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{{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 product formula|Euler product formula]].<ref name=EOM>{{cite encyclopedia |last=Stepanov |first=S. A. |encyclopedia=[[Encyclopedia of Mathematics]] |title=Euler identity |publisher= |url=http://www.encyclopediaofmath.org/index.php?title=Euler_identity&oldid=11612 |date=7 February 2011 |accessdate=7 September 2018}}</ref> |group=n}} (also known as [[List of things named after Leonhard Euler|Euler's equation]]) is the [[Equality (mathematics)|equality]]

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

where
:{{mvar|e}} is [[E (mathematical constant)|Euler's number]], the base of [[natural logarithm]]s,
:{{mvar|i}} is the [[imaginary unit]], which by definition satisfies {{math|''i''<sup>2</sup> {{=}} −1}}, and
:{{mvar|&pi;}} 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 considered to be an exemplar of [[mathematical beauty]] as it shows a profound connection between the most fundamental numbers in mathematics.

==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 |accessdate=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)|number 0]], the additive unit.
* The [[1 (number)|number 1]], the multiplicative unit.
* The [[pi|number {{mvar|&pi;}}]] ({{mvar|&pi;}} = 3.141...).
* The [[e (mathematical constant)|number {{math|''e''}}]] ({{math|''e''}} = 2.718...), a.k.a. Euler's number, which occurs widely in [[mathematical analysis]].
* The [[imaginary unit|number {{math|''i''}}]], the imaginary unit of the [[complex number]]s.

Furthermore, the equation is 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> And [[Paul Nahin]], a professor emeritus at the [[University of New Hampshire]], who has written a book dedicated to [[Euler's formula]] and its applications in [[Fourier analysis]], describes Euler's identity as being "of exquisite beauty".<ref>Nahin, 2006, p. xxxii.</ref>

Mathematics writer [[Constance Reid]] has opined that Euler's identity is "the most famous formula in all mathematics".<ref>Reid, chapter ''e''.</ref> And [[Benjamin Peirce]], a 19th-century American [[philosopher]], mathematician, and professor at [[Harvard University]], after proving Euler's identity during a lecture, stated that the identity "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 poll of readers conducted by ''[[The Mathematical Intelligencer]]'' in 1990 named Euler's identity as the "most beautiful theorem in mathematics".<ref>Wells, 1990.</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>

A study of the brains of sixteen mathematicians found that the "emotional brain" (specifically, the medial [[orbitofrontal cortex]], which lights up for beautiful music, poetry, pictures, etc.) lit up more consistently for Euler's identity than for any other formula.<ref>Zeki et al., 2014.</ref>

At least two books in [[popular mathematics]] have been published about Euler's identity. One is ''A Most Elegant Equation: Euler's formula and the beauty of mathematics'', by David Stipp (2017). Another is ''Euler's Pioneering Equation: The most beautiful theorem in mathematics'', by [[Robin Wilson (mathematician)|Robin Wilson]] (2018).

==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.]]
Fundamentally, 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 + 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 ''r'' is the absolute value of ''z'' (distance from the origin), and <math>\theta</math> is the argument of ''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 ''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 ''z'' is [[Complex number#Multiplication and division in polar form|multiplied]] by <math>e^{i\theta}</math>, it has the effect of rotating ''z'' 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.

==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}}.

In another field of mathematics, by using [[quaternion]] exponentiation, one can show that a similar identity also applies to quaternions. Let {{math|{{mset|''i'', ''j'', ''k''}}}} be the basis elements; then,

:<math>e^{\frac{1}{\sqrt 3}(i \pm j \pm k)\pi} + 1 = 0. </math>

In general, given [[real numbers|real]] {{math|''a''<sub>1</sub>}}, {{math|''a''<sub>2</sub>}}, and {{math|''a''<sub>3</sub>}} such that {{math|''a''<sub>1</sub><sup>2</sup> + ''a''<sub>2</sub><sup>2</sup> + ''a''<sub>3</sub><sup>2</sup> {{=}} 1}}, then,

:<math>e^{\left(a_1i+a_2j+a_3k\right)\pi} + 1 = 0. </math>

For [[octonions]], with real {{math|''a''<sub>''n''</sub>}} such that {{math|''a''<sub>1</sub><sup>2</sup> + ''a''<sub>2</sub><sup>2</sup> + ... + ''a''<sub>7</sub><sup>2</sup> {{=}} 1}}, and with the octonion basis elements {{math|{{mset|''i''<sub>1</sub>, ''i''<sub>2</sub>, ..., ''i''<sub>7</sub>}}}},

:<math>e^{\left(a_1i_1+a_2i_2+\dots+a_7i_7\right)\pi} + 1 = 0. </math>

==History==
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 & 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 {{mvar|e}} with cosine and sine terms in the field of complex numbers, the English mathematician [[Roger Cotes]] (who died in 1716, when Euler was only 9 years old) also knew of this formula and Euler may have acquired the knowledge through his Swiss compatriot [[Johann Bernoulli]].<ref name=Sandifer2007/>

[[Robin Wilson (mathematician)|Robin Wilson]] states the following.<ref>Wilson, p. 151-152.</ref>
{{quote|text= 
We've seen how it [Euler's identity] can easily be deduced from results of Johann Bernoulli and Roger Cotes, but that neither of them seems 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&hellip;.
}}

== See also ==
*[[De Moivre's formula]] 
*[[Exponential function]]
*[[Gelfond's constant]]

==Notes==
{{reflist|group=n}}

==References==
{{Reflist|25em}}

==Sources==
* [[John Horton Conway|Conway, John H.]], and [[Richard K. Guy|Guy, Richard K.]] (1996), ''[https://books.google.com/books?id=0--3rcO7dMYC&pg=PA254 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]]}}
*Wells, David (1990), "Are these the most beautiful?", ''[[The Mathematical Intelligencer]]'', 12: 37–41, {{doi|10.1007/BF03024015}}
*{{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}}
*{{Citation |last1= Zeki |first1= S. |last2= Romaya |first2= J. P. |last3= Benincasa |first3= D. M. T. |last4= Atiyah |first4= M. F. |authorlink1= Semir Zeki |authorlink4= Michael Atiyah |title= The experience of mathematical beauty and its neural correlates |journal= Frontiers in Human Neuroscience |volume= 8 | year= 2014 |doi= 10.3389/fnhum.2014.00068|pmc= 3923150 }}

==External links==
{{Wikiquote|Euler's identity}}
* [https://www.youtube.com/watch?v=UcGDNUDQCc4 Complete derivation of Euler's identity]
* [http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ Intuitive understanding of Euler's formula]

{{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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