The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written 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.
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. The fraction (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator.
Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 30 decimal places:
History
The Babylonian clay tablet YBC 7289 (–1600 BC) gives an approximation of in four sexagesimal figures, , which is accurate to about six decimal digits, and is the closest possible three-place sexagesimal representation of , representing a margin of error of only –0.000042%: See p. 368.Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection High resolution photographs, descriptions, and analysis of the root(2) tablet (YBC 7289) from the Yale Babylonian Collection
Another early approximation is given in ancient Indian mathematical texts, the Sulbasutras (–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, though this has little if any substantial evidence in traditional historical practice. The square root of two is occasionally called Pythagoras's number or Pythagoras's constant.
Ancient Roman architecture In 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.
Decimal value Computation algorithms
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 methodAlthough 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. for computing square roots, an example of Newton's method for computing roots of arbitrary functions. 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 recursive computation:
Each iteration improves the approximation, roughly doubling the number of correct digits. Starting with , the subsequent iterations yield:
Rational approximations The Babylonians had approximated the number as .
The rational approximation (≈ 1.4142857) differs from the correct value by less than (approx. ). Likewise, (≈ 1.4141414...) has a marginally smaller error (approx. ), and (≈ 1.4142012) has an error of approximately .
The rational approximation of the square root of two derived from four iterations of the Babylonian method after starting with () is too large by about ; its square is ≈ .
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. Other mathematical constants whose decimal expansions have been calculated to similarly high precision include , , and the golden ratio. Such computations provide empirical evidence of whether these numbers are normal.
This is a table of recent records in calculating the digits of . Date Name Number of digits 4 April 2025 Teck Por Lim 26 December 2023 Jordan Ranous 5 January 2022 Tizian Hanselmann 28 June 2016 Ron Watkins 3 April 2016 Ron Watkins 20 January 2016 Ron Watkins 9 February 2012 Alexander Yee 22 March 2010 Shigeru Kondo
Proofs of irrationality
Proof by infinite descent One proof of the number's irrationality is the following proof by infinite descent. It is also a proof of a negation by refutation: it proves the statement " is not rational" by assuming that it is rational and then deriving a falsehood.
Assume that is a rational number, meaning that there exists a pair of integers whose ratio is exactly . If the two integers have a common factor, it can be eliminated using the Euclidean algorithm. Then can be written as an irreducible fraction such that and are coprime integers (having no common factor) which additionally means that at least one of or must be odd. It follows that and . ( ) ( are integers) Therefore, is even because it is equal to . ( is necessarily even because it is 2 times another whole number.) It follows that must be even (as squares of odd integers are never even). Because is even, there exists an integer that fulfills . Substituting from step 7 for in the second equation of step 4: , which is equivalent to . Because is divisible by two and therefore even, and because , it follows that is also even which means that is even. By steps 5 and 8, and are both even, which contradicts step 3 (that is irreducible).
Since a falsehood has been derived, the assumption (1) that is a rational number must be false. This means that is not a rational number; that is to say, is irrational.
This proof was hinted at by Aristotle, in his Analytica Priora, §I.23.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". It appeared first as a full proof in Euclid's Elements, as proposition 117 of Book X. However, since the early 19th century, historians have agreed that this proof is an interpolation and not attributable to Euclid.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).
Proof using reciprocals Assume by way of contradiction that were rational. Then we may write as an irreducible fraction in lowest terms, with coprime positive integers . Since , it follows that can be expressed as the irreducible fraction . However, since and differ by an integer, it follows that the denominators of their irreducible fraction representations must be the same, i.e. . This gives the desired contradiction.
Proof by unique factorization 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.
Application of the rational root theorem The irrationality of also follows from the rational root theorem, which states that a rational root of a polynomial, if it exists, must be the quotient of a factor of the constant term and a factor of the leading coefficient. In the case of , the only possible rational roots are and . As is not equal to or , it follows that is irrational. This application also invokes the integer root theorem, a stronger version of the rational root theorem for the case when is a monic polynomial with integer coefficients; for such a polynomial, all roots are necessarily integers (which is not, as 2 is not a perfect square) or irrational.
The rational root theorem (or integer root theorem) may be used to show that any square root of any natural number that is not a perfect square is irrational. For other proofs that the square root of any non-square natural number is irrational, see Quadratic irrational number or Infinite descent.
Geometric proofs Tennenbaum's proof
A simple proof is attributed to Stanley Tennenbaum when he was a student in the early 1950s. Assume that , where and are coprime positive integers. Then and are the smallest positive integers for which . Geometrically, this implies that a square with side length will have an area equal to two squares of (lesser) side length . Call these squares A and B. We can draw these squares and compare their areas - the simplest way to do so is to fit the two B squares into the A squares. When we try to do so, we end up with the arrangement in Figure 1., in which the two B squares overlap in the middle and two uncovered areas are present in the top left and bottom right. In order to assert , we would need to show that the area of the overlap is equal to the area of the two missing areas, i.e. = . In other terms, we may refer to the side lengths of the overlap and missing areas as and , respectively, and thus we have . But since we can see from the diagram that and , and we know that and are integers from their definitions in terms of and , this means that we are in violation of the original assumption that and are the smallest positive integers for which .
Hence, even in assuming that and are the smallest positive integers for which , we may prove that there exists a smaller pair of integers and which satisfy the relation. This contradiction within the definition of and implies that they cannot exist, and thus must be irrational.
Apostol's proof
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 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 Tennebaum's proof, viewed geometrically in another way.
Let be a right isosceles triangle with hypotenuse length and legs as shown in Figure 2. By the Pythagorean theorem, . Suppose and are integers. Let be a ratio given in its lowest terms.
Draw the arcs and with centre . Join . It follows that , and and coincide. Therefore, the triangles and are congruent by SAS.
Because is a right angle and is half a right angle, is also a right isosceles triangle. Hence implies . By symmetry, , and is also a right isosceles triangle. It also follows that .