The Proof That Shook Greece

The most famous proof by contradiction in history: the square root of 2 is irrational. Assume it is rational. Then sqrt(2) = a/b, where a and b are whole numbers with no common factors.

Square both sides: 2 = a²/b², so a² = 2b². This means a² is even, so a must be even. Write a = 2k. Then (2k)² = 2b², so 4k² = 2b², so b² = 2k². But this means b is also even. Now both a and b are even, so they share the factor 2. Contradiction: we assumed they had no common factors.

Legend credits the Pythagorean Hippasus with this discovery around 500 BC. The Pythagoreans believed all numbers were ratios of whole numbers, and his proof destroyed that worldview. According to the story, which is probably a myth, they threw him overboard and drowned him.