Shortest Proof of Irrationality of sqrt(N), where N is not a perfect square
by Nirakar Neo
This proof is really the shortest proof I have ever seen for this theorem, and it’s really witty! I stumbled this proof while browsing the American Mathematical Monthly periodicals, to be specific on page 524 in the June-July 2008 issue. I reproduce the proof here.
Theorem Let be a positive integer. If is a not perfect square, then is irrational.
Proof: Before embarking on the proof, recall that the standard proof uses the method of contradiction. Here we shall prove the contrapositive statement to prove the theorem. That is, we shall prove that if is rational then is a perfect square.
Thus assume that is rational. This means we can write
where are integers, , and that it is in the lowest form. This implies , which we can rearrange so as to write
Since we assumed that is in lowest form, we see that both and must be a integral multiple of and respectively. Taking that constant multiple to be , we may write and . The latter equation says
but since is nothing but , this implies that , which simply means that is a perfect square. This proves the contrapositive statement, and therefore the theorem follows.
Isn’t it beautiful!