Square Root of Two

√2 was the number that broke Greek mathematics. The Pythagoreans held that everything was expressible as a ratio of whole numbers, and the diagonal of a square — the simplest figure imaginable — is not. The proof is short enough to carry in your head: assume √2 = p/q in lowest terms, square it, and you find p and q must both be even, contradicting "lowest terms". Legend has it that Hippasus was drowned at sea for leaking the result, which is almost certainly a later invention, but the crisis it caused was real and pushed Greek mathematics toward geometry and away from arithmetic.

The value stored here is the nearest double to the true constant. Practically, √2 is everywhere: it is the ISO 216 paper ratio, chosen so that halving an A4 sheet gives two sheets of exactly the same proportions, and it is the ratio between RMS and peak voltage in a sinusoid, so mains at 230 V RMS actually peaks at 325 V. A Babylonian tablet, YBC 7289, records the value as 1.41421296 in sexagesimal — accurate to six decimal places, and written more than a thousand years before Pythagoras.