The Pythagorean theorem says the square of the hypotenuse equals the sum of the squares of the other sides.

The longer version

Arrange four copies of a right triangle inside a square so that an empty square appears in the middle, then rearrange the same four triangles to leave two smaller empty squares instead. Since the triangles occupy the same area either way, the leftover regions must match, which says the square on the longest side equals the other two combined.

The relation was known well before the Greek school that carries the name; clay tablets list whole number triples satisfying it, and early Chinese and Indian texts state it too. Its lasting value is as a distance rule: the length of any slanted step is the square root of the sum of squared horizontal and vertical steps.

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.