Page 66 - Start Up Mathematics_7

Test of Equality of Two Rational Numbers Extension
r
p
Two rational numbers and are equal Iff implies if and only if, i.e., either both
the statements are true or both are false.
s
q
iff p × s = r × q. p iff q means:
(i) if p is true, then q is also true.
Example 1: Write the next four rational numbers (ii) if q is true, then p is also true.
in each of the following patterns: Iff can also be represented by ‘⇔’.

–2 –4 –6 –1 –2 –3 1 2 3
(a) , , , … (b) , , , … (c) , , , …
7 14 21 5 10 15 6 12 18

,
,
,
Solution: (a) –8 –10 –12 –14 (b) –4 –5 –6 –7 (c) 4 , 4 , 6 , 7
,
,
,
28 35 42 49 20 25 30 35 24 30 36 42
Example 2: Give four rational numbers equivalent to:
–3 2
(a) (b)
4 9
–6 –9 –12 –15 4 6 8 10
Solution: (a) , , , (b) , , ,
8 8 16 20 18 27 36 45
Example 3: Which of the following pairs represent the same rational number?

(a) 2 and –5 (b) –1 and 1
8 20 –3 3
2 1 –5 –1 –1 1 1 1
Solution: (a) = and = (b) = and =
8 4 20 4 –3 3 3 3
1 –1 1 1
 ≠  =
4 4 3 3
∴ given rational numbers are different. ∴ given rational numbers are same.

Representation of Rational Numbers on a Number Line

We know how to represent integers on a number line. Representation of rational numbers on a
number line is quite similar to the representation of integers.
1 –1
Let’s represent and on a number line.
3 3
(i) Draw a line and mark a point 0 at O.
O

0
(ii) Mark unit lengths OA and OA′ to the right and left of O respectively. Point A represents +1
and A′ represents –1.

A′ O A
–1 0 +1

(iii) Now divide both OA and OA′ into three equal parts as shown below:

A′ O A
–1 0 +1

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