Fun fact I discovered the other day. I'm surprised I wasn't aware of it before.
Take any whole number. Divide it by another whole number satisfying the condition that you have a fraction that may be improper but otherwise cannot be reduced. (The two numbers in the fraction are relatively prime.)
Multiply this fraction by itself. If the bottom number of the fraction is larger, squaring the fraction will make the bottom number grow more than the top number, so you can't get a whole number by squaring this fraction. A similar argument can be made if the top number is larger. There's no way to get a whole number squaring the fraction in that case either.
Since you cannot get a whole number by squaring a fraction, you can't have a fraction of this kind by taking the square root of a whole number. So the only kind of number a square root of a whole number can be is either another whole number, or an irrational. The square root of a prime cannot be a whole number since it has no whole number factors, so the square root of a prime is irrational.
