The square root of 2 (approximately 1.4142) is a positive
real number that, when multiplied by itself, equals the
number 2. It may be written in mathematics as or . It is an
algebraic number, and therefore not a
transcendental number. Technically, it should be called the principal
square root of 2, to distinguish it from the negative number with the same property.
Babylonian clay tablet
YBC 7289 (
c. 1800–1600 BC) gives an approximation of in four
sexagesimal figures, 1 24 51 10, which is accurate to about six
decimal digits, and is the closest possible three-place sexagesimal representation of :
Another early approximation is given in
ancient Indian mathematical texts, the
c. 800–200 BC), as follows: Increase the length [of the side] by its third and this third by its own fourth less the thirty-fourth part of that fourth. That is,
This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of
Pell numbers, which can be derived from the
continued fraction expansion of . Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation.
Pythagoreans discovered that the diagonal of a
square is incommensurable with its side, or in modern language, that the square root of two is
irrational. Little is known with certainty about the time or circumstances of this discovery, but the name of
Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. The square root of two is occasionally called Pythagoras's number or Pythagoras's constant, for example by
Conway & Guy (1996).
Ancient Roman architecture
ancient Roman architecture,
Vitruvius describes the use of the square root of 2 progression or ad quadratum technique. It consists basically in a geometric, rather than arithmetic, method to double a square, in which the diagonal of the original square is equal to the side of the resulting square. Vitruvius attributes the idea to
Plato. The system was employed to build pavements by creating a square
tangent to the corners of the original square at 45 degrees of it. The proportion was also used to design
atria by giving them a length equal to a diagonal taken from a square, whose sides are equivalent to the intended atrium's width.
There are many
algorithms for approximating as a ratio of
integers or as a decimal. The most common algorithm for this, which is used as a basis in many computers and calculators, is the
Babylonian method for computing square roots. It goes as follows:
First, pick a guess, ; 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
The more iterations through the algorithm (that is, the more computations performed and the greater "n"), the better the approximation. Each iteration roughly doubles the number of correct digits. Starting with , the results of the algorithm are as follows:
3/2 = 1.5 (a1)
17/12 = 1.416... (a2)
577/408 = 1.414215... (a3)
665857/470832 = 1.4142135623746... (a4)
The product of pi and the square root of two can be calculated through the following BBT-style formula:
A simple rational approximation 99/70 (≈ 1.4142857) is sometimes used. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000 (approx. +0.72×10−4).
The next two better rational approximations are 140/99 (≈ 1.4141414...) with a marginally smaller error (approx. −0.72×10−4), and 239/169 (≈ 1.4142012) with an error of approx −0.12×10−4.
The rational approximation of the square root of two derived from four iterations of the Babylonian method after starting with a0 = 1 (665,857/470,832) is too large by about 1.6×10−12; its square is ≈ 2.0000000000045.
Records in computation
In 1997, the value of was calculated to 137,438,953,444 decimal places by
Yasumasa Kanada's team. In February 2006, the record for the calculation of was eclipsed with the use of a home computer. Shigeru Kondo calculated one
trillion decimal places in 2010. Among
mathematical constants with computationally challenging decimal expansions, only
e, and the
golden ratio have been calculated more precisely as of March 2022[update]. Such computations aim to check empirically whether such numbers are
This is a table of recent records in calculating the digits of .
Number of digits
January 5, 2022
June 28, 2016
April 3, 2016
January 20, 2016
February 9, 2012
March 22, 2010
Proofs of irrationality
proof of the irrationality of can be obtained from the
rational root theorem, that is, if is a
monic polynomial with integer
coefficients, then any
rationalroot of is necessarily an integer. Applying this to the polynomial , it follows that is either an integer or irrational. Because is not an integer (2 is not a
perfect square), must therefore be irrational. This proof can be generalized to show that any square root of any
natural number that is not a perfect square is irrational.
As with the proof by infinite descent, we obtain . Being the same quantity, each side has the same
prime factorization by the
fundamental theorem of arithmetic, and in particular, would have to have the factor 2 occur the same number of times. However, the factor 2 appears an odd number of times on the right, but an even number of times on the left—a contradiction.
A simple proof is attributed to
Stanley Tennenbaum when he was a student in the early 1950s. Given two squares with integer sides respectively a and b, one of which has twice the
area of the other, place two copies of the smaller square in the larger as shown in Figure 1. The square overlap region in the middle () must equal the sum of the two uncovered squares (). However, these squares on the diagonal have positive integer sides that are smaller than the original squares. Repeating this process, there are arbitrarily small squares one twice the area of the other, yet both having positive integer sides, which is impossible since positive integers cannot be less than 1.
Tom M. Apostol made another geometric reductio ad absurdum argument showing that is irrational. 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 same algebraic proof as in the previous paragraph, viewed geometrically in another way.
Let △ ABC be a right isosceles triangle with hypotenuse length m and legs n as shown in Figure 2. By the
Pythagorean theorem, . Suppose m and n are integers. Let m:n be a
ratio given in its
Draw the arcs BD and CE with centre A. Join DE. It follows that AB = AD, AC = AE and ∠BAC and ∠DAE coincide. Therefore, the
trianglesABC and ADE are
Because ∠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, there is 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, is irrational.
While the proofs by infinite descent are constructively valid when "irrational" is defined to mean "not rational", we can obtain a constructively stronger statement by using a positive definition of "irrational" as "quantifiably apart from every rational". Let a and b be positive integers such that 1<a/b< 3/2 (as 1<2< 9/4 satisfies these bounds). Now 2b2 and a2 cannot be equal, since the first has an odd number of factors 2 whereas the second has an even number of factors 2. Thus |2b2 − a2| ≥ 1. Multiplying the absolute difference |√2 − a/b| by b2(√2 + a/b) in the numerator and denominator, we get
inequality being true because it is assumed that 1<a/b< 3/2, giving a/b + √2 ≤ 3 (otherwise the quantitative apartness can be trivially established). This gives a lower bound of 1/3b2 for the difference |√2 − a/b|, yielding a direct proof of irrationality in its constructively stronger form, not relying on the
law of excluded middle; see
Errett Bishop (1985, p. 18). This proof constructively exhibits an explicit discrepancy between and any rational.
is also the only real number other than 1 whose infinite
tetrate (i.e., infinite exponential tower) is equal to its square. In other words: if for c > 1, x1 = c and xn+1 = cxn for n > 1, the
limit of xn as n → ∞ will be called (if this limit exists) f(c). Then is the only number c > 1 for which f(c) = c2. Or symbolically:
It is not known whether is a
normal number, which is a stronger property than irrationality, but statistical analyses of its
binary expansion are consistent with the hypothesis that it is normal to
Series and product
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
convergentsp/q 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 (i.e., p2 − 2q2 = ±1). The first convergents are: 1/1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408 and the convergent following p/q is p + 2q/p + q. The convergent p/q differs from by almost exactly 1/2√2q2, which follows from:
The following nested square expressions converge to :
In 1786, German physics professor
Georg Christoph Lichtenberg found that any sheet of paper whose long edge is times longer than its short edge could be folded in half and aligned with its shorter side to produce a sheet with exactly the same proportions as the original. This ratio of lengths of the longer over the shorter side guarantees that cutting a sheet in half along a line results in the smaller sheets having the same (approximate) ratio as the original sheet. When Germany standardised
paper sizes at the beginning of the 20th century, they used Lichtenberg's ratio to create the
"A" series of paper sizes. Today, the (approximate)
aspect ratio of paper sizes under
ISO 216 (A4, A0, etc.) is 1:.
Let shorter length and longer length of the sides of a sheet of paper, with
as required by ISO 216.
Let be the analogous ratio of the halved sheet, then
There are some interesting properties involving the square root of 2 in the
^Williams, Kim; Ostwald, Michael (2015). Architecture and Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s. Birkhäuser. p. 204.
^Although the term "Babylonian method" is common in modern usage, there is no direct evidence showing how the Babylonians computed the approximation of seen on tablet YBC 7289. Fowler and Robson offer informed and detailed conjectures. Fowler and Robson, p. 376. Flannery, p. 32, 158.
^All that Aristotle says, while writing about
proofs by contradiction, is that "the diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate".
^The edition of the Greek text of the Elements published by E. F. August in
Berlin in 1826–1829 already relegates this proof to an Appendix. The same thing occurs with
J. L. Heiberg's edition (1883–1888).
Bishop, Errett (1985), Schizophrenia in contemporary mathematics. Errett Bishop: reflections on him and his research (San Diego, Calif., 1983), 1–32, Contemp. Math. 39, Amer. Math. Soc., Providence, RI.