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{{Short description|Mathematical equation linking e, i and π}}
{{Other uses|List of topics named after Leonhard Euler#Identities}}
{{E (mathematical constant)}}
In [[mathematics]], '''Euler's identity'''{{#tag:ref |The term "Euler's identity" (or "Euler identity") is also used elsewhere to refer to other concepts, including the related general formula {{math|''e''<sup>''ix''</sup> {{=}} cos ''x'' + ''i'' sin ''x''}},<ref>Dunham, 1999, [https://books.google.com/books?id=uKOVNvGOkhQC&pg=PR24 p. xxiv].</ref> and the [[Riemann zeta function#Euler's product formula|Euler product formula]].<ref name=EOM>{{Eom| title = Euler identity | author-last1 = Stepanov| author-first1 = S.A. | oldid = 33574}}</ref>  See also [[List of topics named after Leonhard Euler]]. |group=note}} (also known as '''Euler's equation''') is the [[Equality (mathematics)|equality]]
<math display=block>e^{i \pi} + 1 = 0</math>
where
:<math>e</math> is [[E (mathematical constant)|Euler's number]], the base of [[natural logarithm]]s,
:<math>i</math> is the [[imaginary unit]], which by definition satisfies <math>i^2 = -1</math>, and
:<math>\pi</math> is [[pi]], the ratio of the [[circumference]] of a circle to its [[diameter]].
Euler's identity is named after the Swiss mathematician [[Leonhard Euler]]. It is a special case of [[Euler's formula]] <math>e^{ix} = \cos x + i\sin x</math> when evaluated for <math>x = \pi</math>. Euler's identity is considered an exemplar of [[mathematical beauty]], as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in [[Lindemann–Weierstrass theorem#Transcendence of e and π|a proof]]<ref>{{citation|title=The Transcendence of π and the Squaring of the Circle|last1=Milla|first1=Lorenz|arxiv=2003.14035|year=2020}}</ref><ref>{{Cite web|url=https://math.colorado.edu/~rohi1040/expository/eistranscendental.pdf |archive-url=https://web.archive.org/web/20210623215444/https://math.colorado.edu/~rohi1040/expository/eistranscendental.pdf |archive-date=2021-06-23 |url-status=live|title=e is transcendental|last=Hines|first=Robert|website=University of Colorado}}</ref> that {{pi}} is [[Transcendental number|transcendental]], which implies the impossibility of [[squaring the circle]].

==Mathematical beauty==
Euler's identity is often cited as an example of deep [[mathematical beauty]].<ref name=Gallagher2014>{{cite news |last=Gallagher |first=James |title=Mathematics: Why the brain sees maths as beauty |url=https://www.bbc.co.uk/news/science-environment-26151062 |access-date=26 December 2017 |work=[[BBC News Online]] |date=13 February 2014}}</ref> Three of the basic [[arithmetic]] operations occur exactly once each: [[addition]], [[multiplication]], and [[exponentiation]]. The identity also links five fundamental [[mathematical constant]]s:<ref>Paulos, 1992, p. 117.</ref>
* The [[0|number 0]], the [[additive identity]]
* The [[1|number 1]], the [[multiplicative identity]]
* The [[pi|number {{mvar|&pi;}}]] ({{mvar|&pi;}} = 3.14159...), the fundamental [[circle]] constant
* The [[e (mathematical constant)|number {{math|''e''}}]] ({{math|''e''}} = 2.71828...), also known as Euler's number, which occurs widely in [[mathematical analysis]]
* The [[Imaginary unit|number {{math|''i''}}]], the [[imaginary unit]] such that <math>i^2=-1</math>

The equation is often given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.

[[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, [https://books.google.com/books?id=GvSg5HQ7WPcC&pg=PA1 p. 1].</ref> [[Paul Nahin]], a professor emeritus at the [[University of New Hampshire]] who wrote a book dedicated to [[Euler's formula]] and its applications in [[Fourier analysis]], said Euler's identity is "of exquisite beauty".<ref>Nahin, 2006, p. xxxii.</ref>

Mathematics writer [[Constance Reid]] has said that Euler's identity is "the most famous formula in all mathematics".<ref>Reid, chapter ''e''.</ref> [[Benjamin Peirce]], a 19th-century American philosopher, mathematician, and professor at [[Harvard University]], after proving Euler's identity during a lecture, said 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, [https://books.google.com/books?id=eIsyLD_bDKkC&pg=PA160 p. 160], and Kasner & Newman, [https://books.google.com/books?id=Ad8hAx-6m9oC&pg=PA103 p. 103–104].</ref>

A 1990 poll of readers by ''[[The Mathematical Intelligencer]]'' named Euler's identity the "most beautiful theorem in mathematics".<ref>Wells, 1990.</ref> In a 2004 poll of readers by ''[[Physics World]]'', Euler's identity tied with [[Maxwell's equations]] (of [[electromagnetism]]) as the "greatest equation ever".<ref>Crease, 2004.</ref>

At least three books in [[popular mathematics]] have been published about Euler's identity:
*''Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills'', by [[Paul Nahin]] (2011)<ref>{{cite book |last=Nahin |first=Paul |title=Dr. Euler's fabulous formula : cures many mathematical ills |date=2011 |publisher=Princeton University Press |isbn=978-0-691-11822-2 }}</ref>
*''A Most Elegant Equation: Euler's formula and the beauty of mathematics'', by David Stipp (2017)<ref>{{cite book |last=Stipp |first=David |title=A Most Elegant Equation : Euler's Formula and the Beauty of Mathematics |date=2017 |publisher=Basic Books |isbn=978-0-465-09377-9 |edition=First }}</ref>
*''Euler's Pioneering Equation: The most beautiful theorem in mathematics'', by [[Robin Wilson (mathematician)|Robin Wilson]] (2018).<ref>{{cite book |last=Wilson |first=Robin |title=Euler's pioneering equation : the most beautiful theorem in mathematics |date=2018 |publisher=Oxford University Press |location=Oxford |isbn=978-0-19-879493-6 }}</ref>

==Explanations==
===Imaginary exponents===
{{main|Euler's formula}}
{{See also|Exponentiation#Complex_exponents_with_a_positive_real_base|l1=Complex exponents with a positive real base}}
[[File:ExpIPi.gif|thumb|right|In this animation {{mvar|N}} takes various increasing values from 1 to 100. The computation of {{math|(1 + {{sfrac|''iπ''|''N''}})<sup>''N''</sup>}} is displayed as the combined effect of {{mvar|N}} repeated multiplications in the [[complex plane]], with the final point being the actual value of {{math|(1 + {{sfrac|''iπ''|''N''}})<sup>''N''</sup>}}. It can be seen that as {{mvar|N}} gets larger {{math|(1 + {{sfrac|''iπ''|''N''}})<sup>''N''</sup>}} approaches a limit of −1.]]
Euler's identity asserts that <math>e^{i\pi}</math> is equal to −1. The expression <math>e^{i\pi}</math> is a special case of the expression <math>e^z</math>, where {{math|''z''}} is any [[complex number]]. In general, <math>e^z</math> is defined for complex {{math|''z''}} by extending one of the [[characterizations of the exponential function|definitions of the exponential function]] from real exponents to complex exponents. For example, one common definition is:

:<math>e^z = \lim_{n\to\infty} \left(1+\frac z n \right)^n.</math>

Euler's identity therefore states that the limit, as {{math|''n''}} approaches infinity, of <math>(1 + \tfrac {i\pi}{n})^n</math> is equal to −1. This limit is illustrated in the animation to the right.

[[File:Euler's formula.svg|thumb|right|Euler's formula for a general angle]]
Euler's identity is a [[special case]] of [[Euler's formula]], which states that for any [[real number]] {{math|''x''}},

: <math>e^{ix} = \cos x + i\sin x</math>

where the inputs of the [[trigonometry|trigonometric functions]] sine and cosine are given in [[radian]]s.

In particular, when {{math|''x'' {{=}} ''π''}},

: <math>e^{i \pi} = \cos \pi + i\sin \pi.</math>

Since

:<math>\cos \pi = -1</math>

and

:<math>\sin \pi = 0,</math>

it follows that

: <math>e^{i \pi} = -1 + 0 i,</math>

which yields Euler's identity:

: <math>e^{i \pi} +1 = 0.</math>

===Geometric interpretation===
Any complex number <math>z = x + iy</math> can be represented by the point <math>(x, y)</math> on the [[complex plane]]. This point can also be represented in [[Complex_number#Polar_complex_plane|polar coordinates]] as <math>(r, \theta)</math>, where {{Mvar|r}} is the absolute value of {{Mvar|z}} (distance from the origin), and <math>\theta</math> is the argument of {{Mvar|z}} (angle counterclockwise from the positive ''x''-axis). By the definitions of sine and cosine, this point has cartesian coordinates of <math>(r \cos \theta, r \sin \theta)</math>, implying that <math>z = r(\cos \theta + i \sin \theta)</math>. According to Euler's formula, this is equivalent to saying <math>z = r e^{i\theta}</math>.

Euler's identity says that <math>-1 = e^{i\pi}</math>. Since <math>e^{i\pi}</math> is <math>r e^{i\theta}</math> for {{Mvar|r}} = 1 and <math>\theta = \pi</math>, this can be interpreted as a fact about the number −1 on the complex plane: its distance from the origin is 1, and its angle from the positive ''x''-axis is <math>\pi</math> radians.

Additionally, when any complex number {{Mvar|z}} is [[Complex number#Multiplication and division in polar form|multiplied]] by <math>e^{i\theta}</math>, it has the effect of rotating <math>z</math> counterclockwise by an angle of <math>\theta</math> on the complex plane. Since multiplication by −1 reflects a point across the origin, Euler's identity can be interpreted as saying that rotating any point <math>\pi</math> radians around the origin has the same effect as reflecting the point across the origin.  Similarly, setting <math>\theta</math> equal to <math>2\pi</math> yields the related equation <math>e^{2\pi i} = 1,</math> which can be interpreted as saying that rotating any point by one [[turn (angle)|turn]] around the origin returns it to its original position.

==Generalizations==
Euler's identity is also a special case of the more general identity that the {{mvar|n}}th [[roots of unity]], for {{math|''n'' > 1}}, add up to 0:

:<math>\sum_{k=0}^{n-1} e^{2 \pi i \frac{k}{n}} = 0 .</math>

Euler's identity is the case where {{math|''n'' {{=}} 2}}.

A similar identity also applies to [[quaternion#Exponential, logarithm, and power functions|quaternion exponential]]: let {{math|{{mset|''i'', ''j'', ''k''}}}} be the basis [[quaternion]]s; then,
:<math>e^{\frac{1}{\sqrt 3}(i \pm j \pm k)\pi} + 1 = 0. </math>

More generally, let {{mvar|q}} be a quaternion  with a zero real part and a norm equal to 1; that is, <math>q=ai+bj+ck,</math> with <math>a^2+b^2+c^2=1.</math> Then one has
:<math>e^{q\pi} + 1 = 0. </math>

The same formula applies to [[octonion]]s, with a zero real part and a norm equal to 1. These formulas are a direct generalization of Euler's identity, since <math>i</math> and <math>-i</math> are the only complex numbers with a zero real part and a norm (absolute value) equal to 1.

==History==
Euler's identity is a direct result of [[Euler's formula]], published in his monumental 1748 work of mathematical analysis, ''[[Introductio in analysin infinitorum]]'',<ref>Conway & Guy, p. [https://books.google.com/books?id=0--3rcO7dMYC&pg=PA254 254–255].</ref> but it is questionable whether the particular concept of linking five fundamental constants in a compact form can be attributed to Euler himself, as he may never have expressed it.<ref name=Sandifer2007>Sandifer, p. 4.</ref>

[[Robin Wilson (mathematician)|Robin Wilson]] writes:<ref>Wilson, p. 151-152.</ref>
{{quote|text= 
We've seen how [Euler's identity] can easily be deduced from results of [[Johann Bernoulli]] and [[Roger Cotes]], but that neither of them seem to have done so. Even Euler does not seem to have written it down explicitly—and certainly it doesn't appear in any of his publications—though he must surely have realized that it follows immediately from his identity [i.e. [[Euler's formula]]], {{nowrap|''e<sup>ix</sup>'' {{=}} cos ''x'' + ''i'' sin ''x''}}. Moreover, it seems to be unknown who first stated the result explicitly}}

== See also ==
{{Portal|Mathematics}}
*[[De Moivre's formula]] 
*[[Exponential function]]
*[[Gelfond's constant]]

==Notes==
{{reflist|group=note}}

==References==
{{Reflist|colwidth=20em}}

===Sources===
* [[John Horton Conway|Conway, John H.]], and [[Richard K. Guy|Guy, Richard K.]] (1996), ''[[The Book of Numbers (math book)|The Book of Numbers]]'', Springer {{ISBN|978-0-387-97993-9}}
* [[Robert P. Crease|Crease, Robert P.]] (10&nbsp;May 2004), "[http://physicsworld.com/cws/article/print/2004/may/10/the-greatest-equations-ever The greatest equations ever]", ''[[Physics World]]'' [registration required]
* [[William Dunham (mathematician)|Dunham, William]] (1999), ''Euler: The Master of Us All'', [[Mathematical Association of America]] {{ISBN|978-0-88385-328-3}}
* Euler, Leonhard (1922), ''[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
* [[Edward Kasner|Kasner, E.]], and [[James R. Newman|Newman, J.]] (1940), ''[[Mathematics and the Imagination]]'', [[Simon & Schuster]]
* [[Eli Maor|Maor, Eli]] (1998), ''{{mvar|e}}: The Story of a number'', [[Princeton University Press]] {{ISBN|0-691-05854-7}}
* Nahin, Paul J. (2006), ''Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills'', [[Princeton University Press]] {{ISBN|978-0-691-11822-2}}
* [[John Allen Paulos|Paulos, John Allen]] (1992), ''Beyond Numeracy: An Uncommon Dictionary of Mathematics'', [[Penguin Books]] {{ISBN|0-14-014574-5}}
* Reid, Constance (various editions), ''[[From Zero to Infinity]]'', [[Mathematical Association of America]]
* Sandifer, C. Edward (2007), ''[https://books.google.com/books?id=sohHs7ExOsYC&pg=PA4 Euler's Greatest Hits]'', [[Mathematical Association of America]] {{ISBN|978-0-88385-563-8}}
*{{citation |title= A Most Elegant Equation: Euler's formula and the beauty of mathematics |first= David |last= Stipp |year=2017 |publisher= [[Basic Books]]}}
*{{cite journal | author-link= David G. Wells | last1 = Wells | first1 = David | year = 1990 | title = Are these the most beautiful? | journal = [[The Mathematical Intelligencer]] | volume = 12 | issue = 3| pages = 37–41 | doi = 10.1007/BF03024015 | s2cid = 121503263 }}
*{{citation |first= Robin |last= Wilson |author-link= Robin Wilson (mathematician) |title= Euler's Pioneering Equation: The most beautiful theorem in mathematics |publisher= [[Oxford University Press]] |year= 2018 |isbn= 978-0-192-51406-6 }}
*{{Citation |last1= Zeki |first1= S. |last2= Romaya |first2= J. P. |last3= Benincasa |first3= D. M. T. |last4= Atiyah |first4= M. F. |author-link1= Semir Zeki |author-link4= Michael Atiyah |title= The experience of mathematical beauty and its neural correlates |journal= Frontiers in Human Neuroscience |volume= 8 |pages= 68 | year= 2014 |doi= 10.3389/fnhum.2014.00068|pmc= 3923150 |pmid=24592230|doi-access= free }}

==External links==
{{Wikiquote|Euler's identity}}
* [http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ Intuitive understanding of Euler's formula]

{{Leonhard Euler}}
{{DEFAULTSORT:Euler's identity}}
[[Category:Exponentials]]
[[Category:Mathematical identities]]
[[Category:E (mathematical constant)]]
[[Category:Theorems in complex analysis]]
[[Category:Leonhard Euler]]

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