Rational Numbers are Countable?

Rational numbers are the numbers which can be expressed in form of p/q where p & q are integers such that q ≠ 0. If we are given a pair of Rational Numbers say 2/7 and 4/7. If we look at glance on these two rational numbers, we say that there exist only one rational number i.e. ¾ between them. But if the question is find 5 rational numbers b/w the given two rational numbers. Now , we multiply numerator & denominator by 4, we get (2x4)/(7x4) and (4x4)/(7x4) Or 8/28 and 16/28. Now, we have 9/28, 10/28, 11/28, 12/28, 13/28, 14/28 and 15/28 as the rational numbers lying between the two rational numbers 8/28 and 16/28. We can say that they are rational numbers lying between 2/7 and 4/7.

Similarly, if we multiply and divide this pair of rational numbers by 6, we get (2x6) / (7x6) and (4x6) / (7x6) =12/42 and 24/42 Here we find that there 11 rational numbers between the given two rational numbers. Thus we conclude that rational numbers are not countable as there can be any number of rational numbers between the given two rational numbers. Though it is to be remembered that some of them are equivalent but they exist in the different forms.