√2 N decimals Generator

The square root of 2, roughly 1.414214, is the diagonal of a unit square — and it was the first number proved not to be a fraction. The proof, credited to the Pythagorean school, is a short contradiction argument, and the result was so disturbing to a philosophy holding that everything is expressible as ratios of integers that it produced the legend of Hippasus being thrown into the sea for revealing it.

Choose the places and receive up to a thousand digits, computed by integer square root in arbitrary precision using Newton's method, and checked against published values. The algorithm is the same one taught for extracting roots by hand, only operating on giant integers rather than decimals.

The constant has a quiet everyday presence. A4 paper and the whole A series carry the ratio square root of 2 between long and short side — that is what allows folding the sheet in half and getting the same format, a property no other proportion has. It is also the ratio between semitones in equal temperament raised to the sixth power, which makes the tritone exactly half an octave.