Digits of the square root of 2

The oldest surviving computation of √2 is on a clay tablet. YBC 7289, an Old Babylonian exercise from roughly 1800–1600 BCE now in the Yale Babylonian Collection, shows a square with its diagonals labelled in sexagesimal: 1;24,51,10, which is 1.41421296 in decimal. That is wrong by about one part in two million — the most accurate numerical result known from the ancient world, and the scribe shows no sign of thinking the value was anything other than the answer.

Indian geometers of the Sulba Sutras, perhaps 800–600 BCE, gave 1 + 1/3 + 1/(3·4) − 1/(3·4·34), which is exactly 577/408 ≈ 1.4142157 — one of a run of increasingly accurate approximations generated by the Pell numbers.

The Greek contribution was to prove that no such fraction can ever be exact. The discovery is traditionally assigned to the Pythagorean school, often to Hippasus of Metapontum, and the familiar story that he was drowned at sea for divulging it has essentially no evidential support; it is a late anecdote, not a record. Aristotle refers to the reductio argument — assume a fraction in lowest terms, derive that both numerator and denominator are even — as a known example of proof by contradiction. That argument survives as Proposition 117 of Book X of Euclid's Elements, but historians have agreed since the early nineteenth century that it is a later interpolation and not Euclid's own. Plato's Theaetetus credits Theodorus of Cyrene with extending irrationality proofs to the roots of the non-square integers up to 17.

The number then became ordinary and useful. Georg Christoph Lichtenberg pointed out in a letter of 1786 that a sheet with sides in ratio √2 halves into two sheets of the same shape; Walter Porstmann built a metric paper system on that in 1918, published as DIN 476 in 1921 and adopted internationally as ISO 216.

Key properties

• √2 is irrational: if √2 = p/q in lowest terms then p² = 2q², which forces both p and q to be even — a contradiction.
• √2 is algebraic of degree 2, being a root of x² − 2, so it is irrational but not transcendental.
• Its continued fraction is [1; 2, 2, 2, …], with every term after the first equal to 2.
• The convergents are 1/1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, …, each alternately below and above √2, with numerators and denominators drawn from the Pell numbers.
• 1 + 1/3 + 1/(3·4) − 1/(3·4·34) equals exactly 577/408, the eighth convergent above and the value recorded in the Sulba Sutras.
• The Babylonian iteration xₙ₊₁ = (xₙ + 2/xₙ)/2 roughly doubles the number of correct digits each pass; it is Newton's method on x² − 2, and the integer form of it is what computes this page.
• 1/√2 = √2/2 ≈ 0.70710678, the value behind the −3 dB half-power point and the RMS amplitude of a sine wave.

Where they turn up

• ISO 216 paper: every A size has sides in the ratio √2 : 1, so folding a sheet in half yields the next size down with the same proportions. A0 is 841 × 1189 mm and A4 is 210 × 297 mm.
• The diagonal of any square, by Pythagoras — which is why √2 appears whenever a square grid is measured across rather than along.
• Twelve-tone equal temperament: the tritone, six semitones, is a frequency ratio of 2^(6/12) = √2, the only interval that divides the octave exactly in half.
• Signal processing and electronics, where √2 relates a sine wave's peak to its RMS value and 1/√2 marks the −3 dB cutoff of a filter.
• The 1897 Indiana Pi Bill, which as well as implying π = 3.2 implied that √2 equals 10/7 ≈ 1.4286 — a reminder that legislatures cannot settle arithmetic.