If we want to check ‘n’ number of Rational Numbers between any two rational numbers say ‘x’ and ‘y’, then we will first find y - x, and then find the value of ‘d’.
D= (y - x)/ (n + 1),
Now once the value of ‘d’ is known, we will be able to find n rational numbers between ‘x’ and ‘y’ by the formula,
(x + d), (x + 2d), (x + 3d) … …. …, (x + nd),
Also further we see that ‘n’ can be any natural number, so let us say that between 4 and 5, we can find any number of rational numbers, thus we say that these rational numbers between any two rational numbers are countless as the Natural Numbers are countless.
Similarly if we again take another pair of rational numbers, here also again ‘n’ number of rational numbers can be located. So we conclude that the rational numbers between any two numbers are infinite and we cannot count them. So we conclude that the rational numbers are dense.
These rational numbers can also be found by another method. If we have a pair of rational numbers 3/5 and 4/5, and we want to find 5 rational numbers between these two rational numbers, then we multiply the numerator and the denominator of both the numbers by 6 and we get 18/30 and 24/30. Thus it is clear that 19/30, 20/30, 21/30, 22/30 are all rational numbers between the given two rational numbers 3/5 and 4/5. Similarly we can find 10 rational numbers between these two numbers. So we can say that any number of rational numbers can be located between any two given numbers. So rational numbers are dense.