Suppose 2^(1/2) = a/b in lowest terms, where a,b are integers. 2 = a^2/b^2. Implies 2b^2 = a^2. Thus a^2 is even, and so must be a. So express a as 2m, where m is some integer. a^2 = 4m^2. This implies 2b^2 = 4m^2 implies b^2 = 2m^2. So b^2 and b are also even. But a/b was in lowest terms! Contradiction.
Thus 2^(1/2) is irrational. QED.
See also proof.