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| {{Other uses|List of topics named after Leonhard Euler}}
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| [[File:ExpIPi.gif|thumb|right|The [[exponential function]] {{math|''e''}}<sup>{{math|''z''}}</sup> can be defined as the [[limit of a sequence|limit]] of {{nowrap|(1 + {{math|''z''}}/{{math|N}})<sup>{{math|N}}</sup>}}, as {{math|N}} approaches infinity, and thus {{math|''e''}}<sup>{{math|''i''}}{{pi}}</sup> is the limit of {{nowrap|(1 +{{math|''i''}}{{pi}}/{{math|N}})<sup>{{math|N}}</sup>}}. In this animation {{math|N}} takes various increasing values from 1 to 100. The computation of {{nowrap|(1 + {{math|''i''}}{{pi}}/{{math|N}})<sup>{{math|N}}</sup>}} is displayed as the combined effect of {{math|N}} repeated multiplications in the [[complex plane]], with the final point being the actual value of {{nowrap|(1 +{{math|''i''}}{{pi}}/{{math|N}})<sup>{{math|N}}</sup>}}. It can be seen that as {{math|N}} gets larger {{nowrap|(1 +{{math|''i''}}{{pi}}/{{math|N}})<sup>{{math|N}}</sup>}} approaches a limit of −1.]]
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| {{E (mathematical constant)}}
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| In mathematics, '''Euler's identity''' (also known as '''Euler's equation''') is the [[Equality (mathematics)|equality]]
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| :<math>e^{i \pi} + 1 = 0</math>
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| where
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| :'''{{math|''e''}}''' is [[E (mathematical constant)|Euler's number]], the base of [[natural logarithm]]s,
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| :'''{{math|''i''}}''' is the [[imaginary unit]], which satisfies {{math|''i''}}<sup>2</sup> = −1, and
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| :'''{{pi}}''' is [[pi]], the [[ratio]] of the circumference of a [[circle]] to its diameter.
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| Euler's identity is named after the Swiss mathematician [[Leonhard Euler]]. It is considered an example of [[mathematical beauty]].
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| == Explanation ==
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| [[File:Euler's formula.svg|thumb|right|Euler's formula for a general angle]]
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| Euler's [[Identity (mathematics)|identity]] is a [[special case]] of [[Euler's formula]] from [[complex analysis]], which states that for any [[real number]] {{math|''x''}},
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| : <math>e^{ix} = \cos x + i\sin x \,\!</math>
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| where the values of the [[trigonometry|trigonometric functions]] ''sine'' and ''cosine'' are given in ''[[radian]]s''.
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| In particular, when {{math|''x''}} = ''{{pi}}'', or one [[Turn (geometry)|half-turn]] (180°) around a circle:
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| : <math>e^{i \pi} = \cos \pi + i\sin \pi.\,\!</math>
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| Since
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| :<math>\cos \pi = -1 \, \! </math>
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| and
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| :<math>\sin \pi = 0,\,\!</math>
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| it follows that
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| : <math>e^{i \pi} = -1 + 0 i,\,\!</math>
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| which yields Euler's identity:
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| : <math>e^{i \pi} +1 = 0.\,\!</math>
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| The physical explanation of Euler's identity is that it can be viewed as the [[Group theory|group-theoretical]] definition of the number {{pi}}. The following discussion is at the physical level, but can be made mathematically strict. The "group" is the group of rotations of a plane around 0. In fact, one can write
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| : <math>e^{i \pi} = (e^{i \delta})^{\pi / \delta},\,\!</math>
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| with {{delta}} being some small angle.
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| The last equation can be seen as the action of consecutive small shifts along a circle, caused by the application of infinitesimal rotations starting at 1 and continuing through the total length of the arc, connecting points 1 and −1 in the complex plane. Each small shift may then be written as
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| : <math>1 + i \delta \!</math>
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| and the total number of shifts is {{pi}}/{{delta}}. In order to get from 1 to −1, the total transformation would be
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| : <math>(1 + i \delta)^{\pi / \delta}. \!</math>
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| Taking the limit when {{delta}} → 0, denoting ''i''{{delta}} = 1/''n'' and the equation <math>e = \lim_{n\rightarrow \infty}\left(1+ {1 \over n}\right)^n </math>, we arrive at Euler's identity.
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| {{pi}} is defined as the total angle which connects 1 to −1 along the arc. Therefore, the relation between {{pi}} and ''e'' arises because a circle can be defined through the action of the group of shifts which preserve the distance between two points on the circle.
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| This simple argument is the key to understanding other relations involving {{pi}} and ''e''.
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| ==Mathematical beauty==
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| Euler's identity is often cited as an example of deep [[mathematical beauty]]. Three of the basic [[arithmetic]] operations occur exactly once each: [[addition]], [[multiplication]], and [[exponentiation]]. The identity also links five fundamental [[mathematical constant]]s:
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| * The [[0 (number)|number 0]], the additive identity.
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| * The [[1 (number)|number 1]], the multiplicative identity.
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| * The [[pi|number {{pi}}]], which is ubiquitous in [[trigonometry]], the geometry of [[Euclidean space]], and [[mathematical analysis|analytical mathematics]] ({{pi}} = 3.14159265...)
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| * The [[e (mathematical constant)|number {{math|''e''}}]], the base of [[natural logarithm]]s, which occurs widely in mathematical and scientific analysis ({{math|''e''}} = 2.718281828...). Both {{pi}} and e are [[transcendental number]]s.
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| * The [[imaginary unit|number {{math|''i''}}]], the imaginary unit of the [[complex number]]s, a [[field (mathematics)|field of numbers]] that contains the roots of all polynomials (that are not constants), and whose study leads to deeper insights into many areas of [[algebra]] and [[calculus]].
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| Furthermore, in [[algebra]] and other areas of mathematics, [[equation]]s are commonly written with zero on one side of the [[equals sign]].
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| An entire book, ''Dr. Euler's Fabulous Formula'' (published in 2006), written by Paul Nahin (a [[professor]] emeritus at the [[University of New Hampshire]]), is devoted to Euler's identity and its applications in [[Fourier analysis]]. The book states that Euler's identity sets "the gold standard for mathematical beauty".<ref>Cited in Crease, 2007.</ref>
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| After proving Euler's identity during a lecture, [[Benjamin Peirce]], a noted American 19th-century [[philosopher]], mathematician, and professor at [[Harvard University]], stated that "it is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."<ref>Maor [http://books.google.com/books?id=eIsyLD_bDKkC&pg=PA160 p. 160] and Kasner & Newman [http://books.google.com/books?id=Ad8hAx-6m9oC&pg=PA103 p.103–104].</ref> [[Stanford University]] mathematics professor [[Keith Devlin]] has said, "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence."<ref>Nahin, 2006, [http://books.google.com/books?id=GvSg5HQ7WPcC&pg=PA1 p.1].</ref>
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| The German mathematician [[Carl Friedrich Gauss]] was reported to have commented that if this formula was not immediately apparent to a student upon being told it, that student would never be a first-class mathematician.<ref>Derbyshire, p.210.</ref>
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| The mathematics writer [[Constance Reid]] claimed that Euler's identity was "the most famous formula in all mathematics".<ref>Reid, [http://books.google.com/books?id=d3NFIvrTk4sC&pg=PA155 p. 155].</ref> A poll of readers conducted by ''[[The Mathematical Intelligencer]]'' in 1990 named Euler's identity as the "most beautiful theorem in mathematics".<ref>Nahin, 2006, [http://books.google.com/books?id=GvSg5HQ7WPcC&pg=PA2 p.2–3] (poll published in the summer 1990 issue of the magazine).</ref> In another poll of readers that was conducted by ''[[Physics World]]'' in 2004, Euler's identity tied with [[Maxwell's equations]] (of [[electromagnetism]]) as the "greatest equation ever".<ref>Crease, 2004.</ref>
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| ==Generalizations==
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| Euler's identity is also a special case of the more general identity that the ''n''th [[roots of unity]], for ''n'' > 1, add up to 0:
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| :<math>\sum_{k=0}^{n-1} e^{2 \pi i k/n} = 0 .</math>
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| Euler's identity is the case where ''{{math|n}}'' = 2.
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| In another field of mathematics, by using [[quaternion]] exponentiation, one can show that a similar identity also applies to quaternions. Let {''i'', ''j'', ''k''} be the basis elements, then,
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| :<math>e^{\frac{(i \pm j \pm k)}{\sqrt 3}\pi} + 1 = 0. \,</math>
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| In general, given [[real numbers|real]] ''a''<sub>1</sub>, ''a''<sub>2</sub>, and ''a''<sub>3</sub> such that <math>{a_1}^2+{a_2}^2+{a_3}^2 = 1</math>, then,
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| :<math>e^{(a_1i+a_2j+a_3k)\pi} + 1 = 0. \,</math>
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| For [[octonions]], with real ''a''<sub>n</sub> such that <math>{a_1}^2+{a_2}^2+\dots+{a_7}^2 = 1</math> and the octonion basis elements {''i''<sub>1</sub>, ''i''<sub>2</sub>,..., ''i''<sub>7</sub>}, then,
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| :<math>e^{(a_1i_1+a_2i_2+\dots+a_7i_7)\pi} + 1 = 0. \,</math>
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| ==Attribution==
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| It has been claimed that Euler's identity appears in his monumental work of mathematical analysis published in 1748, ''[[Introductio in analysin infinitorum]]''.<ref>Conway and Guy, p.254–255.</ref> However, it is questionable whether this particular concept can be attributed to Euler himself, as he may never have expressed it.<ref name=Sandifer2007>Sandifer, p.4.</ref> (Moreover, while Euler did write in the ''Introductio'' about what we today call "[[Euler's formula]]",<ref>Euler, p.147.</ref> which relates {{math|''e''}} with ''cosine'' and ''sine'' terms in the field of complex numbers, the English mathematician [[Roger Cotes]] also knew of this formula and Euler may have acquired the knowledge through his Swiss compatriot [[Johann Bernoulli]].<ref name=Sandifer2007/>)
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| ==In popular culture==
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| Euler's identity is referenced in at least two episodes of ''[[The Simpsons]]'': [[Treehouse of Horror VI]] (1995);<ref>{{citation|first=David X|last=Cohen|title=The Simpsons: The Complete Seventh Season|chapter=Commentary for "Treehouse of Horror VI|year=2005|publisher=20th Century Fox}}.</ref> and [[MoneyBart]] (2010).<ref>{{cite news |title=The Simpsons' secret formula: it's written by maths geeks|first=Simon|last=Singh|authorlink=Simon Singh |url=http://www.theguardian.com/tv-and-radio/2013/sep/22/the-simpsons-secret-formula-maths-simon-singh|newspaper=The Guardian |date=22 September 2013 |accessdate=22 September 2013}}</ref>
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| ==See also==
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| *[[De Moivre's formula]]
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| *[[Exponential function]]
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| *[[Gelfond's constant]]
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| ==Notes==
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| {{Reflist|2}}
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| ==References==
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| * Conway, John Horton, and Guy, Richard (1996). ''[http://books.google.com/books?id=0--3rcO7dMYC&pg=PA254 The Book of Numbers]'' (Springer, 1996). ISBN 978-0-387-97993-9.
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| * Crease, Robert P., "[http://physicsweb.org/articles/world/17/10/2 The greatest equations ever]", PhysicsWeb, October 2004 (registration required).
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| * Crease, Robert P. "[http://physicsweb.org/articles/world/20/3/3/1 Equations as icons]," PhysicsWeb, March 2007 (registration required).
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| * Derbyshire, J. ''Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics'' (New York: Penguin, 2004).
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| * Euler, Leonhard. ''[http://gallica.bnf.fr/ark:/12148/bpt6k69587.image.r=%22has+celeberrimas+formulas%22.f169.langEN Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus]'' (Leipzig: B. G. Teubneri, 1922).
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| * Kasner, E., and Newman, J., ''[[Mathematics and the Imagination]]'' (Simon & Schuster, 1940).
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| * Maor, Eli, ''e: The Story of a number'' ([[Princeton University Press]], 1998). ISBN 0-691-05854-7
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| * Nahin, Paul J., ''Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills'' (Princeton University Press, 2006). ISBN 978-0-691-11822-2
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| * Reid, Constance, ''From Zero to Infinity'' (Mathematical Association of America, various editions).
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| * Sandifer, C. Edward. ''[http://books.google.co.uk/books?id=sohHs7ExOsYC&pg=PA4 Euler's Greatest Hits]'' (Mathematical Association of America, 2007). ISBN 978-0-88385-563-8
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| ==External links==
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| * [http://www.youtube.com/watch?v=UcGDNUDQCc4 Complete derivation of Euler's identity]
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| * [http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ Intuitive understanding of Euler's formula]
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| {{DEFAULTSORT:Euler's identity}}
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| [[Category:Complex analysis]]
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| [[Category:Exponentials]]
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| [[Category:Mathematical identities]]
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| [[Category:E (mathematical constant)]]
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| [[de:Eulersche Formel#Eulersche Identit.C3.A4t]]
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| [[pl:Wzór Eulera#Tożsamość Eulera]]
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