(iii) False, because
is not an integer.
Example 2 : Find five rational numbers between 1 and 2.
We can approach this problem in at least two ways.
Solution 1 : Recall that to find a rational number between r and s, you can add r and
lies between r and s. So,
is a number
between 1 and 2. You can proceed in this manner to find four more rational numbers
s and divide the sum by 2, that is
between 1 and 2. These four numbers are
5 11 13
4 8 8
Solution 2 : The other option is to find all the five rational numbers in one step. Since
we want five numbers, we write 1 and 2 as rational numbers with denominator 5 + 1,
i.e., 1 =
7 8 9 10
and 2 =
. Then you can check that , , ,
are all rational
6 6 6 6
numbers between 1 and 2. So, the five numbers are
7 4 3 5
, , , and .
6 3 2 3
Remark : Notice that in Example 2, you were asked to find five rational numbers
between 1 and 2. But, you must have realised that in fact there are infinitely many
rational numbers between 1 and 2. In general, there are infinitely many rational
numbers between any two given rational numbers.
Let us take a look at the number line again. Have you picked up all the numbers?
Not, yet. The fact is that there are infinitely many more numbers left on the number
line! There are gaps in between the places of the numbers you picked up, and not just
one or two but infinitely many. The amazing thing is that there are infinitely many
numbers lying between any two of these gaps too!
So we are left with the following questions:
1. What are the numbers, that are left on the number
2. How do we recognise them? That is, how do we
distinguish them from the rationals (rational
These questions will be answered in the next section.