The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself or squared, equals the number 2. It may be written in mathematics as 2 {\displaystyle {\sqrt {2}}} or 2 1 / 2 {\displaystyle 2^{1/2}} . 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.[1] The fraction 99/70 (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator.
Sequence 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 65 decimal places:[2]
The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of 2 {\displaystyle {\sqrt {2}}} in four sexagesimal figures, 1 24 51 10, which is accurate to about six decimal digits,[3] and is the closest possible three-place sexagesimal representation of 2 {\displaystyle {\sqrt {2}}} , representing a margin of error of only –0.000042%:
Another early approximation is given in ancient Indian mathematical texts, the Sulbasutras (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.[4] That is,
This approximation, diverging from the actual value of 2 {\displaystyle {\sqrt {2}}} by approximately +0.07%, 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 2 {\displaystyle {\sqrt {2}}} . 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 to any substantial evidence in traditional historian practice.[5][6] The square root of two is occasionally called Pythagoras's number or Pythagoras's constant.[7]
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.[8]
There are many algorithms for approximating 2 {\displaystyle {\sqrt {2}}} 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[9] for computing square roots, an example of Newton's method for computing roots of arbitrary functions. It goes as follows:
First, pick a guess, a 0 > 0 {\displaystyle 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:
Each iteration improves the approximation, roughly doubling the number of correct digits. Starting with a 0 = 1 {\displaystyle a_{0}=1} , the subsequent iterations yield:
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.
In 1997, the value of 2 {\displaystyle {\sqrt {2}}} was calculated to 137,438,953,444 decimal places by Yasumasa Kanada's team. In February 2006, the record for the calculation of 2 {\displaystyle {\sqrt {2}}} was eclipsed with the use of a home computer. Shigeru Kondo calculated one trillion decimal places in 2010.[10] Other mathematical constants whose decimal expansions have been calculated to similarly high precision include π, e, and the golden ratio.[11] Such computations aim to check empirically whether such numbers are normal.
This is a table of recent records in calculating the digits of 2 {\displaystyle {\sqrt {2}}} .[11]
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 " 2 {\displaystyle {\sqrt {2}}} is not rational" by assuming that it is rational and then deriving a falsehood.
Since we have derived a falsehood, the assumption (1) that 2 {\displaystyle {\sqrt {2}}} is a rational number must be false. This means that 2 {\displaystyle {\sqrt {2}}} is not a rational number; that is to say, 2 {\displaystyle {\sqrt {2}}} is irrational.
This proof was hinted at by Aristotle, in his Analytica Priora, §I.23.[12] 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.[13]
As with the proof by infinite descent, we obtain a 2 = 2 b 2 {\displaystyle a^{2}=2b^{2}} . 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.
The irrationality of 2 {\displaystyle {\sqrt {2}}} 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 p ( x ) = x 2 − − --> 2 {\displaystyle p(x)=x^{2}-2} , the only possible rational roots are ± ± --> 1 {\displaystyle \pm 1} and ± ± --> 2 {\displaystyle \pm 2} . As 2 {\displaystyle {\sqrt {2}}} is not equal to ± ± --> 1 {\displaystyle \pm 1} or ± ± --> 2 {\displaystyle \pm 2} , it follows that 2 {\displaystyle {\sqrt {2}}} is irrational. This application also invokes the integer root theorem, a stronger version of the rational root theorem for the case when p ( x ) {\displaystyle p(x)} is a monic polynomial with integer coefficients; for such a polynomial, all roots are necessarily integers (which 2 {\displaystyle {\sqrt {2}}} 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.
A simple proof is attributed to Stanley Tennenbaum when he was a student in the early 1950s.[14][15] 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 ( ( 2 b − − --> a ) 2 {\displaystyle (2b-a)^{2}} ) must equal the sum of the two uncovered squares ( 2 ( a − − --> b ) 2 {\displaystyle 2(a-b)^{2}} ). 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 2 {\displaystyle {\sqrt {2}}} is irrational.[16] 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, m n = 2 {\displaystyle {\frac {m}{n}}={\sqrt {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 ∠BAC and ∠DAE coincide. Therefore, the triangles ABC and ADE are congruent by SAS.
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, 2 {\displaystyle {\sqrt {2}}} 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[17]
the latter 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 2 {\displaystyle {\sqrt {2}}} and any rational.
This proof uses the following property of primitive Pythagorean triples:
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 2 {\displaystyle {\sqrt {2}}} is rational. Therefore,
Here, (b, b, a) is a primitive Pythagorean triple, and from the lemma a is never even. However, this contradicts the equation 2b2 = a2 which implies that a must be even.
The multiplicative inverse (reciprocal) of the square root of two (i.e., the square root of 1/2) is a widely used constant.
One-half of 2 {\displaystyle {\sqrt {2}}} , also the reciprocal of 2 {\displaystyle {\sqrt {2}}} , 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 2 {\displaystyle {\sqrt {2}}} is
since
This is related to the property of silver ratios.
2 {\displaystyle {\sqrt {2}}} can also be expressed in terms of copies of the imaginary unit i using only the square root and arithmetic operations, if the square root symbol is interpreted suitably for the complex numbers i and −i:
2 {\displaystyle {\sqrt {2}}} 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 2 {\displaystyle {\sqrt {2}}} is the only number c > 1 for which f(c) = c2. Or symbolically:
2 {\displaystyle {\sqrt {2}}} appears in Viète's formula for π,
which is related to the formula[19]
Similar in appearance but with a finite number of terms, 2 {\displaystyle {\sqrt {2}}} appears in various trigonometric constants:[20]
It is not known whether 2 {\displaystyle {\sqrt {2}}} 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.[21]
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 {\displaystyle {\sqrt {2}}} can be represented with a BBP-type formula. BBP-type formulas are known for π√2 and √2 ln(1+√2), however.[22]
The number can be represented by an infinite series of Egyptian fractions, with denominators defined by 2n th terms of a Fibonacci-like recurrence relation a(n) = 34a(n−1) − a(n−2), a(0) = 0, a(1) = 6.[23]
The square root of two has the following continued fraction representation:
The convergents 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 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 2 {\displaystyle {\sqrt {2}}} by almost exactly 1/2√2q2, which follows from:
The following nested square expressions converge to 2 {\textstyle {\sqrt {2}}} :
In 1786, German physics professor Georg Christoph Lichtenberg[24] found that any sheet of paper whose long edge is 2 {\displaystyle {\sqrt {2}}} 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.[24] Today, the (approximate) aspect ratio of paper sizes under ISO 216 (A4, A0, etc.) is 1: 2 {\displaystyle {\sqrt {2}}} .
Proof: Let S = {\displaystyle S=} shorter length and L = {\displaystyle L=} longer length of the sides of a sheet of paper, with
Let R ′ = L ′ S ′ {\displaystyle R'={\frac {L'}{S'}}} be the analogous ratio of the halved sheet, then
There are some interesting properties involving the square root of 2 in the physical sciences:
For the 2018 video game, see Paladins (video game). 1988 video game This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Paladin video game – news · newspapers · books · scholar · JSTOR (December 2020) (Learn how and when to remove this template message) 1988 video gamePaladinPublisher(s)Omnitrend SoftwarePlatf…
King Mongkut's University of Technology Thonburiมหาวิทยาลัยเทคโนโลยีพระจอมเกล้าธนบุรีFormer namesThonburi Technical CollegeKing Mongkut's Institute of Technology, Thonburi CampusKing Mongkut's Institute of Technology ThonburiMottoDanto Seṭṭho ManussesuMotto in EnglishThe trained man winsTypePublic (National) research universityEstablishedFebruary 4, 1960 (1960-02-04)PresidentAssoc. Prof. Dr. Suvit T…
Peruvian politician This biography of a living person needs additional citations for verification. Please help by adding reliable sources. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately from the article and its talk page, especially if potentially libelous.Find sources: José Luis Elías – news · newspapers · books · scholar · JSTOR (May 2021) (Learn how and when to remove this template message) I…
والتر داندي معلومات شخصية الميلاد 6 أبريل 1886[1] سيداليا الوفاة 19 أبريل 1946 (60 سنة) [1] بالتيمور سبب الوفاة نوبة قلبية مواطنة الولايات المتحدة عضو في الأكاديمية الألمانية للعلوم ليوبولدينا الحياة العملية المدرسة الأم مدرسة طب جامعة جونز هوبكنز…
Catherina Davinio in 1990 Caterina Davinio, geboren als Maria Caterina Invidia (Foggia, 25 november 1957), is een Italiaanse dichteres, schrijfster, fotografe, videokunstenares en computerkunstenares. Davinio werd geboren in Foggia, maar groeide op in Rome. Hier studeerde ze ook literatuur en kunst aan de Università La Sapienza. Tentoonstellingen Biennale de Lyon (twee edities) Biënnale van Venetië (zeven edities sinds 1997, waar hij werkte ook als een curator) Athens Biennial Poliphonyx (Bar…
Der Titel dieses Artikels ist mehrdeutig. Für den gleichnamigen Ort in Galizien, siehe Wynnyky. Das Wasserschloss in Seebach, Erkennungszeichen der Gemeinde Weinbergen Weinbergen war eine Gemeinde im Unstrut-Hainich-Kreis in Thüringen. Sie entstand am 30. Juni 1994 im Zuge einer Gebietsreform durch den Zusammenschluss der Gemeinden Bollstedt, Grabe, Höngeda und Seebach[1] und existierte bis zum 31. Dezember 2018. Am 1. Januar 2019 wurde die Gemeinde Weinbergen aufgelös…
Sebuah perhitungan Indeks Pembangunan Manusia (IPM) yang menggunakan metode baru dilaksanakan oleh Badan Pusat Statistik (BPS) dari tahun 2010 hingga sekarang. Berikut ini akan disajikan penjelasan, dimensi dasar, manfaat, dan metodologi perhitungan IPM, serta daftar kabupaten dan kota Jawa Barat menurut IPM tahun 2014. Penjelasan Indeks Pembangunan Manusia (IPM)/Human Development Index (HDI) adalah pengukuran perbandingan dari harapan hidup, melek huruf, pendidikan dan standar hidup untuk semua…
Цей обліковий запис заблоковано безстроково у зв'язку з порушенням правила, що стосується використання додаткових облікових записів. Див. журнал блокувань.
Pour l’article homonyme, voir Bataille de Kosovo. Si ce bandeau n'est plus pertinent, retirez-le. Cliquez ici pour en savoir plus. Cet article ne cite pas suffisamment ses sources (novembre 2009). Si vous disposez d'ouvrages ou d'articles de référence ou si vous connaissez des sites web de qualité traitant du thème abordé ici, merci de compléter l'article en donnant les références utiles à sa vérifiabilité et en les liant à la section « Notes et références » En pratiq…
Para otros usos de este término, véase Tamara. Támara Municipio Paisaje del territorio de Támara. Bandera TámaraLocalización de Támara en Colombia TámaraLocalización de Támara en CasanareCoordenadas 5°49′49″N 72°09′44″O / 5.8302777777778, -72.162222222222Entidad Municipio • País Colombia • Departamento CasanareAlcalde Fernando Gómez Riscanievo (2012-2015) Fernando Mantilla Abreo (2016-2019)Leonel Rodríguez Walteros (2020-2023) Eventos históri…
العلاقات الأوزبكستانية الشمال مقدونية أوزبكستان شمال مقدونيا أوزبكستان شمال مقدونيا تعديل مصدري - تعديل العلاقات الأوزبكستانية الشمال مقدونية هي العلاقات الثنائية التي تجمع بين أوزبكستان وشمال مقدونيا.[1][2][3][4][5] مقارنة بين البلدين هذ…
This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: GE C36-7 – news · newspapers · books · scholar · JSTOR (September 2021) (Learn how and when to remove this template message)GE C36-7C36-7i in Estonia.Type and originPower typeDiesel-electricBuilderGE Transportation Systems (most units)GE do Brazil (15 built for us…
衛志豪2023年2月26日,衛志豪於葵芳新都會廣場L3中庭出席無綫電視劇集《新四十二章》之「終極線索」宣傳活動男艺人本名Wai Chi Ho英文名Michael Wai国籍 中华人民共和国(香港)民族汉族出生 (1974-08-27) 1974年8月27日(49歲)[1] 英屬香港职业主持人、演員语言粵語、英語、國語教育程度中五母校坪石天主教小學瑪利諾神父工業中學父母父:衛國華亲属堂兄:衛世輝代表作…
Kazimierz RutkowskiDFCBorn(1914-05-24)24 May 1914Liszno Russian Empire (present-day Poland)Died3 May 1995(1995-05-03) (aged 80)San Diego, California, U.S.Allegiance Poland United Kingdom United States of AmericaService/branch Polish Air Force Royal Air Force United States Air ForceRankWing commanderService number784518UnitPolish 132nd Fighter EscadrillePolish 36th Observation EscadrilleNo. 306 Polish Fighter SquadronNo. 317 Polish Fighter Squadron61st Fighter Squa…
Lexington Spitzname: Shrine of the South South Main Street, Lexington, VA – Blick nach Norden Lage in Virginia Lexington (Virginia) Lexington Basisdaten Gründung: 1777 Staat: Vereinigte Staaten Bundesstaat: Virginia Koordinaten: 37° 47′ N, 79° 27′ W37.783055555556-79.445277777778324Koordinaten: 37° 47′ N, 79° 27′ W Zeitzone: Eastern (UTC−5/−4) Einwohner: 7.320 (Stand: 2020) Haushalte: 2.066 (Stand: 2020) Fläche: 6,5 km²&…
Puchar Konfederacji w piłce nożnej 2001 1999 2003 Dyscyplina piłka nożna Organizator FIFA Szczegóły turnieju Gospodarze Korea Południowa Japonia Otwarcie 30 maja 2001, Daegu Zamknięcie (finał) 10 czerwca 2001, Jokohama Liczba drużyn 8 (z 6 konfederacji) Liczba aren 6 (w 6 miastach) I miejsce Francja II miejsce Japonia III miejsce Australia Statystyki turnieju Liczba meczów 16 Liczba bramek 31 (1,94 na mecz) Oglądalność 556 723 (34 7…
A Mike SarPoster filmNama lainBurmaအမိုက်စား SutradaraKo Zaw (Ar Yone Oo)Produser Ma Aye Aye Win Ditulis oleh Nay Soe Thaw Pyinsaman Di Yine SkenarioNay Soe ThawPyinsaman Di YineCeritaOhn WinPemeran Khant Si Thu Soe Myat Thuzar Thu Htoo San Thinzar Wint Kyaw Kyaw Kyaw Bo Soe Pyae Thazin Moe Aung Yin Wutt Hmone Shwe Yi Moe Moe (singer) Nan Su Yati Soe Penata musikWin KoSinematograferKyauk Phyu (Padaythar)PenyuntingZaw MinPerusahaanproduksiLucky Seven Film ProductionT…
For other people named William Portman, see William Portman (disambiguation). English judge and politician Arms of Portman:Or, a fleur-de-lis azure Sir William Portman (died 1557) was an English judge, politician and Chief Justice of the King's Bench. He was MP for Taunton in 1529 and 1536.[1] Origins and early career Portman was the son of John Portman, who was buried in the Temple Church on 5 June 1521, by Alice, daughter of William Knoell of Dorset. His family was long established in …
List of events ← 1864 1863 1862 1865 in Portugal → 1866 1867 1868 Centuries: 17th 18th 19th 20th 21st Decades: 1840s 1850s 1860s 1870s 1880s See also:List of years in Portugal Events in the year 1865 in Portugal. Incumbents Monarch: Louis I Prime Minister: Nuno José Severo de Mendoça Rolim de Moura Barreto, 1st Duke of Loulé (until 17 April), Bernardo de Sá Nogueira de Figueiredo, 1st Marquis of Sá da Bandeira (17 April–4 September), Joaquim António de Aguiar (starting 4 Sept…
Planned late-2020s Venus orbiter VERITASIllustration of VERITAS spacecraft in orbit around VenusMission typeVenus OrbiterOperatorJet Propulsion Laboratory / NASAMission duration3 years (planned) Spacecraft propertiesManufacturerLockheed Martin Start of missionLaunch dateNET 2029 InstrumentsVenus Emissivity Mapper (VEM)Venus Interferometric Synthetic Aperture Radar (VISAR)Deep Space Atomic Clock-2 (DSAC-2)Discovery program← PsycheDAVINCI → VERITAS (Venus Emissivity, Ra…
Lokasi Pengunjung: 52.15.101.79