View source for Square root of 2

{{Short description|Unique positive real number which when multiplied by itself gives 2}}
{{use dmy dates |cs1-dates=sy |date=October 2024}}
{{cs1 config |mode=cs1 }}
{{redirect-distinguish|Pythagoras's constant|Pythagoras number}}
{{infobox non-integer number
|image              = Isosceles right triangle with legs length 1.svg
|image_caption      = The square root of 2 is equal to the length of the [[hypotenuse]] of an [[Isosceles triangle|isosceles]] [[right triangle]] with legs of length 1.
|decimal            = {{gaps|1.41421|35623|73095|0488...}}
|continued_fraction = <math>1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}</math>
}}

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 <math>\sqrt{2}</math> or <math>2^{1/2}</math>. 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 [[Unit square|square with sides of one unit of length]]; this follows from the [[Pythagorean theorem]]. It was probably the first number known to be [[irrational number|irrational]].<ref>{{citation |last=Fowler |first=David H. |editor-last1=Gavroglu |editor-first1=Kostas |editor-last2=Christianidis |editor-first2=Jean |editor-last3=Nicolaidis |editor-first3=Efthymios |date=1994 |chapter=The Story of the Discovery of Incommensurability, Revisited |title=Trends in the Historiography of Science |series=Boston Studies in the Philosophy of Science |volume=151 |location=Dortrecht |publisher=Springer |pages=221–236 |doi=10.1007/978-94-017-3596-4 |isbn=978-9048142644}}</ref> The fraction {{sfrac|99|70}} (≈ '''1.4142'''857) is sometimes used as a good [[Diophantine approximation|rational approximation]] with a reasonably small [[denominator]].

Sequence {{OEIS link|A002193}} 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 60 decimal places:<ref>{{cite OEIS |1=A002193 |2= Decimal expansion of square root of 2 |access-date=2020-08-10 }}</ref>

:{{gaps|1.41421|35623|73095|04880|16887|24209|69807|85696|71875|37694|80731|76679}}

==History==
[[File:Ybc7289-bw.jpg|right|thumb|200px|Babylonian clay tablet [[YBC 7289]] with annotations. Besides showing the square root of 2 in [[sexagesimal]] ({{nowrap|1 24 51 10}}), the tablet also gives an example where one side of the square is 30 and the diagonal then is {{nowrap|42 25 35}}. The sexagesimal digit 30 can also stand for {{nowrap|0 30}} = {{sfrac|1|2}}, in which case {{nowrap|0 42 25 35}} is approximately 0.7071065.]]
The [[Babylonia]]n clay tablet [[YBC 7289]] ({{Circa|1800}}–1600 BC) gives an approximation of <math>\sqrt{2}</math> in four [[sexagesimal]] figures, {{nowrap|1 24 51 10}}, which is accurate to about six [[decimal]] digits,<ref>{{cite journal
 | last1 = Fowler | first1 = David | author1-link = David Fowler (mathematician)
 | last2 = Robson | first2 = Eleanor | author2-link = Eleanor Robson
 | doi = 10.1006/hmat.1998.2209
 | issue = 4
 | journal = [[Historia Mathematica]]
 | mr = 1662496
 | pages = 366–378
 | title = Square root approximations in old Babylonian mathematics: YBC 7289 in context
 | volume = 25
 | year = 1998| doi-access = free
 }} See p. 368.<br />[http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html Photograph, illustration, and description of the ''root(2)'' tablet from the Yale Babylonian Collection] {{webarchive|url=https://web.archive.org/web/20120813054036/http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html |date=2012-08-13 }}<br />[http://www.math.ubc.ca/%7Ecass/Euclid/ybc/ybc.html High resolution photographs, descriptions, and analysis of the ''root(2)'' tablet (YBC 7289) from the Yale Babylonian Collection]</ref> and is the closest possible three-place sexagesimal representation of <math>\sqrt{2}</math>, representing a margin of error of only –0.000042%:
:<math>1 + \frac{24}{60} + \frac{51}{60^2} + \frac{10}{60^3} = \frac{305470}{216000} = 1.41421\overline{296}.</math>

Another early approximation is given in [[History of India|ancient Indian]] mathematical texts, the [[Sulba Sutras|Sulbasutras]] ({{Circa|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.''<ref>{{citation |last=Henderson |first=David W. |author-link=David W. Henderson |editor-last=Gorini |editor-first=Catherine A. |date=2000 |chapter=Square roots in the Śulba Sūtras |title=Geometry At Work: Papers in Applied Geometry |series=Mathematical Association of America Notes |volume=53 |location=Washington, D.C. |publisher=[[The Mathematical Association of America]] |pages=39–45 |isbn=978-0883851647 |url=http://www.math.cornell.edu/~dwh/papers/sulba/sulba.html}}</ref> That is,
:<math>1 + \frac{1}{3} + \frac{1}{3 \times 4} - \frac{1}{3 \times4 \times 34} = \frac{577}{408} = 1.41421\overline{56862745098039}.</math>

This approximation, diverging from the actual value of <math>\sqrt{2}</math> by approximately +0.07%, is the seventh in a sequence of increasingly accurate approximations based on the sequence of [[Pell number]]s, which can be derived from the [[simple continued fraction|continued fraction expansion]] of <math>\sqrt{2}</math>. Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation.

[[Pythagoreanism|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 number|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 to any substantial evidence in traditional historian practice.<ref>{{citation |title=The Dangerous Ratio |url=https://nrich.maths.org/2671 |access-date=2023-09-18 |website=nrich.maths.org}}</ref><ref>{{citation |last=Von Fritz |first=Kurt |date=1945 |title=The Discovery of Incommensurability by Hippasus of Metapontum  |journal=Annals of Mathematics |volume=46 |issue=2 |pages=242–264 |doi=10.2307/1969021 |jstor=1969021 |issn=0003-486X}}</ref> The square root of two is occasionally called '''Pythagoras's number'''<ref>{{citation |last1=Conway |first1=John H. |author1-link=John H. Conway |last2=Guy |first2=Richard K. |author2-link = Richard K. Guy |date=1996 |title=The Book of Numbers |title-link=The Book of Numbers (math book) |location=New York |publisher=Copernicus |page=25 |isbn=978-1461240723}}</ref> 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 [[Atrium (architecture)|atria]] by giving them a length equal to a diagonal taken from a square, whose sides are equivalent to the intended atrium's width.<ref>{{citation |title=Architecture and Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s|last1=Williams|first1=Kim|author1-link=Kim Williams (architect)|last2=Ostwald|first2=Michael|publisher=Birkhäuser|year=2015|isbn=9783319001371|pages=204}}</ref>

==Decimal value==
===Computation algorithms===
{{Further|Methods of computing square roots}}
There are many [[algorithm]]s for approximating <math>\sqrt{2}</math> as a ratio of [[integer]]s 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]]<ref>Although the term "Babylonian method" is common in modern usage, there is no direct evidence showing how the Babylonians computed the approximation of <math>\sqrt{2}</math> seen on tablet YBC&nbsp;7289. Fowler and Robson offer informed and detailed conjectures.<br />Fowler and Robson, p. 376. Flannery, p. 32, 158.</ref> for computing square roots, an example of [[Newton's method]] for computing roots of arbitrary functions. It goes as follows:

First, pick a guess, <math>a_0 > 0</math>; 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 [[recursion|recursive]] computation:

:<math>a_{n+1} = \frac12\left(a_n + \dfrac{2}{a_n}\right)=\frac{a_n}{2}+\frac{1}{a_n}. </math>

Each iteration improves the approximation, roughly doubling the number of correct digits. Starting with <math>a_0=1</math>, the subsequent iterations yield:

:<math>\begin{alignat}{3}
a_1 &= \tfrac{3}{2} &&= \mathbf{1}.5, \\
a_2 &= \tfrac{17}{12} &&= \mathbf{1.41}6\ldots, \\
a_3 &= \tfrac{577}{408} &&= \mathbf{1.41421}5\ldots, \\
a_4 &= \tfrac{665857}{470832} &&= \mathbf{1.41421356237}46\ldots, \\
    &\qquad \vdots
\end{alignat}</math>

===Rational approximations===
{{unreferenced section|date=June 2025}}
A simple rational approximation {{sfrac|99|70}} (≈ '''1.4142'''857) is sometimes used. Despite having a denominator of only 70, it differs from the correct value by less than {{sfrac|1|10,000}} (approx. {{val|+0.72e-4}}). 

The next two better rational approximations are {{sfrac|140|99}} (≈ '''1.414'''1414...) with a marginally smaller error (approx. {{val|-0.72e-4}}), and {{sfrac|239|169}} (≈ '''1.4142'''012) with an error of approx {{val|-0.12e-4}}.

The rational approximation of the square root of two derived from four iterations of the Babylonian method after starting with {{math|''a''<sub>0</sub> {{=}} 1}} ({{sfrac|665,857|470,832}}) is too large by about {{val|1.6e-12}}; its square is ≈&thinsp;{{val|2.0000000000045}}.

===Records in computation===
In 1997, the value of <math>\sqrt{2}</math> was calculated to 137,438,953,444 decimal places by [[Yasumasa Kanada]]'s team. In February 2006, the record for the calculation of <math>\sqrt{2}</math> was eclipsed with the use of a home computer. Shigeru Kondo calculated one [[Trillion (short scale)|trillion]] decimal places in 2010.<ref>{{citation |url=http://numbers.computation.free.fr/Constants/Miscellaneous/Records.html |title=Constants and Records of Computation |publisher=Numbers.computation.free.fr |date=2010-08-12 |access-date=2012-09-07 |url-status=live |archive-url=https://web.archive.org/web/20120301190937/http://numbers.computation.free.fr/Constants/Miscellaneous/Records.html |archive-date=2012-03-01 }}</ref> Other [[mathematical constant]]s whose decimal expansions have been calculated to similarly high precision include [[pi|{{pi}}]], [[e (mathematical constant)|{{mvar|e}}]], and the [[golden ratio]].<ref name="y-cruncher">{{citation |url=http://www.numberworld.org/y-cruncher/records.html |title=Records set by y-cruncher |access-date=2022-04-07 |url-status=live |archive-url=https://web.archive.org/web/20220407052022/http://www.numberworld.org/y-cruncher/records.html |archive-date=2022-04-07 }}</ref> Such computations provide empirical evidence of whether these numbers are [[normal number|normal]].

This is a table of recent records in calculating the digits of <math>\sqrt{2}</math>.<ref name="y-cruncher" />
{| class="wikitable sortable"
|-
! data-sort-type="usLongDate" | Date !! Name !! data-sort-type="number" | Number of digits
|-
|style="text-align:right;" | 4 April 2025 || data-sort-value="H" | Teck Por Lim || style="text-align:right;" | {{val|24000000000000}}
|-
|style="text-align:right;" | 26 December 2023 || data-sort-value="H" | Jordan Ranous || style="text-align:right;" | {{val|20000000000000}}
|-
|style="text-align:right;" | 5 January 2022 || data-sort-value="H" | Tizian Hanselmann || style="text-align:right;" | {{val|10000000001000}}
|-
|style="text-align:right;" | 28 June 2016 || data-sort-value="W" | Ron Watkins || style="text-align:right;" | {{val|10000000000000}}
|-
|style="text-align:right;" | 3 April 2016 || data-sort-value="W" | Ron Watkins || style="text-align:right;" | {{val|5000000000000}}
|-
|style="text-align:right;" | 20 January 2016 || data-sort-value="W" | Ron Watkins || style="text-align:right;" | {{val|2000000000100}}
|-
|style="text-align:right;" | 9 February 2012 || data-sort-value="Y" | Alexander Yee || style="text-align:right;" | {{val|2000000000050}}
|-
|style="text-align:right;" | 22 March 2010 || data-sort-value="K" | Shigeru Kondo || style="text-align:right;" | {{val|1000000000000}}
|}

==Proofs of irrationality==
{{refimprove section|date=June 2025}}
===Proof by infinite descent===
One proof of the number's irrationality is the following [[proof by infinite descent]]. It is also a [[Proof by contradiction#Refutation_by_contradiction|proof of a negation by refutation]]: it proves the statement "<math>\sqrt{2}</math> is not rational" by assuming that it is rational and then deriving a falsehood.

# Assume that <math>\sqrt{2}</math> is a rational number, meaning that there exists a pair of integers whose ratio is exactly <math>\sqrt{2}</math>.
# If the two integers have a common [[divisor|factor]], it can be eliminated using the [[Euclidean algorithm]].
# Then <math>\sqrt{2}</math> can be written as an [[irreducible fraction]] <math>\frac{a}{b}</math> such that {{math|''a''}} and {{math|''b''}}  are [[coprime integers]] (having no common factor) which additionally means that at least one of {{math|''a''}} or {{math|''b''}} must be [[parity (mathematics)|odd]].
# It follows that <math>\frac{a^2}{b^2}=2</math> and <math>a^2=2b^2</math>. &emsp; (&thinsp;{{math|[[Exponent#Identities and properties|({{sfrac|''a''|''b''}}){{sup|''n''}} {{=}} {{sfrac|''a''{{sup|''n''}}|''b''{{sup|''n''}}}}]]}}&thinsp;) &emsp; ( {{math|''a''{{sup|2}} and ''b''{{sup|2}}}} are integers)
# Therefore, {{math|''a''{{sup|2}}}} is [[parity (mathematics)|even]] because it is equal to {{math|2''b''{{sup|2}}}}. ({{math|2''b''{{sup|2}}}} is necessarily even because it is 2 times another whole number.)
# It follows that {{math|''a''}} must be even (as squares of odd integers are never even).
# Because {{math|''a''}} is even, there exists an integer {{math|''k''}} that fulfills <math>a = 2k</math>.
# Substituting {{math|2''k''}} from step 7 for {{math|''a''}} in the second equation of step 4: <math>2b^2 = a^2 = (2k)^2 = 4k^2</math>, which is equivalent to <math>b^2=2k^2</math>.
# Because {{math|2''k''{{sup|2}}}} is divisible by two and therefore even, and because <math>2k^2=b^2</math>, it follows that {{math|''b''{{sup|2}}}} is also even which means that {{math|''b''}} is even.
# By steps 5 and 8, {{math|''a''}} and {{math|''b''}} are both even, which contradicts step 3 (that <math>\frac{a}{b}</math> is irreducible).

Since we have derived a falsehood, the assumption (1) that <math>\sqrt{2}</math> is a rational number must be false. This means that <math>\sqrt{2}</math> is not a rational number; that is to say, <math>\sqrt{2}</math> is irrational.

This proof was hinted at by [[Aristotle]], in his ''[[Prior Analytics|Analytica Priora]]'', §I.23.<ref>All that Aristotle says, while writing about [[Proof by contradiction|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".</ref> It appeared first as a full proof in [[Euclid]]'s ''[[Euclid's Elements|Elements]]'', as proposition 117 of Book X. However, since the early 19th century, historians have agreed that this proof is an [[Interpolation (manuscripts)|interpolation]] and not attributable to Euclid.<ref>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 [[Johan Ludvig Heiberg (historian)|J. L. Heiberg's]] edition (1883–1888).</ref>

===Proof using reciprocals===
Assume by way of contradiction that <math>\sqrt 2</math> were rational. Then we may write <math>\sqrt 2 + 1 = \frac{q}{p}</math> as an irreducible fraction in lowest terms, with coprime positive integers <math>q>p</math>. Since <math>(\sqrt 2-1)(\sqrt 2+1)=2-1^2=1</math>, it follows that <math>\sqrt 2-1</math> can be expressed as the irreducible fraction <math>\frac{p}{q}</math>. However, since <math>\sqrt 2-1</math> and <math>\sqrt 2+1</math> differ by an integer, it follows that the denominators of their irreducible fraction representations must be the same, i.e. <math>q=p</math>. This gives the desired contradiction.

===Proof by unique factorization===
As with the proof by infinite descent, we obtain <math>a^2 = 2b^2</math>. 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 <math>\sqrt{2}</math> also follows from the [[rational root theorem]], which states that a rational [[root of a function|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 <math>p(x) = x^2 - 2</math>, the only possible rational roots are <math>\pm 1</math> and <math>\pm 2</math>. As <math>\sqrt{2}</math> is not equal to <math>\pm 1</math> or <math>\pm 2</math>, it follows that <math>\sqrt{2}</math> is irrational. This application also invokes the integer root theorem, a stronger version of the rational root theorem for the case when <math>p(x)</math> is a [[monic polynomial]] with integer [[coefficient]]s; for such a polynomial, all roots are necessarily integers (which <math>\sqrt{2}</math> 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#Square_root_of_non-square_is_irrational|Quadratic irrational number]] or [[proof by infinite descent#Irrationality of √k if it is not an integer|Infinite descent]].

===Geometric proofs===
==== Tennenbaum's proof ====
[[File:NYSqrt2.svg|thumb|Figure 1. Stanley Tennenbaum's geometric proof of the [[Irrational number|irrationality]] of {{math|√2}}]]
A simple proof is attributed to [[Stanley Tennenbaum]] when he was a student in the early 1950s.<ref>{{citation |last1=Miller |first1=Steven J. |last2=Montague |first2=David |date=April 2012 |title=Picturing Irrationality |jstor=10.4169/math.mag.85.2.110 |magazine=[[Mathematics Magazine]] |volume=85 |issue=2 |pages=110–114 |doi=10.4169/math.mag.85.2.110 }}</ref><ref>{{citation |last=Yanofsky |first=Noson S. |date=May–June 2016 |title=Paradoxes, Contradictions, and the Limits of Science |jstor=44808923 |magazine=[[American Scientist]] |volume=103 |issue=3 |pages=166–173 }}</ref> Assume that <math>\sqrt{2} = a/b</math>, where <math>a</math> and <math>b</math> are coprime positive integers. Then <math>a</math> and <math>b</math> are the smallest positive integers for which <math>a^2 = 2b^2</math>. Geometrically, this implies that a square with side length <math>a</math> will have an area equal to two squares of (lesser) side length <math>b</math>. 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 <math>a^2 = 2b^2</math>, we would need to show that the area of the overlap is equal to the area of the two missing areas, i.e. <math>(2b-a)^2</math> = <math>2(a-b)^2</math>. In other terms, we may refer to the side lengths of the overlap and missing areas as <math>p = 2b-a</math> and <math>q = a-b</math>, respectively, and thus we have <math>p^2 = 2q^2</math>. But since we can see from the diagram that <math>p < a</math> and <math>q < b</math>, and we know that <math>p</math> and <math>q</math> are integers from their definitions in terms of <math>a</math> and <math>b</math>, this means that we are in violation of the original assumption that <math>a</math> and <math>b</math> are the smallest positive integers for which <math>a^2 = 2b^2</math>. 

Hence, even in assuming that <math>a</math> and <math>b</math> are the smallest positive integers for which <math>a^2 = 2b^2</math>, we may prove that there exists a smaller pair of integers <math>p</math> and <math>q</math> which satisfy the relation. This contradiction within the definition of <math>a</math> and <math>b</math> implies that they cannot exist, and thus <math>\sqrt{2}</math> must be irrational.

==== Apostol's proof ====
[[File:Irrationality of sqrt2.svg|left|thumb|Figure 2. Tom Apostol's geometric proof of the irrationality of {{math|√2}}]]
[[Tom M. Apostol]] made another geometric ''[[reductio ad absurdum]]'' argument showing that <math>\sqrt{2}</math> is irrational.<ref>{{citation |last=Apostol |first=Tom M. |author-link=Tom M. Apostol |date=2000 |title=Irrationality of The Square Root of Two – A Geometric Proof |jstor=2695741 |journal=[[The American Mathematical Monthly]] |volume=107 |number=9 |pages=841–842 |doi=10.2307/2695741 }}</ref> 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 Tennebaum's proof, viewed geometrically in another way.

Let {{math|△&hairsp;''ABC''}} be a right isosceles triangle with hypotenuse length {{math|''m''}} and legs {{math|''n''}} as shown in Figure 2. By the [[Pythagorean theorem]], <math>\frac{m}{n}=\sqrt{2}</math>. Suppose {{math|''m''}} and {{math|''n''}} are integers. Let {{math|''m'':''n''}} be a [[ratio]] given in its [[lowest terms]].

Draw the arcs {{math|''BD''}} and {{math|''CE''}} with centre {{math|''A''}}. Join {{math|''DE''}}. It follows that {{math|''AB'' {{=}} ''AD''}}, {{math|''AC'' {{=}} ''AE''}} and {{math|∠''BAC''}} and {{math|∠''DAE''}} coincide. Therefore, the [[triangle]]s {{math|''ABC''}} and {{math|''ADE''}} are [[Congruence (geometry)|congruent]] by [[Side-angle-side|SAS]].

Because {{math|∠''EBF''}} is a right angle and {{math|∠''BEF''}} is half a right angle, {{math|△&hairsp;''BEF''}} is also a right isosceles triangle. Hence {{math|''BE'' {{=}} ''m'' − ''n''}} implies {{math|''BF'' {{=}} ''m'' − ''n''}}. By symmetry, {{math|''DF'' {{=}} ''m'' − ''n''}}, and {{math|△&hairsp;''FDC''}} is also a right isosceles triangle. It also follows that {{math|''FC'' {{=}} ''n'' − (''m'' − ''n'') {{=}} 2''n'' − ''m''}}.

Hence, there is an even smaller right isosceles triangle, with hypotenuse length {{math|2''n'' − ''m''}} and legs {{math|''m'' − ''n''}}. These values are integers even smaller than {{math|''m''}} and {{math|''n''}} and in the same ratio, contradicting the hypothesis that {{math|''m'':''n''}} is in lowest terms. Therefore, {{math|''m''}} and {{math|''n''}} cannot be both integers; hence, <math>\sqrt{2}</math> is irrational.
{{clear}}

===Constructive proof===
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  {{math|''a''}} and {{math|''b''}} be positive integers such that {{math|1<{{sfrac|''a''|''b''}}< 3/2}} (as {{math|1<2< 9/4}} satisfies these bounds). Now {{math|2''b''{{sup|2}} }} and {{math|''a''{{sup|2}} }} cannot be equal, since the first has an odd number of factors 2 whereas the second has an even number of factors 2. Thus {{math|{{abs|2''b''{{sup|2}} − ''a''{{sup|2}}}} ≥ 1}}. Multiplying the absolute difference {{math|{{abs|√2 − {{sfrac|''a''|''b''}}}}}} by {{math| ''b''{{sup|2}}(√2 + {{sfrac|''a''|''b''}})}} in the numerator and denominator, we get<ref>See {{citation
 | last1 = Katz | first1 = Karin Usadi
 | last2 = Katz | first2 = Mikhail G.
 | author2-link = Mikhail Katz
 | arxiv = 1110.5456
 | issue = 2
 | journal = [[Intellectica]]
 | pages = 223–302 (see esp. Section 2.3, footnote 15)
 | title = Meaning in Classical Mathematics: Is it at Odds with Intuitionism?
 | volume = 56
 | year = 2011| bibcode = 2011arXiv1110.5456U}}</ref>
:<math>\left|\sqrt2 - \frac{a}{b}\right| = \frac{|2b^2-a^2|}{b^2\!\left(\sqrt{2}+\frac{a}{b}\right)} \ge \frac{1}{b^2\!\left(\sqrt2 + \frac{a}{b}\right)} \ge \frac{1}{3b^2},</math>

the latter [[inequality (mathematics)|inequality]] being true because it is assumed that {{math|1<{{sfrac|''a''|''b''}}< 3/2}}, giving {{math|{{sfrac|''a''|''b''}} + √2 ≤ 3 }} (otherwise the quantitative apartness can be trivially established). This gives a lower bound of {{math|{{sfrac|1|3''b''{{sup|2}}}}}} for the difference {{math|{{abs|√2 − {{sfrac|''a''|''b''}}}}}}, yielding a direct proof of irrationality in its constructively stronger form, not relying on the [[law of excluded middle]].<ref>{{citation |last=Bishop |first=Errett |author-link=Errett Bishop |editor-last=Rosenblatt |editor-first=Murray |editor-link=Murray Rosenblatt |date=1985 |chapter=Schizophrenia in Contemporary Mathematics. |title=Errett Bishop: Reflections on Him and His Research |series=Contemporary Mathematics |volume=39 |location=Providence, RI |publisher=[[American Mathematical Society]] |pages=1–32 |doi=10.1090/conm/039/788163 |isbn=0821850407 |issn=0271-4132}}</ref> This proof constructively exhibits an explicit discrepancy between <math>\sqrt{2}</math> and any rational.

===Proof by Pythagorean triples===
This proof uses the following property of primitive [[Pythagorean triple]]s:

: If {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} are coprime positive integers such that {{math|1=''a''<sup>2</sup> + ''b''<sup>2</sup> = ''c''<sup>2</sup>}}, then {{math|''c''}} is never even.<ref name=Sierpinski>{{citation |last=Sierpiński |first=Wacław |author-link=Waclaw Sierpinski |translator-last=Sharma |translator-first=Ambikeshwa |date=2003 |title=Pythagorean Triangles |location=Mineola, NY |publisher=Dover |pages=4–6 |isbn=978-0486432786}}</ref>

This lemma can be used to show that two identical perfect squares can never be added to produce another perfect square.

Suppose the contrary that <math>\sqrt2</math> is rational. Therefore,

:<math>\sqrt2 = {a \over b}</math>
:where <math>a,b \in \mathbb{Z}</math> and <math>\gcd(a,b) = 1</math>
:Squaring both sides,
:<math>2 = {a^2 \over b^2}</math>
:<math>2b^2 = a^2</math>
:<math>b^2+b^2 = a^2</math>

Here, {{math|(''b'', ''b'', ''a'')}} is a primitive Pythagorean triple, and from the lemma {{math|''a''}} is never even. However, this contradicts the equation {{math|1=2''b''<sup>2</sup> = ''a''<sup>2</sup>}} which implies that {{math|''a''}} must be even.

==Multiplicative inverse==
The [[multiplicative inverse]] (reciprocal) of the square root of two is a widely used [[Mathematical constant|constant]], with the decimal value:<ref>{{cite OEIS |1=A010503 |2=Decimal expansion of 1/sqrt(2) |access-date=3 November 2024}}</ref>

:{{gaps|0.70710|67811|86547|52440|08443|62104|84903|92848|35937|68847|...}}

It is often encountered in [[geometry]] and [[trigonometry]] because the [[unit vector]], which makes a 45° [[angle]] with the axes in a [[Plane (mathematics)|plane]], has the coordinates
:<math>\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\!.</math>
Each coordinate satisfies
:<math>\frac{\sqrt{2}}{2} = \sqrt{\tfrac{1}{2}} = \frac{1}{\sqrt{2}} = \sin 45^\circ = \cos 45^\circ.</math>

==Properties==
[[File:Circular and hyperbolic angle.svg|350px|thumb|[[Angle]] size and sector [[area]] are the same when the conic radius is {{math|√2}}. This diagram illustrates the circular and hyperbolic functions based on sector areas {{math|''u''}}.]]

One interesting property of <math>\sqrt{2}</math> is
:<math>\!\ {1 \over {\sqrt{2} - 1}} = \sqrt{2} + 1</math>
since
:<math>\left(\sqrt{2}+1\right)\!\left(\sqrt{2}-1\right) = 2-1 = 1.</math>
This is related to the property of [[silver ratio]]s.

<math>\sqrt{2}</math> can also be expressed in terms of copies of the [[imaginary unit]] {{math|''i''}} using only the [[square root]] and [[arithmetic operations]], if the square root symbol is interpreted suitably for the [[complex number]]s {{math|''i''}} and {{math|−''i''}}:
:<math>\frac{\sqrt{i}+i \sqrt{i}}{i}\text{ and }\frac{\sqrt{-i}-i \sqrt{-i}}{-i}</math>
<math>\sqrt{2}</math> is also the only real number other than 1 whose infinite [[Tetration|tetrate]] (i.e., infinite exponential tower) is equal to its square. In other words: if for {{math|''c'' > 1}}, {{math|''x''<sub>1</sub> {{=}} ''c''}} and {{math|''x''<sub>''n''+1</sub> {{=}} ''c''<sup>''x''<sub>''n''</sub></sup>}} for {{math|''n'' > 1}}, the [[limit of a sequence|limit]] of {{math|''x''<sub>''n''</sub>}} as {{math|''n'' → ∞}} will be called (if this limit exists) {{math|''f''(''c'')}}. Then <math>\sqrt{2}</math> is the only number {{math|''c'' > 1}} for which {{math|''f''(''c'') {{=}} ''c''<sup>2</sup>}}. Or symbolically:

:<math>\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{~\cdot^{~\cdot^{~\cdot}}}}} = 2.</math>

<math>\sqrt{2}</math> appears in [[Viète's formula]] for {{pi}},

:<math>
\frac2\pi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12\sqrt{\frac12 + \frac12\sqrt\frac12}} \cdots,
</math>

which is related to the formula<ref>{{Citation |title=What is mathematics? An Elementary Approach to Ideas and Methods |first1=Richard |last1=Courant |first2=Herbert |last2=Robbins |location=London |publisher=Oxford University Press |year=1941 |page=124 }}</ref>
:<math>\pi = \lim_{m\to\infty} 2^{m} \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}}}_{m\text{ square roots}}\,.</math>

Similar in appearance but with a finite number of terms, <math>\sqrt{2}</math> appears in various [[exact trigonometric values|trigonometric constants]]:<ref>Julian D. A. Wiseman [http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html Sin and cos in surds] {{webarchive|url=https://web.archive.org/web/20090506080636/http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html |date=2009-05-06 }}</ref>
:<math>\begin{align}
\sin\frac{\pi}{32} &= \tfrac12\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}} &\quad
\sin\frac{3\pi}{16} &= \tfrac12\sqrt{2-\sqrt{2-\sqrt{2}}} &\quad
\sin\frac{11\pi}{32} &= \tfrac12\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2}}}} \\[6pt]
\sin\frac{\pi}{16} &= \tfrac12\sqrt{2-\sqrt{2+\sqrt{2}}} &\quad
\sin\frac{7\pi}{32} &= \tfrac12\sqrt{2-\sqrt{2-\sqrt{2+\sqrt{2}}}} &\quad
\sin\frac{3\pi}{8} &= \tfrac12\sqrt{2+\sqrt{2}} \\[6pt]
\sin\frac{3\pi}{32} &= \tfrac12\sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2}}}} &\quad
\sin\frac{\pi}{4} &= \tfrac12\sqrt{2} &\quad
\sin\frac{13\pi}{32} &= \tfrac12\sqrt{2+\sqrt{2+\sqrt{2-\sqrt{2}}}} \\[6pt]
\sin\frac{\pi}{8} &= \tfrac12\sqrt{2-\sqrt{2}} &\quad
\sin\frac{9\pi}{32} &= \tfrac12\sqrt{2+\sqrt{2-\sqrt{2+\sqrt{2}}}} &\quad
\sin\frac{7\pi}{16} &= \tfrac12\sqrt{2+\sqrt{2+\sqrt{2}}} \\[6pt]
\sin\frac{5\pi}{32} &= \tfrac12\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2}}}} &\quad
\sin\frac{5\pi}{16} &= \tfrac12\sqrt{2+\sqrt{2-\sqrt{2}}} &\quad
\sin\frac{15\pi}{32} &= \tfrac12\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}
\end{align}</math>

It is not known whether <math>\sqrt{2}</math> 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 [[base two]].<ref>{{citation |last1=Good |first1=I. J. |author1-link=I. J. Good |last2=Gover |first2=T. N. |date=1967 |title=The generalized serial test and the binary expansion of <math>\sqrt{2}</math> |journal=Journal of the Royal Statistical Society, Series A |jstor=2344040 |volume=130 |issue=1 |pages=102–107 |doi=10.2307/2344040 }}</ref>

==Representations==
===Series and product===
The identity {{math|cos&thinsp;{{sfrac|π|4}} {{=}} sin&thinsp;{{sfrac|π|4}} {{=}} {{sfrac|1|√2}}}}, along with the infinite product representations for the [[Trigonometric_functions#Infinite_product_expansion|sine and cosine]], leads to products such as
:<math>\frac{1}{\sqrt 2} = \prod_{k=0}^\infty \left(1-\frac{1}{(4k+2)^2}\right) =
\left(1-\frac{1}{4}\right)\!\left(1-\frac{1}{36}\right)\!\left(1-\frac{1}{100}\right) \cdots</math>
and
:<math>\sqrt{2} = \prod_{k=0}^\infty\frac{(4k+2)^2}{(4k+1)(4k+3)} =
\left(\frac{2 \cdot 2}{1 \cdot 3}\right)\!\left(\frac{6 \cdot 6}{5 \cdot 7}\right)\!\left(\frac{10 \cdot 10}{9 \cdot 11}\right)\!\left(\frac{14 \cdot 14}{13 \cdot 15}\right) \cdots</math>
or equivalently,
:<math>\sqrt{2} = \prod_{k=0}^\infty\left(1+\frac{1}{4k+1}\right)\left(1-\frac{1}{4k+3}\right) =
\left(1+\frac{1}{1}\right)\!\left(1-\frac{1}{3}\right)\!\left(1+\frac{1}{5}\right)\!\left(1-\frac{1}{7}\right) \cdots.</math>

The number can also be expressed by taking the [[Taylor series]] of a [[trigonometric function]]. For example, the series for {{math|cos&thinsp;{{sfrac|π|4}}}} gives
:<math>\frac{1}{\sqrt{2}} = \sum_{k=0}^\infty \frac{(-1)^k \left(\frac{\pi}{4}\right)^{2k}}{(2k)!}.</math>
The Taylor series of <math>\sqrt{1 + x}</math> with {{math|''x'' {{=}} 1}} and using the [[double factorial]] {{math|''n''!!}} gives

:<math>\sqrt{2} = \sum_{k=0}^\infty (-1)^{k+1} \frac{(2k-3)!!}{(2k)!!} =
1 + \frac{1}{2} - \frac{1}{2\cdot4} + \frac{1\cdot3}{2\cdot4\cdot6} - \frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8} + \cdots = 1 + \frac{1}{2} - \frac{1}{8} + \frac{1}{16} - \frac{5}{128} + \frac{7}{256} + \cdots.</math>

The [[convergent series|convergence]] of this series can be accelerated with an [[Euler transform]], producing

:<math>\sqrt{2} = \sum_{k=0}^\infty \frac{(2k+1)!}{2^{3k+1}(k!)^2 } = \frac{1}{2} +\frac{3}{8} + \frac{15}{64} + \frac{35}{256} + \frac{315}{4096} + \frac{693}{16384} + \cdots.</math>
It is not known whether <math>\sqrt{2}</math> can be represented with a [[BBP-type formula]]. BBP-type formulas are known for {{math|π√2}} and <math>\sqrt{2} \ln(1+\sqrt{2})</math>, however.<ref>{{citation |url=http://crd.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf |title=A Compendium of BBP-Type Formulas for Mathematical Constants |last1=Bailey |first1=David H. |date=13 February 2011 |access-date=2010-04-30 |url-status=live |archive-url=https://web.archive.org/web/20110610050911/http://crd.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf |archive-date=2011-06-10 }}</ref>

The number can be represented by an infinite series of [[Egyptian fraction]]s, with denominators defined by 2<sup>''n''</sup>th terms of a [[Fibonacci sequence|Fibonacci]]-like [[recurrence relation]] ''a''(''n'') = 34''a''(''n''−1) − ''a''(''n''−2), ''a''(0) = 0, ''a''(1) = 6:<ref>{{Cite OEIS |1=A082405|2=a(n) = 34*a(n-1) - a(n-2); a(0)=0, a(1)=6 |access-date=2016-09-05}}</ref>

:<math>\sqrt{2}=\frac{3}{2}-\frac{1}{2}\sum_{n=0}^\infty \frac{1}{a(2^n)}=\frac{3}{2}-\frac{1}{2}\left(\frac{1}{6}+\frac{1}{204}+\frac{1}{235416}+\dots \right). </math>

===Continued fraction===
[[File:Dedekind cut- square root of two.png|thumb|335px|The square root of 2 and approximations by [[Continued fraction#Infinite continued fractions and convergents|convergents of continued fractions]]]]
The square root of two has the following [[continued fraction]] representation:
:<math>\sqrt2 = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac1\ddots}}}. </math>

The [[convergent (continued fraction)|convergents]] {{math|{{sfrac|''p''|''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 number]]s (i.e., {{Math|1=''p''<sup>2</sup> − 2''q''<sup>2</sup> = ±1}}). The first convergents are: {{math|{{sfrac|1|1}}, {{sfrac|3|2}}, {{sfrac|7|5}}, {{sfrac|17|12}}, {{sfrac|41|29}}, {{sfrac|99|70}}, {{sfrac|239|169}}, {{sfrac|577|408}}}} and the convergent following {{math|{{sfrac|''p''|''q''}}}} is {{math|{{sfrac|''p'' + 2''q''|''p'' + ''q''}}}}. The convergent {{math|{{sfrac|''p''|''q''}}}} differs from <math>\sqrt{2}</math> by almost exactly <math>\frac{1}{2 \sqrt{2}q^2}</math>, which follows from:
:<math>\left|\sqrt2 - \frac{p}{q}\right| = \frac{|2q^2-p^2|}{q^2\!\left(\sqrt{2}+\frac{p}{q}\right)} = \frac{1}{q^2\!\left(\sqrt2 + \frac{p}{q}\right)} \thickapprox \frac{1}{2\sqrt{2}q^2}</math>

===Nested square===
The following nested square expressions converge to {{nobr|<math display=inline>\sqrt2</math>:}}
:<math>\begin{align}
\sqrt{2}
&= \tfrac32 - 2 \left( \tfrac14 - \left( \tfrac14 - \bigl( \tfrac14 - \cdots \bigr)^2 \right)^2 \right)^2 \\[10mu]
&= \tfrac32 - 4 \left( \tfrac18 + \left( \tfrac18 + \bigl( \tfrac18 + \cdots \bigr)^2 \right)^2 \right)^2.
\end{align}</math>

==Applications==
===Paper size===
[[File:A size illustration2.svg|thumb|200px|The A series of paper sizes]]

In 1786, German physics professor [[Georg Christoph Lichtenberg]]<ref name=":0" /> found that any sheet of paper whose long edge is <math>\sqrt{2}</math> 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 size]]s at the beginning of the 20th century, they used Lichtenberg's ratio to create the [[ISO 216#A series|"A" series]] of paper sizes.<ref name=":0">{{citation |title=The Book: A Cover-to-Cover Exploration of the Most Powerful Object of Our Time|last=Houston|first=Keith|publisher=W. W. Norton & Company|year=2016|isbn=978-0393244809|pages=324}}</ref> Today, the (approximate) [[aspect ratio]] of paper sizes under [[ISO 216]] (A4, A0, etc.) is 1:<math>\sqrt{2}</math>.

Proof:

Let <math>S = </math> shorter length and <math>L = </math> longer length of the sides of a sheet of paper, with
:<math>R = \frac{L}{S} = \sqrt{2}</math> as required by ISO 216.
Let <math>R' = \frac{L'}{S'}</math> be the analogous ratio of the halved sheet, then
:<math>R' = \frac{S}{L/2} = \frac{2S}{L} = \frac{2}{(L/S)} = \frac{2}{\sqrt{2}} = \sqrt{2} = R. </math>

===Physical sciences===
There are some interesting properties involving the square root of 2 in the [[physical sciences]]:
* The square root of two is the [[interval (music)#Frequency ratios|frequency ratio]] of a [[tritone]] interval in twelve-tone [[equal temperament]] music.
* The square root of two forms the relationship of [[F-number|f-stops]] in photographic lenses, which in turn means that the ratio of ''areas'' between two successive [[aperture]]s is 2.
* The celestial latitude (declination) of the Sun during a planet's astronomical [[cross-quarter day]] points equals the tilt of the planet's axis divided by <math>\sqrt{2}</math>.
{{distances_between_double_cube_corners.svg}}
* In the brain there are lattice cells, discovered in 2005 by a group led by May-Britt and Edvard Moser.  "The grid cells were found in the cortical area located right next to the hippocampus [...] At one end of this cortical area the mesh size is small and at the other it is very large. However, the increase in mesh size is not left to chance, but increases by the squareroot of two from one area to the next."<ref name="Hjernen er stjernen" >{{citation |title=The Book: Hjernen er sternen|last=Nordengen|first=Kaja|publisher=2016 Kagge Forlag AS|year=2016|isbn=978-82-489-2018-2|page=81}}</ref>

==See also==
* [[List of mathematical constants]]
* [[Square root of 3]]
* [[Square root of 5]]
* [[Gelfond–Schneider constant]], {{math|2<sup>√2</sup>}}
* [[Silver ratio]], {{math|1 + √2}}

{{clear}}

==Notes==
{{Reflist|25em}}

==References==
* {{Citation
  | author = Aristotle | author-link = Aristotle
  | year = 1938 | orig-year = c. 350 BC
  | title = Categories; On Interpretation; Prior Analytics. Greek text with translation
  | translator1 = H. P. Cooke
  | translator2 = Hugh Tredennick
  | series = Loeb Classical Library | volume = 325
  | place = Cambridge, MA | publisher = Harvard University Press
  | isbn = 9780674993594 
  | at = ''Prior Analytics'' § I.23
  | url = https://archive.org/details/categoriesoninte0000aris/page/320/mode/2up?q=%22proves+that+the+diagonal%22 | url-access = limited }}
* {{citation
  | last = Flannery | first = David
  | year = 2006
  | title = The Square Root of 2: A Dialogue Concerning a Number and a Sequence
  | location = New York
  | publisher = Copernicus Books
  | isbn = 978-0387202204
  }}
* {{citation
  | last1 = Fowler | first1 = David | author1-link = David Fowler (mathematician)
  | last2 = Robson | first2 = Eleanor | author2-link = Eleanor Robson
  | year = 1998
  | title = Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context
  | journal = [[Historia Mathematica]]
  | volume = 25 | issue = 4 | pages = 366–378
  | doi = 10.1006/hmat.1998.2209 | doi-access = free
  }}

==External links==
* {{Citation
 | last1 = Gourdon | first1 = X.
 | last2 = Sebah | first2 = P.
 | contribution = Pythagoras' Constant: <math>\sqrt{2}</math>
 | title = Numbers, Constants and Computation
 | url = http://numbers.computation.free.fr/Constants/Sqrt2/sqrt2.html
 | year = 2001}}.
* [https://www.gutenberg.org/ebooks/129 The Square Root of Two to 5 million digits] by Jerry Bonnell and [[Robert J. Nemiroff]]. May, 1994.
* [http://www.cut-the-knot.org/proofs/sq_root.shtml Square root of 2 is irrational], a collection of proofs
* {{citation |last=Haran |first=Brady |author-link=Brady Haran |others=featuring Grime, James; Bowley, Roger |title=Root 2 |url=https://www.numberphile.com/videos/root-2 |series=Numberphile |type=video |date=27 January 2012 }}
* [http://pisearch.org/sqrt2 {{tmath|\sqrt2}} Search Engine] 2 billion searchable digits of √2, π, and {{mvar|e}}
{{Algebraic numbers}}
{{Irrational number}}
{{Authority control}}

{{DEFAULTSORT:Square Root Of Two}}
[[Category:Quadratic irrational numbers]]
[[Category:Mathematical constants]]
[[Category:Pythagorean theorem]]
[[Category:Articles containing proofs]]