Imagine that √2 is rational, then it can be written as the ratio of integers, p and q.
(1) √2 = p/q
It may be presumed that p and q have no common factors, and thus cannot be reduced. Square both sides to get:
(2) 2 = p²/q²
which leads to
(3) p² = 2q²
Therefore, p² is an even number. It can only be even if p is an even number. Then p² is divisible by 4. Hence q² and (ergo) q must also be even. Since p and q are both even, the assumption that p and q have no common factors is contradicted. Therefore, √2 must be irrational. Q.E.D. (quod erat demonstrandum).