Editing Euler's identity (section)

{{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]].