What We Cannot Know: Explorations at the Edge of Knowledge

by Marcus du Sautoy

Pythagoras’ theorem about right-angled triangles implied that this long side had length the square root of 2 times the length of the short sides. But Hippasus could prove that there was no fraction whose square was exactly 2. The proof uses one of the classic tools in the mathematician’s arsenal: proof by contradiction. Hippasus began by assuming there was a fraction whose square was 2. By some deft manipulation this always led to the contradictory statement that there was a number that was both odd and even. The only way to resolve this contradiction was to realize that the original assumption must have been false: there can be no fraction whose square is 2.