What is √2 (Square Root of 2)?
√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.
| 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. The digits shown below are verified against the continued fraction.
Pi
Memorize pi, e, and 38 mathematical constants using the numpad path method
Play nowWorks on any device.
