√2 is the length of the diagonal of a unit square. Place a square with sides of length 1 on a table. The distance from one corner to the opposite corner is exactly √2. This is the Pythagorean theorem: 1² + 1² = (√2)².
The Pythagoreans discovered around 500 BC that √2 cannot be expressed as a fraction p/q where p and q are integers. The proof by contradiction is elegant: assume √2 = p/q in lowest terms. Then 2q² = p², so p² is even, so p is even, write p = 2k. Then 2q² = 4k², so q² = 2k², so q is also even. This contradicts p/q being in lowest terms. √2 is irrational.
Convergents from the continued fraction [1; 2, 2, 2, …]. Each fraction is the best rational approximation with that denominator.
Convergents of square root of 2 from continued fraction
| fraction | decimal | error |
|---|---|---|
| 1/1 | 1.000 | 0.41421 |
| 3/2 | 1.500 | 0.08579 |
| 7/5 | 1.400 | 0.01421 |
| 17/12 | 1.41667 | 0.00246 |
| 99/70 | 1.41429 | 0.0000849 |
√2 is algebraic (it satisfies x² = 2) but irrational. In trigonometry: sin(45°) = cos(45°) = 1/√2. The A-paper series (A4, A3, A2…) uses the ratio 1:√2, so that folding a sheet in half gives the same proportions. Computed to full precision: 1.41421356237309504880168872…
Each right triangle has one leg equal to the previous hypotenuse and one leg equal to 1. The hypotenuses are √1, √2, √3, √4, √5… Most are irrational. √2 (red) was the first proved irrational, by the Pythagoreans around 500 BC.
The square root of 2 is approximately 1.41421356237309504880. It was the first number ever proved irrational, by the ancient Greeks around 500 BCE. It is algebraic, satisfying x² = 2. It appears as the diagonal length of a unit square, in equal-temperament music tuning (each semitone multiplies frequency by the 12th root of 2), in A-series paper dimensions (A4 folded gives A5, same proportions), and in the Pythagorean theorem whenever legs are equal.
Square Root of 2 is irrational. Its decimal expansion never ends and never repeats. Every digit shown below is computed from the continued fraction.