Pythagorean proof of irrationality of sqrt(2)
Assume sqrt 2 is rational:

Then two is of the form (a / b)^2, a and b have no common factors

so a^2 = 2 b^2
Thus a^2 must be even
Thus a must also be even
So a = 2c for some c
(2c)^2 = 2  b^2
b^2 = 2c^2
So b^2 is even, b is even

But if a and b are even then they have common factors
So there is no a and b s.t. 2 = (a / b)^2
Therefore sqrt(2) is irrational