating or non-repeating. Moreover, a
number whose decimal expansion is non-
terminating or non-repeating is irrational. For
e.g. sqrt(2), 0.10100100010000… etc.

8. Operations on Real Numbers
 Rational numbers satisfy the commutative,
associative and distributive property for addition,
multiplication and division. Also, rational numbers
are closed in addition, subtraction, multiplication
and division.
 In case of irrational numbers, commutative,
associative and distributive property holds good.
But the sum, difference, quotient and product of
two irrational numbers may not be always
irrational.

9. Operations on Real Numbers (Cont.)
 E.g. √7 + (- √7)=0= Rational number
√5 * √5 =5= Rational number
5*5=25= Rational number
2-2=0 = Rational number
 Sum or product of an irrational number and
rational number is an irrational number.

10. Rationalization Of Denominator
If we have to rationalize the denominator
of i.e., to make the denominator
as the rational number, then multiply or
divide by (√a+b), where ‘a’ and ‘b’ are
integers.

11. Laws Of Exponents For Real Numbers
If x>0 is any real number, p and q are Rational
numbers, m and n are rational numbers then

12. Real Numbers and Number Line
All the real numbers can be represented on number
line.
1.
Number line representing natural numbers.
2.
Number line representing whole numbers.

13. 3.
Number Line representing integers.
4.
Number Line representing rational numbers.

14. 5.
we can represent irrational number on number line
using spiral method.