The square root of two was one of the first numbers proven irrational, which unsettled ancient Greek mathematicians.

The longer version

Suppose the square root of two were the fraction a over b in lowest terms. Squaring gives a squared equals two b squared, so a must be even, and substituting back forces b to be even too. Both being even contradicts lowest terms, so no such fraction exists.

The diagonal of a unit square is exactly that length, so a ruler-and-compass construction produces something no ratio of whole numbers can state. Ancient Greek geometers had built their picture of quantity on ratios, and this result cracked it; stories say the discovery was met with dismay. Geometry and arithmetic have been treated as separate provinces ever since.

Where this entry sits

This section collects odd corners of mathematics, from stubborn prime numbers to counting systems older than modern schools. Expect famous constants, probability puzzles, shapes that break everyday intuition, and numbers far too large to picture. The full list sits on the Numbers and Math page, where every entry is listed in order.