Page 14 - ICSE Math 8

r Maths Info
q ←→ s
p ←→
r
If ps > qr, then p > . • Every positive rational
s
q
number is greater than 0.
r
II. Let p , , u , ..., be rational numbers. To compare more than two • Every negative rational
q s v number is less than 0.
rational number, follow these steps:
Step 1: Express each rational number with a positive denominator.
Step 2: Find the LCM of the denominators.

Step 3: Write each of the given rational numbers with LCM as the common denominator.
Step 4: The rational number with the smallest numerator, amongst all the equivalent rational numbers,
is the smallest.
−4 5 −7 2
Example 1: Compare , , , . 3 9, 12,18, 3
9 −12 18 −3
Solution: Write each of the given rational number with the positive denominator. 3 3, 4, 6, 1
− −4 −5 −7 2
We have , , , . 2 1, 4, 2, 1
9 12 18 3 2 1, 2, 1, 1
Now, find the LCM of 9, 12, 18 and 3.
LCM = 3 × 3 × 2 × 2 = 36 1 1, 1, 1, 1
Write all the rational numbers with 36 (LCM) as the common denominator.
−4 −×4 4 −16 −5 −×53 −15
= = = =
9 9 × 4 36 12 12 × 3 36
−7 −×72 −14 −2 −×212 −24
18 = 18 × 2 = 36 3 = 312 = 36
×
Compare the numerators and arrange in ascending order.
−24 −16 −15 −14 2 − 4 5 − 7
< < < or < < <
36 36 36 36 − 3 9 − 12 18

Standard Form of a Rational Number

The standard form or lowest form of a rational number is p , where p and q have no common divisor other than
12 q
1 and q is always positive. For example, − 18 is a rational number which is not in its standard or lowest form.
Since 12 and 18 have a common divisor 6,
12 = 12 6÷ = 2
− 18 − 18 6÷ − 3
Now 2 and 3 have no common divisor except 1. But the denominator is negative. To write in standard form,
2 2×−() − 2
1
the denominator should be positive. So, = = .
3
− 3 −× −() 3
1
Equivalent Rational Numbers
p p pn× pn
I. If is a rational number and ‘n’ is an integer, where n ≠ 0, then = = .
q q qn× qn
p 3 3 3 × 2 6
For example, if q = 7 , n = 2, then 7 = 7 × 2 = 14 .

So, p and pn are equivalent rational numbers.
q qn

2