Page 9 - Start Up Mathematics_8 (Non CCE)

1 Rational Numbers

We have already learnt about natural numbers, whole numbers integers and fractions. The need to extend our
number system arises when we have to divide two integers. For example, if we have to divide 2 by –3, then
2
–3 is neither an integer nor a fraction. Such numbers are known as rational numbers.
p
A rational number is a number that can be expressed as where p and q are both integers and q ≠ 0. In
q p
other words, a rational number is also the quotient of two integers p and q in the form where q ≠ 0.
q
−4 7 6 −5
Examples of rational numbers are , , , .
5 12 −11 −8
p
Since all integers can be expressed in the form q , where q = 1, therefore all integers are rational numbers. If

the numerator p and the denominator q have the same sign, it is a positive rational number, otherwise, it is a
negative rational number.
−3 3 +3 3
For example, = ; = are positive rational numbers.
−4 4 +4 4
−3 −3 +3 3
= ; =− 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

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. So, any real number
that is not rational is irrational.
For example, 2 and π are irrational numbers as their decimal equivalent are non-terminating, non-repeating
decimals.
3 3 − 3 3
Absolute Value of a rational number is its numerical value with no regards to its sign. So, = and = .
Absolute value of a rational number is always non-negative. 7 7 7 7

Comparison of Rational Numbers
p r
I. Let q and be two rational numbers. Extension
s
p r
q ←→ s The value of 2 is 1.414213562373095 ...
←→
r
If ps > qr, then p > . and π is 3.1415926535...
s
q
p r u
II. Let q , , v , ..., be rational numbers.
s
Follow these steps to compare three or more rational numbers:
Step 1: Take LCM of the denominators q, s, v, ..., etc.
Step 2: Convert all rational numbers into their equivalent rational numbers with LCM as the denominator.
Step 3: The rational number with the smallest numerator, amongst all the equivalent rational numbers, is the
smallest rational number and so on.