The square root of 2, also known as Pythagoras' constant, is the positive real number that, when multiplied by itself, gives the number 2.
Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. Its numerical value truncated to 65 decimal places^{[1]} is:
The quick approximation for the square root of two is most frequently used. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000.
List of numbers – Irrational and suspected irrational numbers γ – ζ(3) – √2 – √3 – √5 – φ – ρ – δ_{S} – α – e – π – δ 

Binary  1.0110101000001001111... 
Decimal  1.4142135623730950488... 
Hexadecimal  1.6A09E667F3BCC908B2F... 
Continued fraction 
Contents 
The Babylonian clay tablet YBC 7289 (c. 1800–1600 BCE) gives an approximation of in four sexagesimal figures, which is about six decimal figures:^{[2]}
Another early close approximation of this number is given in ancient Indian mathematical texts, the Sulbasutras (c. 800–200 BCE) as follows: Increase the length [of the side] by its third and this third by its own fourth less the thirtyfourth part of that fourth.^{[3]} That is,
This ancient Indian approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, that can be derived from the continued fraction expansion of
The discovery of the irrational numbers is usually attributed to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2. According to one legend, Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but his beliefs would not accept the existence of irrational numbers and so he sentenced Hippasus to death by drowning.^{[4]} Other legends report that Hippasus was drowned by some Pythagoreans,^{[5]} or merely expelled from their circle.^{[4]}
There are a number of algorithms for approximating the square root of 2, which in expressions as a ratio of integers or as a decimal can only be approximated. The most common algorithm for this, one used as a basis in many computers and calculators, is the Babylonian method^{[6]} of computing square roots, which is one of many methods of computing square roots. It goes as follows:
First, pick a guess, a_{0} > 0; the value of the guess affects only how many iterations are required to reach an approximation of a certain accuracy. Then, using that guess, iterate through the following recursive computation:
The more iterations through the algorithm (that is, the more computations performed and the greater "n"), the better approximation of the square root of 2 is achieved. Each iteration approximately doubles the number of correct digits. Starting with a_{0} = 1 the next approximations are
The value of √2 was calculated to 137,438,953,444 decimal places by Yasumasa Kanada's team in 1997.
In February 2006 the record for the calculation of √2 was eclipsed with the use of a home computer. Shigeru Kondo calculated 200,000,000,000 decimal places in slightly over 13 days and 14 hours using a 3.6 GHz PC with 16 GiB of memory.^{[7]}
Among mathematical constants with computationally challenging decimal expansions, only π has been calculated more precisely.^{[8]}
A short proof of this result is to obtain it from Gauss's lemma, that if p(x) is a monic polynomial with integer coefficients, then any rational root of p(x) is necessarily an integer. Applying this to the polynomial p(x) = x^{2} − 2, it follows that √2 is either an integer or irrational. Since √2 is not an integer (2 is not a perfect square), √2 must therefore be irrational.
See quadratic irrational for a proof that the square root of any nonsquare natural number is irrational.
It is not known whether √2 is a normal number, a stronger property than irrationality, but statistical analyses of its binary expansion are consistent with the hypothesis that is normal to base two.^{[9]}
One proof of the number's irrationality is the following proof by infinite descent. It is also a proof by contradiction, which means the proposition is proved by assuming that the opposite of the proposition is true and showing that this assumption is false, thereby implying that the proposition must be true.
Since there is a contradiction, the assumption (1) that √2 is a rational number must be false. The opposite is proven: √2 is irrational.
An alternative proof uses the same approach with the fundamental theorem of arithmetic which says every integer greater than 1 has a unique factorization into powers of primes:
This proof can be generalized to show that any root of any natural number which is not the square of a natural number is irrational. The article quadratic irrational gives a proof of the same result but not using the fundamental theorem of arithmetic.
The following reductio ad absurdum argument showing the irrationality of √2 is less wellknown. It uses the additional information 2 > √2 > 1 so that 1 > √2 − 1 > 0.
Another reductio ad absurdum showing that √2 is irrational is less wellknown.^{[10]} It is also an example of proof by infinite descent. It makes use of classic compass and straightedge construction, proving the theorem by a method similar to that employed by ancient Greek geometers. It is essentially the previous proof viewed geometrically.
Let ABC be a right isosceles triangle with hypotenuse length m and legs n. By the Pythagorean theorem, m/n = √2. Suppose m and n are integers. Let m:n be a ratio given in its lowest terms.
Draw the arcs BD and CE with centre A. Join DE. It follows that AB = AD, AC = AE and the ∠BAC and ∠DAE coincide. Therefore the triangles ABC and ADE are congruent by SAS.
Since ∠EBF is a right angle and ∠BEF is half a right angle, BEF is also a right isosceles triangle. Hence BE = m − n implies BF = m − n. By symmetry, DF = m − n, and FDC is also a right isosceles triangle. It also follows that FC = n − (m − n) = 2n − m.
Hence we have an even smaller right isosceles triangle, with hypotenuse length 2n − m and legs m − n. These values are integers even smaller than m and n and in the same ratio, contradicting the hypothesis that m:n is in lowest terms. Therefore m and n cannot be both integers, hence √2 is irrational.
Then α is irrational.
Proof: suppose α = a/b with a, b ∈ N^{+}.
For sufficiently big n,
then
but aq_{n} − bp_{n} is an integer, absurd, then α is irrational.
Proof: let p_{1} = q_{1} = 1 and
for all n ∈ N.
By induction,
for all n ∈ N. For n = 1,
and if is true for n then is true for n + 1. In fact
By lemma 1 applications √2 is irrational.
Onehalf of √2, approximately 0.70710 67811 86548, is a common quantity in geometry and trigonometry because the unit vector that makes a 45° angle with the axes in a plane has the coordinates
This number satisfies
One interesting property of the square root of two is as follows:
This is a result of a property of silver means.
Another interesting property of the square root of two:
The square root of two can also be expressed in terms of the copies of the imaginary unit i using only the square root and arithmetic operations:
The square root of two is also the only real number other than 1 whose infinite tetrate is equal to its square.
The square root of two can also be used to approximate π:
for m square roots and only one minus sign.^{[11]}
The identity cos(π/4) = sin(π/4) = 1/√2, along with the infinite product representations for the sine and cosine, leads to products such as
and
or equivalently,
The number can also be expressed by taking the Taylor series of a trigonometric function. For example, the series for cos(π/4) gives
The Taylor series of √(1 + x) with x = 1 and using the double factorial n!! gives
The convergence of this series can be accelerated with an Euler transform, producing
It is not known whether √2 can be represented with a BBPtype formula. BBPtype formulas are known for π√2 and √2 ln(1+√2), however. [1]
The square root of two has the following continued fraction representation:
The convergents formed by truncating this representation form a sequence of fractions that approximate the square root of two to increasing accuracy, and that are described by the Pell numbers (known as side and diameter numbers to the ancient Greeks due to their use in approximating the ratio between the sides and diagonal of a square). The first convergents are: 1/1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408. The convergent p/q differs from the square root of 2 by almost exactly and then the next convergent is (p + 2q)/(p + q).
The square root of two is the aspect ratio of paper sizes under ISO 216. This ratio guarantees that cutting in half a sheet by a line parallel to its short side results in two sheets having the same ratio.
Indeed, if a rectangle has sides x and x√2, its half has sides x and x√2/2, the latter being the same as x/√2. Therefore, the proportion between the long side (x√2/√2) and the short side (x/√2) is again √2.
