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.
FAQ
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Who proved \u221A2 is irrational?
The Pythagoreans, possibly Hippasus of Metapontum, proved it around 500-450 BC. The discovery was shocking because their philosophy held all numbers are ratios of whole numbers. Legend holds Hippasus was drowned at sea for revealing the proof. -
How was the irrationality of \u221A2 proved?
By contradiction. Assume \u221A2 = p/q in lowest terms. Then p^2 = 2q^2, so p is even - write p = 2m. Then 4m^2 = 2q^2, so q^2 = 2m^2, meaning q is also even. But if both are even they share a factor of 2, contradicting lowest terms. -
Did ancient civilizations know about \u221A2?
Yes. The Yale Babylonian Collection tablet YBC 7289, dating to around 1800-1600 BC, shows an approximation of \u221A2 accurate to five decimal places. The Babylonians computed diagonal lengths without any formal proof of irrationality. -
Why was the 1:\u221A2 ratio chosen for A-paper sizes?
German engineer Walter Porstmann formalized the German DIN 476 standard in 1922 (later adopted internationally as ISO 216 in 1975). The key insight, first noted by Lichtenberg in 1786: if a sheet has sides in ratio 1:\u221A2, folding it in half produces a sheet with the exact same proportions. This allows all sizes from A0 to A10 to share one standard. -
Where else does \u221A2 appear in mathematics?
In a 45-45-90 right triangle, the hypotenuse is always \u221A2 times the legs. The face diagonal of a unit cube is \u221A2. It appears as the continued fraction [1;2,2,2,...] and in quantum mechanics wave function normalization.
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