√2 is the positive number that multiplied by itself gives 2. Geometrically, it is the length of the diagonal of a square with side 1. A direct consequence of the Pythagorean theorem: 1² + 1² = (√2)².
The Pythagoreans discovered around 500 BC that √2 cannot be expressed as a ratio of integers: it is irrational. This was deeply troubling to a school of thought that believed all magnitudes were rationally related. The proof by contradiction is one of the oldest in mathematics.
√2 appears in trigonometry (sin 45° = cos 45° = 1/√2), in the diagonal of rectangles, in the A-series paper format (A4, A3…), and as a fundamental constant in quantum mechanics.