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Representation of Rational Numbers on a Number Line

We know that on a number line, the points to the right of 0 denote the positive integers, while the
points to the left of 0 denote the negative integers. To represent fractions on a number line, we
divide the distance between two positive numbers into the required number of equal divisions. In
a similar manner, we can represent positive rational numbers on a number line as shown below.

0 1 2 3 4 5 6 7 8

8 8 8 8 8 8 8 8
For negative rational numbers, we divide the distance between two negative integers into equal
parts according to the denominator of the rational number.
−8 2 −3 11
Let us represent some rational numbers such as , , , and on a number line.
5 5 5 5
Firstly, we draw a number line and mark integers –1, –2, –3, 0, 1, 2, 3 on it. Since the denominator
of each rational number is 5, so we divide the distance between two consecutive integers in 5
equal parts.
−3
In , numerator is –3, so we mark a point ‘A’ at 3rd equal part between 0 and –1.
5

So, point A represents −3 .
5
−8
In , numerator is –8, so we mark a point ‘B’ at 3rd equal part between –1 and –2.
5
−8 2 11
Thus, point B represents . Similarly, point C and D will represent and , respectively.
5 5 5
–8 –3 2 11
5 5 5 5

–3 –2 B –1 A 0 C 1 2 D 3

Equivalent Rational Numbers

Equivalent rational numbers are obtained in the same way as equivalent fractions.

By multiplying or dividing the numerator and denominator of a rational number by the same
non-zero integer, we obtain another rational number equivalent to the given rational number.
÷
×
p pm pm
If is a rational number and m is a non-zero integer, then or is a rational number
×
÷
q p qm qm
equivalent to .
q
3
−3 −×32 −6 −3 −× − ( ) 3 9
For example: (a) = = and = =
2 22 4 2 2 ×− ( ) 3 −6
×
−6 9 −3
Here, and are equivalent to the rational number .
4 − 6 2
12 12 2 6 12 12 ÷− ( 4) − 3
÷
(b) = = and = =
− 16 − 16 2 − 8 − 16 − 16 ÷ − ( 4) 4
÷
6 −3 12
Here, and are equivalent to the rational number .
− 8 4 − 16
Mathematics-7 56