Page 13 - ICSE Math 8

1 Rational Numbers

Key Concepts

• Positive and Negative Rational Numbers • Multiplication and Division of Rational Numbers
• Comparison of Rational Numbers • Rational Numbers on a Number Line
• Properties of Rational Numbers • Rational Numbers Between Two Given Rational
• Addition and Subtraction of Rational Numbers Numbers

We have already learnt about natural numbers, whole numbers, integers and fractions. The need to extend our
2
number system arises when we divide two integers. For example, if we divide 2 by –3, then –3 is neither an
integer nor a fraction. Such numbers are known as rational numbers.

A rational number is a number that can be expressed as p , where p and q are both integers and q ≠ 0.
q
In other words, a rational number is also the quotient of two integers p and q in the form p , where q ≠ 0.
−4 7 6 −5 q
Examples of rational numbers are , , , .
5 12 −11 −8

Since all integers can be expressed in the form p , where q = 1, therefore all integers are rational numbers.
q
Positive and Negative Rational Numbers
A rational number is said to be a positive rational number if its numerator and denominator are both positive
−3 3 +3 3
or both negative. For example, = ; = are positive rational numbers.
−4 4 +4 4
A rational number is said to be a negative rational number if its numerator and denominator have opposite
−3 −3 +3 3
signs. For example, = ; =− are negative rational numbers.
+4 4 −4 4 0 0
As per the definition of rational number, 0 can be expressed as ; , etc. Hence, zero is also a rational
number. Zero is neither positive nor negative. 1 − 5
Irrational numbers
The decimal representation of a rational number either terminates or repeats the sequence of digits. If, however
the decimal representation continues forever without repeating, it is an irrational number. For example, 2
and π are irrational numbers as their decimal equivalent are non-terminating, non-repeating decimals.
Absolute value
3
Absolute value of a rational number is its numerical value irrespective of its sign. So, 3 = 3 and − 3 = .
Absolute value of a rational number is always non-negative. 7 7 7 7

Real numbers
Real numbers are a combination of all the rational and irrational numbers. That means a set of real numbers
consists of all the numbers that we have studied till now. The set of real numbers is denoted by R.

Comparison of Rational Numbers
p r
I. Let and be two rational numbers. To compare these rational numbers, we compare their products
q s
as follows:

1