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Let us represent them on number line and write the numbers between these two rational numbers.

−3 = −1 4 = 2
6 2 6 3 To write the rational numbers
between two rational
Note: numbers always covert the
–1 −5 −4 −3 −2 −1 0 1 2 3 4 5 1
6 6 6 6 6 6 6 6 6 6 given rational numbers with
same denominators.
−2 −1 1 2 3 −1 2
Clearly, , 0 , ,, , are the rational numbers between and .
6 6 6 6 6 2 3

−15 20
We can also write these given rational numbers as and , respectively.
30 30
−15 −14 −13 −12 1 2 3 19 20
Now, we can write < < < ... < < < < < ... < < rational numbers
0
30 30 30 30 30 30 30 30 30
between them.
−1 2 −30 40
Similarly, the given rational numbers and can also be written as and and so on
2 3 60 60
and we can find many more rational numbers between them.
Hence, we can find unlimited (infinite) rational numbers between any two rational numbers.
Example 7: Write five rational numbers between:

−4 7 −5 −3
(a) and (b) and
13 13 7 8

Solution: (a) We know that, –4 < –3 < –2 < –1 < 0 < 1 < 2 < 3 < 4 < 5 < 6 < 7
−4 7 −3 −2 −1 0 1
Hence, five rational numbers between and are: , , , ,
13 13 13 13 13 13 13

(b) The LCM of 7 and 8 is 56.
Now, write both the rational numbers with denominator 56.

−5 = −×58 = −40 −3 −×37 −21
7 78 56 and 8 = 87 = 56
×
×
−40 −21
Clearly, <
56 56
So, the rational numbers between −40 and −21 are:
56 56
−39 < −38 < −37 < −36 < −35 < .... < −20
56 56 56 56 56 56
−5 −3 −39 −38 −37 −36 −35
Thus, the five rational numbers between and are: , , , , .
7 8 56 56 56 56 56

Think and Answer

3 4
Write six rational numbers between and .
4 12

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