This was insipired by Treadstone71.
Proof that for
Then, is rational or irrational.
It suffices to prove that the square of a non-integer rational number cannot be an integer.
Let non-integer and let it be in its lowest terms. That is, the intersection of the set of prime factors of n and m is empty. Let n/m be squared. However, since the set of prime factors of n and m did not change, the interesction is still empty, and therefore no cancellation can be made. is therefore a rational number in its lowest terms with its denominator not equal to 1, and is therefore not an integer. QED.