If

were rational then

would also be rational (for consider that

).

But from this it would follow that both

and

are rational, which is not the case.

Whether the square root of a natural number is rational or not is determined by wether or not its (essentially) unique prime factor decomposition consists only of

*even* powers of prime numbers or not. In the former case its square root is a natural number, in the latter case its square root is irrational. So

are all irrational, whereas

are natural numbers.