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1. Shard Calculation

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Response:

Calculated Shard: 152 (from laksa030)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa152.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Euler%27s_identity\')), getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Euler%27s_identity\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity","crawl_time":1775427286,"first_indexed_time":1376022059,"http_code":200,"src_unparsed":"org,wikipedia!en,\/wiki\/Euler%27s_identity s443","src_root_hash":"17790707453426894952","history_drop_reason":null,"meta_title":"Euler's identity - Wikipedia","meta_descriptions":[],"attrs_boilerpipe_text":"From Wikipedia, the free encyclopedia\nIn\nmathematics\n,\nEuler's identity\n[\nnote 1\n]\n(also known as\nEuler's equation\n) is the\nequality\nwhere\nEuler's identity is named after the Swiss mathematician\nLeonhard Euler\n. It is a special case of\nEuler's formula\nwhen evaluated for\n. Euler's identity is considered an exemplar of\nmathematical beauty\n, as it shows a profound connection between the most fundamental numbers in mathematics.\nMathematical beauty\n[\nedit\n]\nEuler's identity is often cited as an example of deep\nmathematical beauty\n.\n[\n3\n]\nThree of the basic\narithmetic\noperations occur exactly once each:\naddition\n,\nmultiplication\n, and\nexponentiation\n. The identity also links five fundamental\nmathematical constants\n:\n[\n4\n]\nThe\nnumber\n0\n, the\nadditive identity\nThe\nnumber\n1\n, the\nmultiplicative identity\nThe\nnumber\nπ\n(\nπ = 3.14159...\n), the fundamental\ncircle\nconstant\nThe\nnumber\ne\n(\ne\n= 2.71828...\n), also known as Euler's number, which occurs widely in\nmathematical analysis\nThe\nnumber\ni\n, the\nimaginary unit\nsuch that\ni\n2\n= −1\nThe equation is often given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.\nStanford University\nmathematics professor\nKeith Devlin\nhas said, \"like a Shakespearean\nsonnet\nthat 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\".\n[\n5\n]\nPaul Nahin\n, a professor emeritus at the\nUniversity of New Hampshire\nwho wrote a book dedicated to\nEuler's formula\nand its applications in\nFourier analysis\n, said Euler's identity is \"of exquisite beauty\".\n[\n6\n]\nMathematics writer\nConstance Reid\nhas said that Euler's identity is \"the most famous formula in all mathematics\".\n[\n7\n]\nBenjamin Peirce\n, a 19th-century American philosopher, mathematician, and professor at\nHarvard University\n, 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\".\n[\n8\n]\nA 1990 poll of readers by\nThe Mathematical Intelligencer\nnamed Euler's identity the \"most beautiful theorem in mathematics\".\n[\n9\n]\nIn a 2004 poll of readers by\nPhysics World\n, Euler's identity tied with\nMaxwell's equations\n(of\nelectromagnetism\n) as the \"greatest equation ever\".\n[\n10\n]\nAt least three books in\npopular mathematics\nhave been published about Euler's identity:\nDr. Euler's Fabulous Formula: Cures Many Mathematical Ills\n, by\nPaul Nahin\n(2011)\n[\n11\n]\nA Most Elegant Equation: Euler's formula and the beauty of mathematics\n, by David Stipp (2017)\n[\n12\n]\nEuler's Pioneering Equation: The most beautiful theorem in mathematics\n, by\nRobin Wilson\n(2018)\n[\n13\n]\nImaginary exponents\n[\nedit\n]\nIn this animation\nN\ntakes various increasing values from 1 to 100. The computation of\n(1 +\n⁠\niπ\n\/\nN\n⁠\n)\nN\nis displayed as the combined effect of\nN\nrepeated multiplications in the\ncomplex plane\n, with the final point being the actual value of\n(1 +\n⁠\niπ\n\/\nN\n⁠\n)\nN\n. It can be seen that as\nN\ngets larger\n(1 +\n⁠\niπ\n\/\nN\n⁠\n)\nN\napproaches a limit of −1.\nEuler's identity asserts that\nis equal to −1. The expression\nis a special case of the expression\n, where\nz\nis any\ncomplex number\n. In general,\nis defined for complex\nz\nby extending one of the\ndefinitions of the exponential function\nfrom real exponents to complex exponents. For example, one common definition is:\nEuler's identity therefore states that the limit, as\nn\napproaches infinity, of\nis equal to −1. This limit is illustrated in the animation to the right.\nEuler's formula for a general angle\nEuler's identity is a\nspecial case\nof\nEuler's formula\n, which states that for any\nreal number\nx\n,\nwhere the inputs of the\ntrigonometric functions\nsine and cosine are given in\nradians\n.\nIn particular, when\nx\n=\nπ\n,\nSince\nand\nit follows that\nwhich yields Euler's identity:\nGeometric interpretation\n[\nedit\n]\nAny complex number\ncan be represented by the point\non the\ncomplex plane\n. This point can also be represented in\npolar coordinates\nas\n,\nwhere\nr\nis the absolute value of\nz\n(distance from the origin), and\nis the argument of\nz\n(angle counterclockwise from the positive\nx\n-axis). By the definitions of sine and cosine, this point has cartesian coordinates of\n,\nimplying that\n.\nAccording to Euler's formula, this is equivalent to saying\n.\nEuler's identity says that\n.\nSince\nis\nfor\nr\n= 1 and\n,\nthis 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\nx\n-axis is\nradians.\nAdditionally, when any complex number\nz\nis\nmultiplied\nby\n,\nit has the effect of rotating\ncounterclockwise by an angle of\non 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\nradians around the origin has the same effect as reflecting the point across the origin.  Similarly, setting\nequal to\nyields the related equation\n,\nwhich can be interpreted as saying that rotating any point by one\nturn\naround the origin returns it to its original position.\nEuler's identity is also a special case of the more general identity that the\nn\nth\nroots of unity\n, for\nn\n> 1\n, add up to 0:\nEuler's identity is the case where\nn\n= 2\n.\nA similar identity also applies to\nquaternion exponential\n: let\n{\ni\n,\nj\n,\nk\n}\nbe the basis\nquaternions\n; then,\nMore generally, let\nq\nbe a quaternion  with a zero real part and a norm equal to 1; that is,\nwith\nThen one has\nThe same formula applies to\noctonions\n, with a zero real part and a norm equal to 1. These formulas are a direct generalization of Euler's identity, since\nand\nare the only complex numbers with a zero real part and a norm (absolute value) equal to 1.\nEuler's identity is a direct result of\nEuler's formula\n, first published in his monumental 1748 work of mathematical analysis,\nIntroductio in analysin infinitorum\n,\n[\n14\n]\nbut 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.\n[\n15\n]\nRobin Wilson\nwrites:\n[\n16\n]\nWe've seen how [Euler's identity] can easily be deduced from results of\nJohann Bernoulli\nand\nRoger Cotes\n, 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.\nEuler's formula\n],\ne\nix\n= cos\nx\n+\ni\nsin\nx\n. Moreover, it seems to be unknown who first stated the result explicitly\nDe Moivre's formula\nExponential function\nGelfond's constant\n^\nThe term \"Euler's identity\" (or \"Euler identity\") is also used elsewhere to refer to other concepts, including the related general formula\ne\nix\n= cos\nx\n+\ni\nsin\nx\n,\n[\n1\n]\nand the\nEuler product formula\n.\n[\n2\n]\nSee also\nList of topics named after Leonhard Euler\n.\n^\nDunham, 1999,\np. xxiv\n.\n^\nStepanov, S. A. (2001) [1994].\n\"Euler identity\"\n.\nEncyclopedia of Mathematics\n.\nEMS Press\n.\n^\nGallagher, James (13 February 2014).\n\"Mathematics: Why the brain sees maths as beauty\"\n.\nBBC News\n. Retrieved\n26 December\n2017\n.\n^\nPaulos, 1992, p. 117.\n^\nNahin, 2006,\np. 1\n.\n^\nNahin, 2006, p. xxxii.\n^\nReid, chapter\ne\n.\n^\nMaor,\np. 160\n, and Kasner & Newman,\np. 103–104\n.\n^\nWells, 1990.\n^\nCrease, 2004.\n^\nNahin, Paul (2011).\nDr. Euler's fabulous formula: cures many mathematical ills\n. Princeton University Press.\nISBN\n \n978-0-691-11822-2\n.\n^\nStipp, David (2017).\nA Most Elegant Equation: Euler's Formula and the Beauty of Mathematics\n(First ed.). Basic Books.\nISBN\n \n978-0-465-09377-9\n.\n^\nWilson, Robin (2018).\nEuler's pioneering equation: the most beautiful theorem in mathematics\n. Oxford: Oxford University Press.\nISBN\n \n978-0-19-879493-6\n.\n^\nConway & Guy, p.\n254–255\n.\n^\nSandifer, p. 4.\n^\nWilson, p. 151-152.\nConway, John H.\n, and\nGuy, Richard K.\n(1996),\nThe Book of Numbers\n, Springer\nISBN\n \n978-0-387-97993-9\nCrease, Robert P.\n(10 May 2004), \"\nThe greatest equations ever\n\",\nPhysics World\n[registration required]\nDunham, William\n(1999),\nEuler: The Master of Us All\n,\nMathematical Association of America\nISBN\n \n978-0-88385-328-3\nEuler, Leonhard\n,\nComplete work.\nLeonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus\n, Leipzig: B. G. Teubneri\nKasner, E.\n, and\nNewman, J.\n(1940),\nMathematics and the Imagination\n,\nSimon & Schuster\nMaor, Eli\n(1998),\ne\n: The Story of a number\n,\nPrinceton University Press\nISBN\n \n0-691-05854-7\nNahin, Paul J. (2006),\nDr. Euler's Fabulous Formula: Cures Many Mathematical Ills\n,\nPrinceton University Press\nISBN\n \n978-0-691-11822-2\nPaulos, John Allen\n(1992),\nBeyond Numeracy: An Uncommon Dictionary of Mathematics\n,\nPenguin Books\nISBN\n \n0-14-014574-5\nReid, Constance (various editions),\nFrom Zero to Infinity\n,\nMathematical Association of America\nSandifer, C. Edward (2007),\nEuler's Greatest Hits\n,\nMathematical Association of America\nISBN\n \n978-0-88385-563-8\nStipp, David (2017).\nA Most Elegant Equation: Euler's formula and the beauty of mathematics\n.\nBasic Books\n.\nWells, David\n(1990). \"Are these the most beautiful?\".\nThe Mathematical Intelligencer\n.\n12\n(3):\n37–\n41.\ndoi\n:\n10.1007\/BF03024015\n.\nS2CID\n \n121503263\n.\nWilson, Robin\n(2018).\nEuler's Pioneering Equation: The most beautiful theorem in mathematics\n.\nOxford University Press\n.\nISBN\n \n978-0-192-51406-6\n.\nZeki, S.\n; Romaya, J. P.; Benincasa, D. M. T.;\nAtiyah, M. F.\n(2014). \"The experience of mathematical beauty and its neural correlates\".\nFrontiers in Human Neuroscience\n.\n8\n: 68.\ndoi\n:\n10.3389\/fnhum.2014.00068\n.\nPMC\n \n3923150\n.\nPMID\n \n24592230\n.\nIntuitive understanding of Euler's formula","attrs_markdown":"[Jump to content](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#bodyContent)\n\nMain menu\n\nMain menu\n\nmove to sidebar\n\nhide\n\nNavigation\n\n- [Main page](https:\/\/en.wikipedia.org\/wiki\/Main_Page \"Visit the main page [z]\")\n- [Contents](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contents \"Guides to browsing Wikipedia\")\n- [Current events](https:\/\/en.wikipedia.org\/wiki\/Portal:Current_events \"Articles related to current events\")\n- [Random article](https:\/\/en.wikipedia.org\/wiki\/Special:Random \"Visit a randomly selected article [x]\")\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About \"Learn about Wikipedia and how it works\")\n- [Contact us](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us \"How to contact Wikipedia\")\n\nContribute\n\n- [Help](https:\/\/en.wikipedia.org\/wiki\/Help:Contents \"Guidance on how to use and edit Wikipedia\")\n- [Learn to edit](https:\/\/en.wikipedia.org\/wiki\/Help:Introduction \"Learn how to edit Wikipedia\")\n- [Community portal](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Community_portal \"The hub for editors\")\n- [Recent changes](https:\/\/en.wikipedia.org\/wiki\/Special:RecentChanges \"A list of recent changes to Wikipedia [r]\")\n- [Upload file](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:File_upload_wizard \"Add images or other media for use on Wikipedia\")\n- [Special pages](https:\/\/en.wikipedia.org\/wiki\/Special:SpecialPages \"A list of all special pages [q]\")\n\n[![](https:\/\/en.wikipedia.org\/static\/images\/icons\/enwiki-25.svg) ![Wikipedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-wordmark-en-25.svg) ![The Free Encyclopedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-tagline-en-25.svg)](https:\/\/en.wikipedia.org\/wiki\/Main_Page)\n\n[Search](https:\/\/en.wikipedia.org\/wiki\/Special:Search \"Search Wikipedia [f]\")\n\nAppearance\n\n- [Donate](https:\/\/donate.wikimedia.org\/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en)\n- [Create account](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CreateAccount&returnto=Euler%27s+identity \"You are encouraged to create an account and log in; however, it is not mandatory\")\n- [Log in](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UserLogin&returnto=Euler%27s+identity \"You're encouraged to log in; however, it's not mandatory. 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[o]\")\n\n## Contents\nmove to sidebar\n\nhide\n\n- [(Top)](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity)\n- [1 Mathematical beauty](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#Mathematical_beauty)\n- [2 Explanations](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#Explanations)\n  Toggle Explanations subsection\n  - [2\\.1 Imaginary exponents](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#Imaginary_exponents)\n  - [2\\.2 Geometric interpretation](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#Geometric_interpretation)\n- [3 Generalizations](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#Generalizations)\n- [4 History](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#History)\n- [5 See also](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#See_also)\n- [6 Notes](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#Notes)\n- [7 References](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#References)\n  Toggle References subsection\n  - [7\\.1 Sources](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#Sources)\n- [8 External links](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#External_links)\n\nToggle the table of contents\n\n# Euler's identity\n54 languages\n\n- [العربية](https:\/\/ar.wikipedia.org\/wiki\/%D9%85%D8%AA%D8%B7%D8%A7%D8%A8%D9%82%D8%A9_%D8%A3%D9%88%D9%8A%D9%84%D8%B1 \"متطابقة أويلر – Arabic\")\n- [Asturianu](https:\/\/ast.wikipedia.org\/wiki\/Identid%C3%A1_d%27Euler \"Identidá d'Euler – Asturian\")\n- [Башҡортса](https:\/\/ba.wikipedia.org\/wiki\/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80_%D1%82%D0%BE%D0%B6%D0%B4%D0%B5%D1%81%D1%82%D0%B2%D0%BE%D2%BB%D1%8B_\\(%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BB%D1%8B_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7\\) \"Эйлер тождествоһы (комплекслы анализ) – Bashkir\")\n- [Български](https:\/\/bg.wikipedia.org\/wiki\/%D0%A0%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%BD%D0%B0_%D0%9E%D0%B9%D0%BB%D0%B5%D1%80 \"Равенство на Ойлер – Bulgarian\")\n- [বাংলা](https:\/\/bn.wikipedia.org\/wiki\/%E0%A6%85%E0%A6%AF%E0%A6%BC%E0%A6%B2%E0%A6%BE%E0%A6%B0%E0%A7%87%E0%A6%B0_%E0%A6%85%E0%A6%AD%E0%A7%87%E0%A6%A6 \"অয়লারের অভেদ – Bangla\")\n- [Català](https:\/\/ca.wikipedia.org\/wiki\/Identitat_d%27Euler \"Identitat d'Euler – Catalan\")\n- [کوردی](https:\/\/ckb.wikipedia.org\/wiki\/%DA%BE%D8%A7%D9%88%D8%A6%DB%95%D9%86%D8%AC%D8%A7%D9%85%DB%8C_%D8%A6%DB%86%DB%8C%D9%84%DB%95%D8%B1 \"ھاوئەنجامی ئۆیلەر – Central Kurdish\")\n- [Čeština](https:\/\/cs.wikipedia.org\/wiki\/Eulerova_rovnost \"Eulerova rovnost – Czech\")\n- [Чӑвашла](https:\/\/cv.wikipedia.org\/wiki\/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80_%C3%A7%D0%B0%D0%B2%D0%B0%D1%85%D0%BB%C4%83%D1%85%C4%95_\\(%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BB%C4%83_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7\\) \"Эйлер çавахлăхĕ (комплекслă анализ) – Chuvash\")\n- [Deutsch](https:\/\/de.wikipedia.org\/wiki\/Eulersche_Formel#Eulersche_Identit.C3.A4t \"Eulersche Formel – German\")\n- [Ελληνικά](https:\/\/el.wikipedia.org\/wiki\/%CE%A4%CE%B1%CF%85%CF%84%CF%8C%CF%84%CE%B7%CF%84%CE%B1_%CF%84%CE%BF%CF%85_%CE%8C%CE%B9%CE%BB%CE%B5%CF%81 \"Ταυτότητα του Όιλερ – Greek\")\n- [Esperanto](https:\/\/eo.wikipedia.org\/wiki\/E%C5%ADlera_idento \"Eŭlera idento – Esperanto\")\n- [Español](https:\/\/es.wikipedia.org\/wiki\/Identidad_de_Euler \"Identidad de Euler – Spanish\")\n- [Euskara](https:\/\/eu.wikipedia.org\/wiki\/Eulerren_identitatea \"Eulerren identitatea – Basque\")\n- [فارسی](https:\/\/fa.wikipedia.org\/wiki\/%D8%A7%D8%AA%D8%AD%D8%A7%D8%AF_%D8%A7%D9%88%DB%8C%D9%84%D8%B1 \"اتحاد اویلر – Persian\")\n- [Suomi](https:\/\/fi.wikipedia.org\/wiki\/Eulerin_identiteetti \"Eulerin identiteetti – Finnish\")\n- [Français](https:\/\/fr.wikipedia.org\/wiki\/Identit%C3%A9_d%27Euler \"Identité d'Euler – French\")\n- [Arpetan](https:\/\/frp.wikipedia.org\/wiki\/Identit%C3%A2t_d%27Euler \"Identitât d'Euler – Arpitan\")\n- [Nordfriisk](https:\/\/frr.wikipedia.org\/wiki\/Euler_sin_identiteet \"Euler sin identiteet – Northern Frisian\")\n- [Gaeilge](https:\/\/ga.wikipedia.org\/wiki\/C%C3%A9annacht_Euler \"Céannacht Euler – Irish\")\n- [Galego](https:\/\/gl.wikipedia.org\/wiki\/Identidade_de_Euler \"Identidade de Euler – Galician\")\n- [עברית](https:\/\/he.wikipedia.org\/wiki\/%D7%96%D7%94%D7%95%D7%AA_%D7%90%D7%95%D7%99%D7%9C%D7%A8 \"זהות אוילר – Hebrew\")\n- [हिन्दी](https:\/\/hi.wikipedia.org\/wiki\/%E0%A4%86%E0%A4%AF%E0%A4%B2%E0%A4%B0_%E0%A4%B8%E0%A4%B0%E0%A5%8D%E0%A4%B5%E0%A4%B8%E0%A4%AE%E0%A4%BF%E0%A4%95%E0%A4%BE \"आयलर सर्वसमिका – Hindi\")\n- [Hrvatski](https:\/\/hr.wikipedia.org\/wiki\/Eulerov_identitet \"Eulerov identitet – Croatian\")\n- [Magyar](https:\/\/hu.wikipedia.org\/wiki\/Euler-%C3%B6sszef%C3%BCgg%C3%A9s \"Euler-összefüggés – Hungarian\")\n- [Հայերեն](https:\/\/hy.wikipedia.org\/wiki\/%D4%B7%D5%B5%D5%AC%D5%A5%D6%80%D5%AB_%D5%B6%D5%B8%D6%82%D5%B5%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6 \"Էյլերի նույնություն – Armenian\")\n- [Bahasa Indonesia](https:\/\/id.wikipedia.org\/wiki\/Identitas_Euler \"Identitas Euler – Indonesian\")\n- [Italiano](https:\/\/it.wikipedia.org\/wiki\/Identit%C3%A0_di_Eulero \"Identità di Eulero – Italian\")\n- [日本語](https:\/\/ja.wikipedia.org\/wiki\/%E3%82%AA%E3%82%A4%E3%83%A9%E3%83%BC%E3%81%AE%E7%AD%89%E5%BC%8F \"オイラーの等式 – Japanese\")\n- [한국어](https:\/\/ko.wikipedia.org\/wiki\/%EC%98%A4%EC%9D%BC%EB%9F%AC_%ED%95%AD%EB%93%B1%EC%8B%9D \"오일러 항등식 – Korean\")\n- [Latina](https:\/\/la.wikipedia.org\/wiki\/Euleri_identitas \"Euleri identitas – Latin\")\n- [Lombard](https:\/\/lmo.wikipedia.org\/wiki\/Identit%C3%A0_de_Euler \"Identità de Euler – Lombard\")\n- [Bahasa Melayu](https:\/\/ms.wikipedia.org\/wiki\/Identiti_Euler \"Identiti Euler – Malay\")\n- [Nederlands](https:\/\/nl.wikipedia.org\/wiki\/Identiteit_van_Euler \"Identiteit van Euler – Dutch\")\n- [Norsk bokmål](https:\/\/no.wikipedia.org\/wiki\/Eulers_likhet \"Eulers likhet – Norwegian Bokmål\")\n- [Occitan](https:\/\/oc.wikipedia.org\/wiki\/Identitat_d%27Euler \"Identitat d'Euler – Occitan\")\n- [Polski](https:\/\/pl.wikipedia.org\/wiki\/Wz%C3%B3r_Eulera#To%C5%BCsamo%C5%9B%C4%87_Eulera \"Wzór Eulera – Polish\")\n- [Piemontèis](https:\/\/pms.wikipedia.org\/wiki\/Identit%C3%A0_d%27Euler \"Identità d'Euler – Piedmontese\")\n- [Português](https:\/\/pt.wikipedia.org\/wiki\/Identidade_de_Euler \"Identidade de Euler – Portuguese\")\n- [Русский](https:\/\/ru.wikipedia.org\/wiki\/%D0%A2%D0%BE%D0%B6%D0%B4%D0%B5%D1%81%D1%82%D0%B2%D0%BE_%D0%AD%D0%B9%D0%BB%D0%B5%D1%80%D0%B0_\\(%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D1%8B%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7\\) \"Тождество Эйлера (комплексный анализ) – Russian\")\n- [Simple English](https:\/\/simple.wikipedia.org\/wiki\/Euler%27s_identity \"Euler's identity – Simple English\")\n- [Slovenščina](https:\/\/sl.wikipedia.org\/wiki\/Eulerjeva_ena%C4%8Dba \"Eulerjeva enačba – Slovenian\")\n- [Српски \/ srpski](https:\/\/sr.wikipedia.org\/wiki\/%D0%9E%D1%98%D0%BB%D0%B5%D1%80%D0%BE%D0%B2_%D0%B8%D0%B4%D0%B5%D0%BD%D1%82%D0%B8%D1%82%D0%B5%D1%82 \"Ојлеров идентитет – Serbian\")\n- [Svenska](https:\/\/sv.wikipedia.org\/wiki\/Eulers_identitet \"Eulers identitet – Swedish\")\n- [தமிழ்](https:\/\/ta.wikipedia.org\/wiki\/%E0%AE%86%E0%AE%AF%E0%AF%8D%E0%AE%B2%E0%AE%B0%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%8A%E0%AE%B0%E0%AF%81%E0%AE%AE%E0%AF%88 \"ஆய்லரின் முற்றொருமை – Tamil\")\n- [ไทย](https:\/\/th.wikipedia.org\/wiki\/%E0%B9%80%E0%B8%AD%E0%B8%81%E0%B8%A5%E0%B8%B1%E0%B8%81%E0%B8%A9%E0%B8%93%E0%B9%8C%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%AD%E0%B9%87%E0%B8%AD%E0%B8%A2%E0%B9%80%E0%B8%A5%E0%B8%AD%E0%B8%A3%E0%B9%8C \"เอกลักษณ์ของอ็อยเลอร์ – Thai\")\n- [Türkçe](https:\/\/tr.wikipedia.org\/wiki\/Euler_%C3%B6zde%C5%9Fli%C4%9Fi \"Euler özdeşliği – Turkish\")\n- [Татарча \/ tatarça](https:\/\/tt.wikipedia.org\/wiki\/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80_%D0%B1%D0%B5%D1%80%D0%B4%D3%99%D0%B9%D0%BB%D0%B5%D0%B3%D0%B5 \"Эйлер бердәйлеге – Tatar\")\n- [Українська](https:\/\/uk.wikipedia.org\/wiki\/%D0%A2%D0%BE%D1%82%D0%BE%D0%B6%D0%BD%D1%96%D1%81%D1%82%D1%8C_%D0%95%D0%B9%D0%BB%D0%B5%D1%80%D0%B0 \"Тотожність Ейлера – Ukrainian\")\n- [Tiếng Việt](https:\/\/vi.wikipedia.org\/wiki\/%C4%90%E1%BB%93ng_nh%E1%BA%A5t_th%E1%BB%A9c_Euler \"Đồng nhất thức Euler – Vietnamese\")\n- [吴语](https:\/\/wuu.wikipedia.org\/wiki\/%E6%AC%A7%E6%8B%89%E6%81%92%E7%AD%89%E5%BC%8F \"欧拉恒等式 – Wu\")\n- [ייִדיש](https:\/\/yi.wikipedia.org\/wiki\/%D7%90%D7%95%D7%99%D7%9C%D7%A2%D7%A8-%D7%90%D7%99%D7%93%D7%A2%D7%A0%D7%98%D7%99%D7%98%D7%A2%D7%98 \"אוילער-אידענטיטעט – Yiddish\")\n- [粵語](https:\/\/zh-yue.wikipedia.org\/wiki\/%E6%AD%90%E6%8B%89%E6%81%86%E7%AD%89%E5%BC%8F \"歐拉恆等式 – Cantonese\")\n- [中文](https:\/\/zh.wikipedia.org\/wiki\/%E6%AD%90%E6%8B%89%E6%81%86%E7%AD%89%E5%BC%8F \"歐拉恆等式 – Chinese\")\n\n[Edit links](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q204819#sitelinks-wikipedia \"Edit interlanguage links\")\n\n- [Article](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity \"View the content page [c]\")\n- [Talk](https:\/\/en.wikipedia.org\/wiki\/Talk:Euler%27s_identity \"Discuss improvements to the content page [t]\")\n\nEnglish\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=edit \"Edit this page [e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=history \"Past revisions of this page [h]\")\n\nTools\n\nTools\n\nmove to sidebar\n\nhide\n\nActions\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity)\n- 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[g]\")\n\nAppearance\n\nmove to sidebar\n\nhide\n\nFrom Wikipedia, the free encyclopedia\n\nMathematical equation linking e, i and π\n\nFor other uses, see [List of topics named after Leonhard Euler § Identities](https:\/\/en.wikipedia.org\/wiki\/List_of_topics_named_after_Leonhard_Euler#Identities \"List of topics named after Leonhard Euler\").\n\n|   |\n|---|\n| Part of [a series of articles](https:\/\/en.wikipedia.org\/wiki\/Category:E_\\(mathematical_constant\\) \"Category:E (mathematical constant)\") on the |\n| mathematical constant [e](https:\/\/en.wikipedia.org\/wiki\/E_\\(mathematical_constant\\) \"E (mathematical constant)\") |\n| [![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/7\/71\/Euler%27s_formula.svg\/250px-Euler%27s_formula.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Euler%27s_formula.svg) |\n| Properties |\n| [Natural logarithm](https:\/\/en.wikipedia.org\/wiki\/Natural_logarithm \"Natural logarithm\") [Exponential function](https:\/\/en.wikipedia.org\/wiki\/Exponential_function \"Exponential function\") |\n| Applications |\n| [Compound interest](https:\/\/en.wikipedia.org\/wiki\/Compound_interest \"Compound interest\") [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\") [Euler's identity]() [Half-lifes](https:\/\/en.wikipedia.org\/wiki\/Half-life \"Half-life\") Exponential [growth](https:\/\/en.wikipedia.org\/wiki\/Exponential_growth \"Exponential growth\") and [decay](https:\/\/en.wikipedia.org\/wiki\/Exponential_decay \"Exponential decay\") [Probability theory](https:\/\/en.wikipedia.org\/wiki\/Probability_theory \"Probability theory\") [Bernoulli trial](https:\/\/en.wikipedia.org\/wiki\/Bernoulli_trial \"Bernoulli trial\") [Derangement](https:\/\/en.wikipedia.org\/wiki\/Derangement \"Derangement\") [Probability density function](https:\/\/en.wikipedia.org\/wiki\/Probability_density_function \"Probability density function\") |\n| Defining e |\n| [proof that e is irrational](https:\/\/en.wikipedia.org\/wiki\/Proof_that_e_is_irrational \"Proof that e is irrational\") [representations of e](https:\/\/en.wikipedia.org\/wiki\/List_of_representations_of_e \"List of representations of e\") [Lindemann–Weierstrass theorem](https:\/\/en.wikipedia.org\/wiki\/Lindemann%E2%80%93Weierstrass_theorem \"Lindemann–Weierstrass theorem\") |\n| People |\n| [Jakob Bernoulli](https:\/\/en.wikipedia.org\/wiki\/Jakob_Bernoulli \"Jakob Bernoulli\") [John Napier](https:\/\/en.wikipedia.org\/wiki\/John_Napier \"John Napier\") [Leonhard Euler](https:\/\/en.wikipedia.org\/wiki\/Leonhard_Euler \"Leonhard Euler\") |\n| Related topics |\n| [Schanuel's conjecture](https:\/\/en.wikipedia.org\/wiki\/Schanuel%27s_conjecture \"Schanuel's conjecture\") |\n| [v](https:\/\/en.wikipedia.org\/wiki\/Template:E_\\(mathematical_constant\\) \"Template:E (mathematical constant)\") [t](https:\/\/en.wikipedia.org\/wiki\/Template_talk:E_\\(mathematical_constant\\) \"Template talk:E (mathematical constant)\") [e](https:\/\/en.wikipedia.org\/wiki\/Special:EditPage\/Template:E_\\(mathematical_constant\\) \"Special:EditPage\/Template:E (mathematical constant)\") |\n\nIn [mathematics](https:\/\/en.wikipedia.org\/wiki\/Mathematics \"Mathematics\"), **Euler's identity**[\\[note 1\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-3) (also known as **Euler's equation**) is the [equality](https:\/\/en.wikipedia.org\/wiki\/Equality_\\(mathematics\\) \"Equality (mathematics)\") e i π \\+ 1 \\= 0 {\\\\displaystyle e^{i\\\\pi }+1=0} ![{\\\\displaystyle e^{i\\\\pi }+1=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a7464809a40f9e486de3a454745f572fbf8bb256) where\n\n- e\n  \n  {\\\\displaystyle e}\n  \n  ![{\\\\displaystyle e}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/cd253103f0876afc68ebead27a5aa9867d927467)\n  is [Euler's number](https:\/\/en.wikipedia.org\/wiki\/E_\\(mathematical_constant\\) \"E (mathematical constant)\"), the base of [natural logarithms](https:\/\/en.wikipedia.org\/wiki\/Natural_logarithm \"Natural logarithm\"),\n- i\n  \n  {\\\\displaystyle i}\n  \n  ![{\\\\displaystyle i}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/add78d8608ad86e54951b8c8bd6c8d8416533d20)\n  is the [imaginary unit](https:\/\/en.wikipedia.org\/wiki\/Imaginary_unit \"Imaginary unit\"), which by definition satisfies\n  i\n  \n  2\n  \n  \\=\n  \n  −\n  \n  1\n  \n  {\\\\displaystyle i^{2}=-1}\n  \n  ![{\\\\displaystyle i^{2}=-1}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/88e98a401d352e5037d5043028e2d7f449e83fa6)\n  , and\n- π\n  \n  {\\\\displaystyle \\\\pi }\n  \n  ![{\\\\displaystyle \\\\pi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9be4ba0bb8df3af72e90a0535fabcc17431e540a)\n  is [pi](https:\/\/en.wikipedia.org\/wiki\/Pi \"Pi\"), the ratio of the [circumference](https:\/\/en.wikipedia.org\/wiki\/Circumference \"Circumference\") of a circle to its [diameter](https:\/\/en.wikipedia.org\/wiki\/Diameter \"Diameter\").\n\nEuler's identity is named after the Swiss mathematician [Leonhard Euler](https:\/\/en.wikipedia.org\/wiki\/Leonhard_Euler \"Leonhard Euler\"). It is a special case of [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\") e i x \\= cos ⁡ x \\+ i sin ⁡ x {\\\\displaystyle e^{ix}=\\\\cos x+i\\\\sin x} ![{\\\\displaystyle e^{ix}=\\\\cos x+i\\\\sin x}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4907c0489ab08ce550c7700a1587d4634801dff8) when evaluated for x \\= π {\\\\displaystyle x=\\\\pi } ![{\\\\displaystyle x=\\\\pi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9b4512a97fa6b7772825e2c887e010a99e217005). Euler's identity is considered an exemplar of [mathematical beauty](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty \"Mathematical beauty\"), as it shows a profound connection between the most fundamental numbers in mathematics.\n\n## Mathematical beauty\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=edit&section=1 \"Edit section: Mathematical beauty\")\\]\n\nEuler's identity is often cited as an example of deep [mathematical beauty](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty \"Mathematical beauty\").[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-Gallagher2014-4) Three of the basic [arithmetic](https:\/\/en.wikipedia.org\/wiki\/Arithmetic \"Arithmetic\") operations occur exactly once each: [addition](https:\/\/en.wikipedia.org\/wiki\/Addition \"Addition\"), [multiplication](https:\/\/en.wikipedia.org\/wiki\/Multiplication \"Multiplication\"), and [exponentiation](https:\/\/en.wikipedia.org\/wiki\/Exponentiation \"Exponentiation\"). The identity also links five fundamental [mathematical constants](https:\/\/en.wikipedia.org\/wiki\/Mathematical_constant \"Mathematical constant\"):[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-5)\n\n- The [number 0](https:\/\/en.wikipedia.org\/wiki\/0 \"0\"), the [additive identity](https:\/\/en.wikipedia.org\/wiki\/Additive_identity \"Additive identity\")\n- The [number 1](https:\/\/en.wikipedia.org\/wiki\/1 \"1\"), the [multiplicative identity](https:\/\/en.wikipedia.org\/wiki\/Multiplicative_identity \"Multiplicative identity\")\n- The [number π](https:\/\/en.wikipedia.org\/wiki\/Pi \"Pi\") (π = 3.14159...), the fundamental [circle](https:\/\/en.wikipedia.org\/wiki\/Circle \"Circle\") constant\n- The [number e](https:\/\/en.wikipedia.org\/wiki\/E_\\(mathematical_constant\\) \"E (mathematical constant)\") (*e* = 2.71828...), also known as Euler's number, which occurs widely in [mathematical analysis](https:\/\/en.wikipedia.org\/wiki\/Mathematical_analysis \"Mathematical analysis\")\n- The [number i](https:\/\/en.wikipedia.org\/wiki\/Imaginary_unit \"Imaginary unit\"), the [imaginary unit](https:\/\/en.wikipedia.org\/wiki\/Imaginary_unit \"Imaginary unit\") such that *i*2 = −1\n\nThe equation is often given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.\n\n[Stanford University](https:\/\/en.wikipedia.org\/wiki\/Stanford_University \"Stanford University\") mathematics professor [Keith Devlin](https:\/\/en.wikipedia.org\/wiki\/Keith_Devlin \"Keith Devlin\") has said, \"like a Shakespearean [sonnet](https:\/\/en.wikipedia.org\/wiki\/Sonnet \"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\".[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-6) [Paul Nahin](https:\/\/en.wikipedia.org\/wiki\/Paul_Nahin \"Paul Nahin\"), a professor emeritus at the [University of New Hampshire](https:\/\/en.wikipedia.org\/wiki\/University_of_New_Hampshire \"University of New Hampshire\") who wrote a book dedicated to [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\") and its applications in [Fourier analysis](https:\/\/en.wikipedia.org\/wiki\/Fourier_analysis \"Fourier analysis\"), said Euler's identity is \"of exquisite beauty\".[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-7)\n\nMathematics writer [Constance Reid](https:\/\/en.wikipedia.org\/wiki\/Constance_Reid \"Constance Reid\") has said that Euler's identity is \"the most famous formula in all mathematics\".[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-8) [Benjamin Peirce](https:\/\/en.wikipedia.org\/wiki\/Benjamin_Peirce \"Benjamin Peirce\"), a 19th-century American philosopher, mathematician, and professor at [Harvard University](https:\/\/en.wikipedia.org\/wiki\/Harvard_University \"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\".[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-9)\n\nA 1990 poll of readers by *[The Mathematical Intelligencer](https:\/\/en.wikipedia.org\/wiki\/The_Mathematical_Intelligencer \"The Mathematical Intelligencer\")* named Euler's identity the \"most beautiful theorem in mathematics\".[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-10) In a 2004 poll of readers by *[Physics World](https:\/\/en.wikipedia.org\/wiki\/Physics_World \"Physics World\")*, Euler's identity tied with [Maxwell's equations](https:\/\/en.wikipedia.org\/wiki\/Maxwell%27s_equations \"Maxwell's equations\") (of [electromagnetism](https:\/\/en.wikipedia.org\/wiki\/Electromagnetism \"Electromagnetism\")) as the \"greatest equation ever\".[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-11)\n\nAt least three books in [popular mathematics](https:\/\/en.wikipedia.org\/wiki\/Popular_mathematics \"Popular mathematics\") have been published about Euler's identity:\n\n- *Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills*, by [Paul Nahin](https:\/\/en.wikipedia.org\/wiki\/Paul_Nahin \"Paul Nahin\") (2011)[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-12)\n- *A Most Elegant Equation: Euler's formula and the beauty of mathematics*, by David Stipp (2017)[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-13)\n- *Euler's Pioneering Equation: The most beautiful theorem in mathematics*, by [Robin Wilson](https:\/\/en.wikipedia.org\/wiki\/Robin_Wilson_\\(mathematician\\) \"Robin Wilson (mathematician)\") (2018)[\\[13\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-14)\n\n## Explanations\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=edit&section=2 \"Edit section: Explanations\")\\]\n\n### Imaginary exponents\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=edit&section=3 \"Edit section: Imaginary exponents\")\\]\n\nMain article: [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\")\n\nSee also: [Complex exponents with a positive real base](https:\/\/en.wikipedia.org\/wiki\/Exponentiation#Complex_exponents_with_a_positive_real_base \"Exponentiation\")\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/0\/0e\/ExpIPi.gif\/250px-ExpIPi.gif)](https:\/\/en.wikipedia.org\/wiki\/File:ExpIPi.gif)\n\nIn this animation N takes various increasing values from 1 to 100. The computation of (1 + ⁠*iπ*\/*N*⁠)*N* is displayed as the combined effect of N repeated multiplications in the [complex plane](https:\/\/en.wikipedia.org\/wiki\/Complex_plane \"Complex plane\"), with the final point being the actual value of\n\n(1 +\n\n⁠*iπ*\/*N*⁠)*N*\n\n. It can be seen that as N gets larger\n\n(1 +\n\n⁠*iπ*\/*N*⁠)*N*\n\napproaches a limit of −1.\n\nEuler's identity asserts that e i π {\\\\displaystyle e^{i\\\\pi }} ![{\\\\displaystyle e^{i\\\\pi }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/14a02b8464db88c495fd187a3e181865822b21de) is equal to −1. The expression e i π {\\\\displaystyle e^{i\\\\pi }} ![{\\\\displaystyle e^{i\\\\pi }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/14a02b8464db88c495fd187a3e181865822b21de) is a special case of the expression e z {\\\\displaystyle e^{z}} ![{\\\\displaystyle e^{z}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f4772def31b56e642df3e4d1160cadff3d80ba45), where *z* is any [complex number](https:\/\/en.wikipedia.org\/wiki\/Complex_number \"Complex number\"). In general, e z {\\\\displaystyle e^{z}} ![{\\\\displaystyle e^{z}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f4772def31b56e642df3e4d1160cadff3d80ba45) is defined for complex *z* by extending one of the [definitions of the exponential function](https:\/\/en.wikipedia.org\/wiki\/Characterizations_of_the_exponential_function \"Characterizations of the exponential function\") from real exponents to complex exponents. For example, one common definition is:\n\ne z \\= lim n → ∞ ( 1 \\+ z n ) n . {\\\\displaystyle e^{z}=\\\\lim \\_{n\\\\to \\\\infty }\\\\left(1+{\\\\frac {z}{n}}\\\\right)^{n}.} ![{\\\\displaystyle e^{z}=\\\\lim \\_{n\\\\to \\\\infty }\\\\left(1+{\\\\frac {z}{n}}\\\\right)^{n}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/40ce3c19f0325e83628f164efe1da5a60bc80b49)\n\nEuler's identity therefore states that the limit, as *n* approaches infinity, of ( 1 \\+ i π n ) n {\\\\displaystyle (1+{\\\\tfrac {i\\\\pi }{n}})^{n}} ![{\\\\displaystyle (1+{\\\\tfrac {i\\\\pi }{n}})^{n}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/25296a2d06e81ecdc74113c1c10cd0b2649aa59a) is equal to −1. This limit is illustrated in the animation to the right.\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/7\/71\/Euler%27s_formula.svg\/250px-Euler%27s_formula.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Euler%27s_formula.svg)\n\nEuler's formula for a general angle\n\nEuler's identity is a [special case](https:\/\/en.wikipedia.org\/wiki\/Special_case \"Special case\") of [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\"), which states that for any [real number](https:\/\/en.wikipedia.org\/wiki\/Real_number \"Real number\") *x*,\n\ne i x \\= cos ⁡ x \\+ i sin ⁡ x {\\\\displaystyle e^{ix}=\\\\cos x+i\\\\sin x} ![{\\\\displaystyle e^{ix}=\\\\cos x+i\\\\sin x}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4907c0489ab08ce550c7700a1587d4634801dff8)\n\nwhere the inputs of the [trigonometric functions](https:\/\/en.wikipedia.org\/wiki\/Trigonometry \"Trigonometry\") sine and cosine are given in [radians](https:\/\/en.wikipedia.org\/wiki\/Radian \"Radian\").\n\nIn particular, when *x* = *π*,\n\ne i π \\= cos ⁡ π \\+ i sin ⁡ π . {\\\\displaystyle e^{i\\\\pi }=\\\\cos \\\\pi +i\\\\sin \\\\pi .} ![{\\\\displaystyle e^{i\\\\pi }=\\\\cos \\\\pi +i\\\\sin \\\\pi .}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/809c56fce05afb471ebb280a3588c6f8e14198fd)\n\nSince\n\ncos ⁡ π \\= − 1 {\\\\displaystyle \\\\cos \\\\pi =-1} ![{\\\\displaystyle \\\\cos \\\\pi =-1}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2e009a140f4ee530848ede1ead96aeac3fc54a23)\n\nand\n\nsin ⁡ π \\= 0 , {\\\\displaystyle \\\\sin \\\\pi =0,} ![{\\\\displaystyle \\\\sin \\\\pi =0,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f7ca27f5c501370bbc23e7203d7ca1fe4764b1fa)\n\nit follows that\n\ne i π \\= − 1 \\+ 0 i , {\\\\displaystyle e^{i\\\\pi }=-1+0i,} ![{\\\\displaystyle e^{i\\\\pi }=-1+0i,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/24995de3ca85e21e47689f00348ae2661e091317)\n\nwhich yields Euler's identity:\n\ne i π \\+ 1 \\= 0\\. {\\\\displaystyle e^{i\\\\pi }+1=0.} ![{\\\\displaystyle e^{i\\\\pi }+1=0.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e76caf050d8bc37cd2350c40517face26de5ecb7)\n\n### Geometric interpretation\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=edit&section=4 \"Edit section: Geometric interpretation\")\\]\n\nAny complex number z \\= x \\+ i y {\\\\displaystyle z=x+iy} ![{\\\\displaystyle z=x+iy}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/08e90bb6b36fef59c6113eed2a08f10d77240741) can be represented by the point ( x , y ) {\\\\displaystyle (x,y)} ![{\\\\displaystyle (x,y)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/41cf50e4a314ca8e2c30964baa8d26e5be7a9386) on the [complex plane](https:\/\/en.wikipedia.org\/wiki\/Complex_plane \"Complex plane\"). This point can also be represented in [polar coordinates](https:\/\/en.wikipedia.org\/wiki\/Complex_number#Polar_complex_plane \"Complex number\") as ( r , θ ) {\\\\displaystyle (r,\\\\theta )} ![{\\\\displaystyle (r,\\\\theta )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ed8396fdc359fb06c93722137c959e7496e47ed6) , where r is the absolute value of z (distance from the origin), and θ {\\\\displaystyle \\\\theta } ![{\\\\displaystyle \\\\theta }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af) 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 ( r cos ⁡ θ , r sin ⁡ θ ) {\\\\displaystyle (r\\\\cos \\\\theta ,r\\\\sin \\\\theta )} ![{\\\\displaystyle (r\\\\cos \\\\theta ,r\\\\sin \\\\theta )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/560ed9ef39aed01ca74a50d7b5cae2cced5de2e1) , implying that z \\= r ( cos ⁡ θ \\+ i sin ⁡ θ ) {\\\\displaystyle z=r(\\\\cos \\\\theta +i\\\\sin \\\\theta )} ![{\\\\displaystyle z=r(\\\\cos \\\\theta +i\\\\sin \\\\theta )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/350014e34fca54c3ea03a63825d6afa7a743ffa7) . According to Euler's formula, this is equivalent to saying z \\= r e i θ {\\\\displaystyle z=re^{i\\\\theta }} ![{\\\\displaystyle z=re^{i\\\\theta }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e5b36ddd965193c2b7d6ea24a7c3678814d0dc8d) .\n\nEuler's identity says that − 1 \\= e i π {\\\\displaystyle -1=e^{i\\\\pi }} ![{\\\\displaystyle -1=e^{i\\\\pi }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/127dc145108d2d2b72f37efba2ce9e924be890a1) . Since e i π {\\\\displaystyle e^{i\\\\pi }} ![{\\\\displaystyle e^{i\\\\pi }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/14a02b8464db88c495fd187a3e181865822b21de) is r e i θ {\\\\displaystyle re^{i\\\\theta }} ![{\\\\displaystyle re^{i\\\\theta }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/304e85daab31cb0b8918d182576f69764bf0b3f3) for r = 1 and θ \\= π {\\\\displaystyle \\\\theta =\\\\pi } ![{\\\\displaystyle \\\\theta =\\\\pi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ab4db588619489e27efb50a1d0d5aa016c49ce15) , 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 π {\\\\displaystyle \\\\pi } ![{\\\\displaystyle \\\\pi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9be4ba0bb8df3af72e90a0535fabcc17431e540a) radians.\n\nAdditionally, when any complex number z is [multiplied](https:\/\/en.wikipedia.org\/wiki\/Complex_number#Multiplication_and_division_in_polar_form \"Complex number\") by e i θ {\\\\displaystyle e^{i\\\\theta }} ![{\\\\displaystyle e^{i\\\\theta }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7b4b8e67ee479d68e0e5040aaf87eff99214c90f) , it has the effect of rotating z {\\\\displaystyle z} ![{\\\\displaystyle z}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bf368e72c009decd9b6686ee84a375632e11de98) counterclockwise by an angle of θ {\\\\displaystyle \\\\theta } ![{\\\\displaystyle \\\\theta }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af) 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 π {\\\\displaystyle \\\\pi } ![{\\\\displaystyle \\\\pi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9be4ba0bb8df3af72e90a0535fabcc17431e540a) radians around the origin has the same effect as reflecting the point across the origin. Similarly, setting θ {\\\\displaystyle \\\\theta } ![{\\\\displaystyle \\\\theta }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af) equal to 2 π {\\\\displaystyle 2\\\\pi } ![{\\\\displaystyle 2\\\\pi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/73efd1f6493490b058097060a572606d2c550a06) yields the related equation e 2 π i \\= 1 {\\\\displaystyle e^{2\\\\pi i}=1} ![{\\\\displaystyle e^{2\\\\pi i}=1}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6da4586a614d29da6632bc76b2b4417996d62e88) , which can be interpreted as saying that rotating any point by one [turn](https:\/\/en.wikipedia.org\/wiki\/Turn_\\(angle\\) \"Turn (angle)\") around the origin returns it to its original position.\n\n## Generalizations\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=edit&section=5 \"Edit section: Generalizations\")\\]\n\nEuler's identity is also a special case of the more general identity that the nth [roots of unity](https:\/\/en.wikipedia.org\/wiki\/Roots_of_unity \"Roots of unity\"), for *n* \\> 1, add up to 0:\n\n∑ k \\= 0 n − 1 e 2 π i k n \\= 0\\. {\\\\displaystyle \\\\sum \\_{k=0}^{n-1}e^{2\\\\pi i{\\\\frac {k}{n}}}=0.} ![{\\\\displaystyle \\\\sum \\_{k=0}^{n-1}e^{2\\\\pi i{\\\\frac {k}{n}}}=0.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7f6bd22aa5cd424ee8298ca5090892d6ac900082)\n\nEuler's identity is the case where *n* = 2.\n\nA similar identity also applies to [quaternion exponential](https:\/\/en.wikipedia.org\/wiki\/Quaternion#Exponential,_logarithm,_and_power_functions \"Quaternion\"): let {*i*, *j*, *k*} be the basis [quaternions](https:\/\/en.wikipedia.org\/wiki\/Quaternion \"Quaternion\"); then, e 1 3 ( i ± j ± k ) π \\+ 1 \\= 0\\. {\\\\displaystyle e^{{\\\\frac {1}{\\\\sqrt {3}}}(i\\\\pm j\\\\pm k)\\\\pi }+1=0.} ![{\\\\displaystyle e^{{\\\\frac {1}{\\\\sqrt {3}}}(i\\\\pm j\\\\pm k)\\\\pi }+1=0.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b7425d865b67e05665dfa275bd0515129138f460)\n\nMore generally, let q be a quaternion with a zero real part and a norm equal to 1; that is, q \\= a i \\+ b j \\+ c k , {\\\\displaystyle q=ai+bj+ck,} ![{\\\\displaystyle q=ai+bj+ck,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f29e3114ba4e82b5c0f75a8c1141e2843715cdb7) with a 2 \\+ b 2 \\+ c 2 \\= 1\\. {\\\\displaystyle a^{2}+b^{2}+c^{2}=1.} ![{\\\\displaystyle a^{2}+b^{2}+c^{2}=1.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/11f2994d705edbe9d0ad3ffcb854ca63d3fb6745) Then one has e q π \\+ 1 \\= 0\\. {\\\\displaystyle e^{q\\\\pi }+1=0.} ![{\\\\displaystyle e^{q\\\\pi }+1=0.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1df35eeca09cd9e9557ac9218c992857a00fb3b7)\n\nThe same formula applies to [octonions](https:\/\/en.wikipedia.org\/wiki\/Octonion \"Octonion\"), with a zero real part and a norm equal to 1. These formulas are a direct generalization of Euler's identity, since i {\\\\displaystyle i} ![{\\\\displaystyle i}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/add78d8608ad86e54951b8c8bd6c8d8416533d20) and − i {\\\\displaystyle -i} ![{\\\\displaystyle -i}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/91fddb9f89a520937db3a8821575068cdcc76f60) are the only complex numbers with a zero real part and a norm (absolute value) equal to 1.\n\n## History\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=edit&section=6 \"Edit section: History\")\\]\n\nEuler's identity is a direct result of [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\"), first published in his monumental 1748 work of mathematical analysis, *[Introductio in analysin infinitorum](https:\/\/en.wikipedia.org\/wiki\/Introductio_in_analysin_infinitorum \"Introductio in analysin infinitorum\")*,[\\[14\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-15) 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.[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-Sandifer2007-16)\n\n[Robin Wilson](https:\/\/en.wikipedia.org\/wiki\/Robin_Wilson_\\(mathematician\\) \"Robin Wilson (mathematician)\") writes:[\\[16\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-17)\n\n> We've seen how \\[Euler's identity\\] can easily be deduced from results of [Johann Bernoulli](https:\/\/en.wikipedia.org\/wiki\/Johann_Bernoulli \"Johann Bernoulli\") and [Roger Cotes](https:\/\/en.wikipedia.org\/wiki\/Roger_Cotes \"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](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\")\\], *eix* = cos *x* + *i* sin *x*. Moreover, it seems to be unknown who first stated the result explicitly\n\n## See also\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=edit&section=7 \"Edit section: See also\")\\]\n\n- [![icon](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/3e\/Nuvola_apps_edu_mathematics_blue-p.svg\/40px-Nuvola_apps_edu_mathematics_blue-p.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Nuvola_apps_edu_mathematics_blue-p.svg)[Mathematics portal](https:\/\/en.wikipedia.org\/wiki\/Portal:Mathematics \"Portal:Mathematics\")\n\n- [De Moivre's formula](https:\/\/en.wikipedia.org\/wiki\/De_Moivre%27s_formula \"De Moivre's formula\")\n- [Exponential function](https:\/\/en.wikipedia.org\/wiki\/Exponential_function \"Exponential function\")\n- [Gelfond's constant](https:\/\/en.wikipedia.org\/wiki\/Gelfond%27s_constant \"Gelfond's constant\")\n\n## Notes\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=edit&section=8 \"Edit section: Notes\")\\]\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-3)** The term \"Euler's identity\" (or \"Euler identity\") is also used elsewhere to refer to other concepts, including the related general formula *e**ix* = cos *x* + *i* sin *x*,[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-1) and the [Euler product formula](https:\/\/en.wikipedia.org\/wiki\/Riemann_zeta_function#Euler's_product_formula \"Riemann zeta function\").[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-EOM-2) See also [List of topics named after Leonhard Euler](https:\/\/en.wikipedia.org\/wiki\/List_of_topics_named_after_Leonhard_Euler \"List of topics named after Leonhard Euler\").\n\n## References\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=edit&section=9 \"Edit section: References\")\\]\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-1)** Dunham, 1999, [p. xxiv](https:\/\/books.google.com\/books?id=uKOVNvGOkhQC&pg=PR24).\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-EOM_2-0)**\n   Stepanov, S. A. (2001) \\[1994\\]. [\"Euler identity\"](https:\/\/www.encyclopediaofmath.org\/index.php?title=Euler_identity&oldid=33574). *[Encyclopedia of Mathematics](https:\/\/en.wikipedia.org\/wiki\/Encyclopedia_of_Mathematics \"Encyclopedia of Mathematics\")*. [EMS Press](https:\/\/en.wikipedia.org\/wiki\/European_Mathematical_Society \"European Mathematical Society\").\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-Gallagher2014_4-0)**\n   Gallagher, James (13 February 2014). [\"Mathematics: Why the brain sees maths as beauty\"](https:\/\/www.bbc.co.uk\/news\/science-environment-26151062). *[BBC News](https:\/\/en.wikipedia.org\/wiki\/BBC_News \"BBC News\")*. Retrieved 26 December 2017.\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-5)** Paulos, 1992, p. 117.\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-6)** Nahin, 2006, [p. 1](https:\/\/books.google.com\/books?id=GvSg5HQ7WPcC&pg=PA1).\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-7)** Nahin, 2006, p. xxxii.\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-8)** Reid, chapter *e*.\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-9)** Maor, [p. 160](https:\/\/books.google.com\/books?id=eIsyLD_bDKkC&pg=PA160), and Kasner & Newman, [p. 103–104](https:\/\/books.google.com\/books?id=Ad8hAx-6m9oC&pg=PA103).\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-10)** Wells, 1990.\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-11)** Crease, 2004.\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-12)**\n    Nahin, Paul (2011). *Dr. Euler's fabulous formula: cures many mathematical ills*. Princeton University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-691-11822-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-691-11822-2 \"Special:BookSources\/978-0-691-11822-2\")\n    \n    .\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-13)**\n    Stipp, David (2017). *A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics* (First ed.). Basic Books. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-465-09377-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-465-09377-9 \"Special:BookSources\/978-0-465-09377-9\")\n    \n    .\n13. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-14)**\n    Wilson, Robin (2018). *Euler's pioneering equation: the most beautiful theorem in mathematics*. Oxford: Oxford University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-19-879493-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-19-879493-6 \"Special:BookSources\/978-0-19-879493-6\")\n    \n    .\n14. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-15)** Conway & Guy, p. [254–255](https:\/\/books.google.com\/books?id=0--3rcO7dMYC&pg=PA254).\n15. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-Sandifer2007_16-0)** Sandifer, p. 4.\n16. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-17)** Wilson, p. 151-152.\n\n### Sources\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=edit&section=10 \"Edit section: Sources\")\\]\n\n- [Conway, John H.](https:\/\/en.wikipedia.org\/wiki\/John_Horton_Conway \"John Horton Conway\"), and [Guy, Richard K.](https:\/\/en.wikipedia.org\/wiki\/Richard_K._Guy \"Richard K. Guy\") (1996), *[The Book of Numbers](https:\/\/en.wikipedia.org\/wiki\/The_Book_of_Numbers_\\(math_book\\) \"The Book of Numbers (math book)\")*, Springer\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [978-0-387-97993-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-387-97993-9 \"Special:BookSources\/978-0-387-97993-9\")\n- [Crease, Robert P.](https:\/\/en.wikipedia.org\/wiki\/Robert_P._Crease \"Robert P. Crease\") (10 May 2004), \"[The greatest equations ever](http:\/\/physicsworld.com\/cws\/article\/print\/2004\/may\/10\/the-greatest-equations-ever)\", *[Physics World](https:\/\/en.wikipedia.org\/wiki\/Physics_World \"Physics World\")* \\[registration required\\]\n- [Dunham, William](https:\/\/en.wikipedia.org\/wiki\/William_Dunham_\\(mathematician\\) \"William Dunham (mathematician)\") (1999), *Euler: The Master of Us All*, [Mathematical Association of America](https:\/\/en.wikipedia.org\/wiki\/Mathematical_Association_of_America \"Mathematical Association of America\")\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [978-0-88385-328-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-88385-328-3 \"Special:BookSources\/978-0-88385-328-3\")\n- [Euler, Leonhard](https:\/\/en.wikipedia.org\/wiki\/Euler,_Leonhard \"Euler, Leonhard\"), *Complete work.* *[Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus](http:\/\/gallica.bnf.fr\/ark:\/12148\/bpt6k69587.image.r=%22has+celeberrimas+formulas%22.f169.langEN)*, Leipzig: B. G. Teubneri\n- [Kasner, E.](https:\/\/en.wikipedia.org\/wiki\/Edward_Kasner \"Edward Kasner\"), and [Newman, J.](https:\/\/en.wikipedia.org\/wiki\/James_R._Newman \"James R. Newman\") (1940), *[Mathematics and the Imagination](https:\/\/en.wikipedia.org\/wiki\/Mathematics_and_the_Imagination \"Mathematics and the Imagination\")*, [Simon & Schuster](https:\/\/en.wikipedia.org\/wiki\/Simon_%26_Schuster \"Simon & Schuster\")\n- [Maor, Eli](https:\/\/en.wikipedia.org\/wiki\/Eli_Maor \"Eli Maor\") (1998), *e: The Story of a number*, [Princeton University Press](https:\/\/en.wikipedia.org\/wiki\/Princeton_University_Press \"Princeton University Press\")\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-691-05854-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-691-05854-7 \"Special:BookSources\/0-691-05854-7\")\n- Nahin, Paul J. (2006), *Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills*, [Princeton University Press](https:\/\/en.wikipedia.org\/wiki\/Princeton_University_Press \"Princeton University Press\")\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [978-0-691-11822-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-691-11822-2 \"Special:BookSources\/978-0-691-11822-2\")\n- [Paulos, John Allen](https:\/\/en.wikipedia.org\/wiki\/John_Allen_Paulos \"John Allen Paulos\") (1992), *Beyond Numeracy: An Uncommon Dictionary of Mathematics*, [Penguin Books](https:\/\/en.wikipedia.org\/wiki\/Penguin_Books \"Penguin Books\")\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-14-014574-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-14-014574-5 \"Special:BookSources\/0-14-014574-5\")\n- Reid, Constance (various editions), *[From Zero to Infinity](https:\/\/en.wikipedia.org\/wiki\/From_Zero_to_Infinity \"From Zero to Infinity\")*, [Mathematical Association of America](https:\/\/en.wikipedia.org\/wiki\/Mathematical_Association_of_America \"Mathematical Association of America\")\n- Sandifer, C. Edward (2007), *[Euler's Greatest Hits](https:\/\/books.google.com\/books?id=sohHs7ExOsYC&pg=PA4)*, [Mathematical Association of America](https:\/\/en.wikipedia.org\/wiki\/Mathematical_Association_of_America \"Mathematical Association of America\")\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [978-0-88385-563-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-88385-563-8 \"Special:BookSources\/978-0-88385-563-8\")\n- Stipp, David (2017). *A Most Elegant Equation: Euler's formula and the beauty of mathematics*. [Basic Books](https:\/\/en.wikipedia.org\/wiki\/Basic_Books \"Basic Books\").\n- [Wells, David](https:\/\/en.wikipedia.org\/w\/index.php?title=David_G._Wells&action=edit&redlink=1 \"David G. Wells (page does not exist)\") (1990). \"Are these the most beautiful?\". *[The Mathematical Intelligencer](https:\/\/en.wikipedia.org\/wiki\/The_Mathematical_Intelligencer \"The Mathematical Intelligencer\")*. **12** (3): 37–41\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/BF03024015](https:\/\/doi.org\/10.1007%2FBF03024015). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [121503263](https:\/\/api.semanticscholar.org\/CorpusID:121503263).\n- [Wilson, Robin](https:\/\/en.wikipedia.org\/wiki\/Robin_Wilson_\\(mathematician\\) \"Robin Wilson (mathematician)\") (2018). *Euler's Pioneering Equation: The most beautiful theorem in mathematics*. [Oxford University Press](https:\/\/en.wikipedia.org\/wiki\/Oxford_University_Press \"Oxford University Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-192-51406-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-192-51406-6 \"Special:BookSources\/978-0-192-51406-6\")\n  \n  .\n- [Zeki, S.](https:\/\/en.wikipedia.org\/wiki\/Semir_Zeki \"Semir Zeki\"); Romaya, J. P.; Benincasa, D. M. T.; [Atiyah, M. F.](https:\/\/en.wikipedia.org\/wiki\/Michael_Atiyah \"Michael Atiyah\") (2014). \"The experience of mathematical beauty and its neural correlates\". *Frontiers in Human Neuroscience*. **8**: 68. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.3389\/fnhum.2014.00068](https:\/\/doi.org\/10.3389%2Ffnhum.2014.00068). [PMC](https:\/\/en.wikipedia.org\/wiki\/PMC_\\(identifier\\) \"PMC (identifier)\") [3923150](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3923150). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [24592230](https:\/\/pubmed.ncbi.nlm.nih.gov\/24592230).\n\n## External links\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=edit&section=11 \"Edit section: External links\")\\]\n\n![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/f\/fa\/Wikiquote-logo.svg\/40px-Wikiquote-logo.svg.png)\n\nWikiquote has quotations related to ***[Euler's identity](https:\/\/en.wikiquote.org\/wiki\/Euler%27s_identity \"q:Euler's identity\")***.\n\n- [Intuitive understanding of Euler's formula](http:\/\/betterexplained.com\/articles\/intuitive-understanding-of-eulers-formula\/)\n\n| [v](https:\/\/en.wikipedia.org\/wiki\/Template:Leonhard_Euler \"Template:Leonhard Euler\") [t](https:\/\/en.wikipedia.org\/wiki\/Template_talk:Leonhard_Euler \"Template talk:Leonhard Euler\") [e](https:\/\/en.wikipedia.org\/wiki\/Special:EditPage\/Template:Leonhard_Euler \"Special:EditPage\/Template:Leonhard Euler\")[Leonhard Euler](https:\/\/en.wikipedia.org\/wiki\/Leonhard_Euler \"Leonhard Euler\") |\n|---|\n| [Euler–Lagrange equation](https:\/\/en.wikipedia.org\/wiki\/Euler%E2%80%93Lagrange_equation \"Euler–Lagrange equation\") [Euler–Lotka equation](https:\/\/en.wikipedia.org\/wiki\/Euler%E2%80%93Lotka_equation \"Euler–Lotka equation\") [Euler–Maclaurin formula](https:\/\/en.wikipedia.org\/wiki\/Euler%E2%80%93Maclaurin_formula \"Euler–Maclaurin formula\") [Euler–Maruyama method](https:\/\/en.wikipedia.org\/wiki\/Euler%E2%80%93Maruyama_method \"Euler–Maruyama method\") [Euler–Mascheroni constant](https:\/\/en.wikipedia.org\/wiki\/Euler%E2%80%93Mascheroni_constant \"Euler–Mascheroni constant\") [Euler–Poisson–Darboux equation](https:\/\/en.wikipedia.org\/wiki\/Euler%E2%80%93Poisson%E2%80%93Darboux_equation \"Euler–Poisson–Darboux equation\") [Euler–Rodrigues formula](https:\/\/en.wikipedia.org\/wiki\/Euler%E2%80%93Rodrigues_formula \"Euler–Rodrigues formula\") [Euler–Tricomi equation](https:\/\/en.wikipedia.org\/wiki\/Euler%E2%80%93Tricomi_equation \"Euler–Tricomi equation\") [Euler's continued fraction formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_continued_fraction_formula \"Euler's continued fraction formula\") [Euler's critical load](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_critical_load \"Euler's critical load\") [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\") [Euler's four-square identity](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_four-square_identity \"Euler's four-square identity\") [Euler's identity]() [Euler's pump and turbine equation](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_pump_and_turbine_equation \"Euler's pump and turbine equation\") [Euler's rotation theorem](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_rotation_theorem \"Euler's rotation theorem\") [Euler's sum of powers conjecture](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_sum_of_powers_conjecture \"Euler's sum of powers conjecture\") [Euler's theorem](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_theorem \"Euler's theorem\") [Euler equations (fluid dynamics)](https:\/\/en.wikipedia.org\/wiki\/Euler_equations_\\(fluid_dynamics\\) \"Euler equations (fluid dynamics)\") [Euler function](https:\/\/en.wikipedia.org\/wiki\/Euler_function \"Euler function\") [Euler method](https:\/\/en.wikipedia.org\/wiki\/Euler_method \"Euler method\") [Euler numbers](https:\/\/en.wikipedia.org\/wiki\/Euler_numbers \"Euler numbers\") [Euler number (physics)](https:\/\/en.wikipedia.org\/wiki\/Euler_number_\\(physics\\) \"Euler number (physics)\") [Euler–Bernoulli beam theory](https:\/\/en.wikipedia.org\/wiki\/Euler%E2%80%93Bernoulli_beam_theory \"Euler–Bernoulli beam theory\") [Namesakes](https:\/\/en.wikipedia.org\/wiki\/List_of_things_named_after_Leonhard_Euler \"List of things named after Leonhard Euler\") |\n| 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Wikipedia® is a registered trademark of the [Wikimedia Foundation, Inc.](https:\/\/wikimediafoundation.org\/), a non-profit organization.\n\n- [Privacy policy](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Privacy_policy)\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About)\n- [Disclaimers](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:General_disclaimer)\n- [Contact Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us)\n- [Legal & safety contacts](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Legal:Wikimedia_Foundation_Legal_and_Safety_Contact_Information)\n- [Code of Conduct](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Universal_Code_of_Conduct)\n- [Developers](https:\/\/developer.wikimedia.org\/)\n- [Statistics](https:\/\/stats.wikimedia.org\/#\/en.wikipedia.org)\n- [Cookie statement](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Cookie_statement)\n- [Mobile view](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&mobileaction=toggle_view_mobile)\n\n- [![Wikimedia Foundation](https:\/\/en.wikipedia.org\/static\/images\/footer\/wikimedia.svg)](https:\/\/www.wikimedia.org\/)\n- [![Powered by MediaWiki](https:\/\/en.wikipedia.org\/w\/resources\/assets\/mediawiki_compact.svg)](https:\/\/www.mediawiki.org\/)\n\nSearch\n\nToggle the table of contents\n\nEuler's identity\n\n54 languages\n\n[Add topic](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity)","attrs_readable_markdown":"From Wikipedia, the free encyclopedia\n\nIn [mathematics](https:\/\/en.wikipedia.org\/wiki\/Mathematics \"Mathematics\"), **Euler's identity**[\\[note 1\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-3) (also known as **Euler's equation**) is the [equality](https:\/\/en.wikipedia.org\/wiki\/Equality_\\(mathematics\\) \"Equality (mathematics)\") ![{\\\\displaystyle e^{i\\\\pi }+1=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a7464809a40f9e486de3a454745f572fbf8bb256) where\n\nEuler's identity is named after the Swiss mathematician [Leonhard Euler](https:\/\/en.wikipedia.org\/wiki\/Leonhard_Euler \"Leonhard Euler\"). It is a special case of [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\") ![{\\\\displaystyle e^{ix}=\\\\cos x+i\\\\sin x}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4907c0489ab08ce550c7700a1587d4634801dff8) when evaluated for ![{\\\\displaystyle x=\\\\pi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9b4512a97fa6b7772825e2c887e010a99e217005). Euler's identity is considered an exemplar of [mathematical beauty](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty \"Mathematical beauty\"), as it shows a profound connection between the most fundamental numbers in mathematics.\n\n## Mathematical beauty\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=edit&section=1 \"Edit section: Mathematical beauty\")\\]\n\nEuler's identity is often cited as an example of deep [mathematical beauty](https:\/\/en.wikipedia.org\/wiki\/Mathematical_beauty \"Mathematical beauty\").[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-Gallagher2014-4) Three of the basic [arithmetic](https:\/\/en.wikipedia.org\/wiki\/Arithmetic \"Arithmetic\") operations occur exactly once each: [addition](https:\/\/en.wikipedia.org\/wiki\/Addition \"Addition\"), [multiplication](https:\/\/en.wikipedia.org\/wiki\/Multiplication \"Multiplication\"), and [exponentiation](https:\/\/en.wikipedia.org\/wiki\/Exponentiation \"Exponentiation\"). The identity also links five fundamental [mathematical constants](https:\/\/en.wikipedia.org\/wiki\/Mathematical_constant \"Mathematical constant\"):[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-5)\n\n- The [number 0](https:\/\/en.wikipedia.org\/wiki\/0 \"0\"), the [additive identity](https:\/\/en.wikipedia.org\/wiki\/Additive_identity \"Additive identity\")\n- The [number 1](https:\/\/en.wikipedia.org\/wiki\/1 \"1\"), the [multiplicative identity](https:\/\/en.wikipedia.org\/wiki\/Multiplicative_identity \"Multiplicative identity\")\n- The [number π](https:\/\/en.wikipedia.org\/wiki\/Pi \"Pi\") (π = 3.14159...), the fundamental [circle](https:\/\/en.wikipedia.org\/wiki\/Circle \"Circle\") constant\n- The [number e](https:\/\/en.wikipedia.org\/wiki\/E_\\(mathematical_constant\\) \"E (mathematical constant)\") (*e* = 2.71828...), also known as Euler's number, which occurs widely in [mathematical analysis](https:\/\/en.wikipedia.org\/wiki\/Mathematical_analysis \"Mathematical analysis\")\n- The [number i](https:\/\/en.wikipedia.org\/wiki\/Imaginary_unit \"Imaginary unit\"), the [imaginary unit](https:\/\/en.wikipedia.org\/wiki\/Imaginary_unit \"Imaginary unit\") such that *i*2 = −1\n\nThe equation is often given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.\n\n[Stanford University](https:\/\/en.wikipedia.org\/wiki\/Stanford_University \"Stanford University\") mathematics professor [Keith Devlin](https:\/\/en.wikipedia.org\/wiki\/Keith_Devlin \"Keith Devlin\") has said, \"like a Shakespearean [sonnet](https:\/\/en.wikipedia.org\/wiki\/Sonnet \"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\".[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-6) [Paul Nahin](https:\/\/en.wikipedia.org\/wiki\/Paul_Nahin \"Paul Nahin\"), a professor emeritus at the [University of New Hampshire](https:\/\/en.wikipedia.org\/wiki\/University_of_New_Hampshire \"University of New Hampshire\") who wrote a book dedicated to [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\") and its applications in [Fourier analysis](https:\/\/en.wikipedia.org\/wiki\/Fourier_analysis \"Fourier analysis\"), said Euler's identity is \"of exquisite beauty\".[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-7)\n\nMathematics writer [Constance Reid](https:\/\/en.wikipedia.org\/wiki\/Constance_Reid \"Constance Reid\") has said that Euler's identity is \"the most famous formula in all mathematics\".[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-8) [Benjamin Peirce](https:\/\/en.wikipedia.org\/wiki\/Benjamin_Peirce \"Benjamin Peirce\"), a 19th-century American philosopher, mathematician, and professor at [Harvard University](https:\/\/en.wikipedia.org\/wiki\/Harvard_University \"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\".[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-9)\n\nA 1990 poll of readers by *[The Mathematical Intelligencer](https:\/\/en.wikipedia.org\/wiki\/The_Mathematical_Intelligencer \"The Mathematical Intelligencer\")* named Euler's identity the \"most beautiful theorem in mathematics\".[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-10) In a 2004 poll of readers by *[Physics World](https:\/\/en.wikipedia.org\/wiki\/Physics_World \"Physics World\")*, Euler's identity tied with [Maxwell's equations](https:\/\/en.wikipedia.org\/wiki\/Maxwell%27s_equations \"Maxwell's equations\") (of [electromagnetism](https:\/\/en.wikipedia.org\/wiki\/Electromagnetism \"Electromagnetism\")) as the \"greatest equation ever\".[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-11)\n\nAt least three books in [popular mathematics](https:\/\/en.wikipedia.org\/wiki\/Popular_mathematics \"Popular mathematics\") have been published about Euler's identity:\n\n- *Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills*, by [Paul Nahin](https:\/\/en.wikipedia.org\/wiki\/Paul_Nahin \"Paul Nahin\") (2011)[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-12)\n- *A Most Elegant Equation: Euler's formula and the beauty of mathematics*, by David Stipp (2017)[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-13)\n- *Euler's Pioneering Equation: The most beautiful theorem in mathematics*, by [Robin Wilson](https:\/\/en.wikipedia.org\/wiki\/Robin_Wilson_\\(mathematician\\) \"Robin Wilson (mathematician)\") (2018)[\\[13\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-14)\n\n### Imaginary exponents\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=edit&section=3 \"Edit section: Imaginary exponents\")\\]\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/0\/0e\/ExpIPi.gif\/250px-ExpIPi.gif)](https:\/\/en.wikipedia.org\/wiki\/File:ExpIPi.gif)\n\nIn this animation N takes various increasing values from 1 to 100. The computation of (1 + ⁠*iπ*\/*N*⁠)*N* is displayed as the combined effect of N repeated multiplications in the [complex plane](https:\/\/en.wikipedia.org\/wiki\/Complex_plane \"Complex plane\"), with the final point being the actual value of (1 + ⁠*iπ*\/*N*⁠)*N*. It can be seen that as N gets larger (1 + ⁠*iπ*\/*N*⁠)*N* approaches a limit of −1.\n\nEuler's identity asserts that ![{\\\\displaystyle e^{i\\\\pi }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/14a02b8464db88c495fd187a3e181865822b21de) is equal to −1. The expression ![{\\\\displaystyle e^{i\\\\pi }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/14a02b8464db88c495fd187a3e181865822b21de) is a special case of the expression ![{\\\\displaystyle e^{z}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f4772def31b56e642df3e4d1160cadff3d80ba45), where *z* is any [complex number](https:\/\/en.wikipedia.org\/wiki\/Complex_number \"Complex number\"). In general, ![{\\\\displaystyle e^{z}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f4772def31b56e642df3e4d1160cadff3d80ba45) is defined for complex *z* by extending one of the [definitions of the exponential function](https:\/\/en.wikipedia.org\/wiki\/Characterizations_of_the_exponential_function \"Characterizations of the exponential function\") from real exponents to complex exponents. For example, one common definition is:\n\n![{\\\\displaystyle e^{z}=\\\\lim \\_{n\\\\to \\\\infty }\\\\left(1+{\\\\frac {z}{n}}\\\\right)^{n}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/40ce3c19f0325e83628f164efe1da5a60bc80b49)\n\nEuler's identity therefore states that the limit, as *n* approaches infinity, of ![{\\\\displaystyle (1+{\\\\tfrac {i\\\\pi }{n}})^{n}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/25296a2d06e81ecdc74113c1c10cd0b2649aa59a) is equal to −1. This limit is illustrated in the animation to the right.\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/7\/71\/Euler%27s_formula.svg\/250px-Euler%27s_formula.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Euler%27s_formula.svg)\n\nEuler's formula for a general angle\n\nEuler's identity is a [special case](https:\/\/en.wikipedia.org\/wiki\/Special_case \"Special case\") of [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\"), which states that for any [real number](https:\/\/en.wikipedia.org\/wiki\/Real_number \"Real number\") *x*,\n\n![{\\\\displaystyle e^{ix}=\\\\cos x+i\\\\sin x}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4907c0489ab08ce550c7700a1587d4634801dff8)\n\nwhere the inputs of the [trigonometric functions](https:\/\/en.wikipedia.org\/wiki\/Trigonometry \"Trigonometry\") sine and cosine are given in [radians](https:\/\/en.wikipedia.org\/wiki\/Radian \"Radian\").\n\nIn particular, when *x* = *π*,\n\n![{\\\\displaystyle e^{i\\\\pi }=\\\\cos \\\\pi +i\\\\sin \\\\pi .}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/809c56fce05afb471ebb280a3588c6f8e14198fd)\n\nSince\n\n![{\\\\displaystyle \\\\cos \\\\pi =-1}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2e009a140f4ee530848ede1ead96aeac3fc54a23)\n\nand\n\n![{\\\\displaystyle \\\\sin \\\\pi =0,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f7ca27f5c501370bbc23e7203d7ca1fe4764b1fa)\n\nit follows that\n\n![{\\\\displaystyle e^{i\\\\pi }=-1+0i,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/24995de3ca85e21e47689f00348ae2661e091317)\n\nwhich yields Euler's identity:\n\n![{\\\\displaystyle e^{i\\\\pi }+1=0.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e76caf050d8bc37cd2350c40517face26de5ecb7)\n\n### Geometric interpretation\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Euler%27s_identity&action=edit&section=4 \"Edit section: Geometric interpretation\")\\]\n\nAny complex number ![{\\\\displaystyle z=x+iy}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/08e90bb6b36fef59c6113eed2a08f10d77240741) can be represented by the point ![{\\\\displaystyle (x,y)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/41cf50e4a314ca8e2c30964baa8d26e5be7a9386) on the [complex plane](https:\/\/en.wikipedia.org\/wiki\/Complex_plane \"Complex plane\"). This point can also be represented in [polar coordinates](https:\/\/en.wikipedia.org\/wiki\/Complex_number#Polar_complex_plane \"Complex number\") as ![{\\\\displaystyle (r,\\\\theta )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ed8396fdc359fb06c93722137c959e7496e47ed6), where r is the absolute value of z (distance from the origin), and ![{\\\\displaystyle \\\\theta }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af) 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 ![{\\\\displaystyle (r\\\\cos \\\\theta ,r\\\\sin \\\\theta )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/560ed9ef39aed01ca74a50d7b5cae2cced5de2e1), implying that ![{\\\\displaystyle z=r(\\\\cos \\\\theta +i\\\\sin \\\\theta )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/350014e34fca54c3ea03a63825d6afa7a743ffa7). According to Euler's formula, this is equivalent to saying ![{\\\\displaystyle z=re^{i\\\\theta }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e5b36ddd965193c2b7d6ea24a7c3678814d0dc8d).\n\nEuler's identity says that ![{\\\\displaystyle -1=e^{i\\\\pi }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/127dc145108d2d2b72f37efba2ce9e924be890a1). Since ![{\\\\displaystyle e^{i\\\\pi }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/14a02b8464db88c495fd187a3e181865822b21de) is ![{\\\\displaystyle re^{i\\\\theta }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/304e85daab31cb0b8918d182576f69764bf0b3f3) for r = 1 and ![{\\\\displaystyle \\\\theta =\\\\pi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ab4db588619489e27efb50a1d0d5aa016c49ce15), 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 ![{\\\\displaystyle \\\\pi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9be4ba0bb8df3af72e90a0535fabcc17431e540a) radians.\n\nAdditionally, when any complex number z is [multiplied](https:\/\/en.wikipedia.org\/wiki\/Complex_number#Multiplication_and_division_in_polar_form \"Complex number\") by ![{\\\\displaystyle e^{i\\\\theta }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7b4b8e67ee479d68e0e5040aaf87eff99214c90f), it has the effect of rotating ![{\\\\displaystyle z}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bf368e72c009decd9b6686ee84a375632e11de98) counterclockwise by an angle of ![{\\\\displaystyle \\\\theta }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af) 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 ![{\\\\displaystyle \\\\pi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9be4ba0bb8df3af72e90a0535fabcc17431e540a) radians around the origin has the same effect as reflecting the point across the origin. Similarly, setting ![{\\\\displaystyle \\\\theta }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af) equal to ![{\\\\displaystyle 2\\\\pi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/73efd1f6493490b058097060a572606d2c550a06) yields the related equation ![{\\\\displaystyle e^{2\\\\pi i}=1}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6da4586a614d29da6632bc76b2b4417996d62e88), which can be interpreted as saying that rotating any point by one [turn](https:\/\/en.wikipedia.org\/wiki\/Turn_\\(angle\\) \"Turn (angle)\") around the origin returns it to its original position.\n\nEuler's identity is also a special case of the more general identity that the nth [roots of unity](https:\/\/en.wikipedia.org\/wiki\/Roots_of_unity \"Roots of unity\"), for *n* \\> 1, add up to 0:\n\n![{\\\\displaystyle \\\\sum \\_{k=0}^{n-1}e^{2\\\\pi i{\\\\frac {k}{n}}}=0.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7f6bd22aa5cd424ee8298ca5090892d6ac900082)\n\nEuler's identity is the case where *n* = 2.\n\nA similar identity also applies to [quaternion exponential](https:\/\/en.wikipedia.org\/wiki\/Quaternion#Exponential,_logarithm,_and_power_functions \"Quaternion\"): let {*i*, *j*, *k*} be the basis [quaternions](https:\/\/en.wikipedia.org\/wiki\/Quaternion \"Quaternion\"); then, ![{\\\\displaystyle e^{{\\\\frac {1}{\\\\sqrt {3}}}(i\\\\pm j\\\\pm k)\\\\pi }+1=0.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b7425d865b67e05665dfa275bd0515129138f460)\n\nMore generally, let q be a quaternion with a zero real part and a norm equal to 1; that is, ![{\\\\displaystyle q=ai+bj+ck,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f29e3114ba4e82b5c0f75a8c1141e2843715cdb7) with ![{\\\\displaystyle a^{2}+b^{2}+c^{2}=1.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/11f2994d705edbe9d0ad3ffcb854ca63d3fb6745) Then one has ![{\\\\displaystyle e^{q\\\\pi }+1=0.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1df35eeca09cd9e9557ac9218c992857a00fb3b7)\n\nThe same formula applies to [octonions](https:\/\/en.wikipedia.org\/wiki\/Octonion \"Octonion\"), with a zero real part and a norm equal to 1. These formulas are a direct generalization of Euler's identity, since ![{\\\\displaystyle i}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/add78d8608ad86e54951b8c8bd6c8d8416533d20) and ![{\\\\displaystyle -i}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/91fddb9f89a520937db3a8821575068cdcc76f60) are the only complex numbers with a zero real part and a norm (absolute value) equal to 1.\n\nEuler's identity is a direct result of [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\"), first published in his monumental 1748 work of mathematical analysis, *[Introductio in analysin infinitorum](https:\/\/en.wikipedia.org\/wiki\/Introductio_in_analysin_infinitorum \"Introductio in analysin infinitorum\")*,[\\[14\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-15) 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.[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-Sandifer2007-16)\n\n[Robin Wilson](https:\/\/en.wikipedia.org\/wiki\/Robin_Wilson_\\(mathematician\\) \"Robin Wilson (mathematician)\") writes:[\\[16\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-17)\n\n> We've seen how \\[Euler's identity\\] can easily be deduced from results of [Johann Bernoulli](https:\/\/en.wikipedia.org\/wiki\/Johann_Bernoulli \"Johann Bernoulli\") and [Roger Cotes](https:\/\/en.wikipedia.org\/wiki\/Roger_Cotes \"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](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\")\\], *eix* = cos *x* + *i* sin *x*. Moreover, it seems to be unknown who first stated the result explicitly\n\n- [De Moivre's formula](https:\/\/en.wikipedia.org\/wiki\/De_Moivre%27s_formula \"De Moivre's formula\")\n- [Exponential function](https:\/\/en.wikipedia.org\/wiki\/Exponential_function \"Exponential function\")\n- [Gelfond's constant](https:\/\/en.wikipedia.org\/wiki\/Gelfond%27s_constant \"Gelfond's constant\")\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-3)** The term \"Euler's identity\" (or \"Euler identity\") is also used elsewhere to refer to other concepts, including the related general formula *e**ix* = cos *x* + *i* sin *x*,[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-1) and the [Euler product formula](https:\/\/en.wikipedia.org\/wiki\/Riemann_zeta_function#Euler's_product_formula \"Riemann zeta function\").[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_note-EOM-2) See also [List of topics named after Leonhard Euler](https:\/\/en.wikipedia.org\/wiki\/List_of_topics_named_after_Leonhard_Euler \"List of topics named after Leonhard Euler\").\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-1)** Dunham, 1999, [p. xxiv](https:\/\/books.google.com\/books?id=uKOVNvGOkhQC&pg=PR24).\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-EOM_2-0)**\n   Stepanov, S. A. (2001) \\[1994\\]. [\"Euler identity\"](https:\/\/www.encyclopediaofmath.org\/index.php?title=Euler_identity&oldid=33574). *[Encyclopedia of Mathematics](https:\/\/en.wikipedia.org\/wiki\/Encyclopedia_of_Mathematics \"Encyclopedia of Mathematics\")*. [EMS Press](https:\/\/en.wikipedia.org\/wiki\/European_Mathematical_Society \"European Mathematical Society\").\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-Gallagher2014_4-0)**\n   Gallagher, James (13 February 2014). [\"Mathematics: Why the brain sees maths as beauty\"](https:\/\/www.bbc.co.uk\/news\/science-environment-26151062). *[BBC News](https:\/\/en.wikipedia.org\/wiki\/BBC_News \"BBC News\")*. Retrieved 26 December 2017.\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-5)** Paulos, 1992, p. 117.\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-6)** Nahin, 2006, [p. 1](https:\/\/books.google.com\/books?id=GvSg5HQ7WPcC&pg=PA1).\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-7)** Nahin, 2006, p. xxxii.\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-8)** Reid, chapter *e*.\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-9)** Maor, [p. 160](https:\/\/books.google.com\/books?id=eIsyLD_bDKkC&pg=PA160), and Kasner & Newman, [p. 103–104](https:\/\/books.google.com\/books?id=Ad8hAx-6m9oC&pg=PA103).\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-10)** Wells, 1990.\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-11)** Crease, 2004.\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-12)**\n    Nahin, Paul (2011). *Dr. Euler's fabulous formula: cures many mathematical ills*. Princeton University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-691-11822-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-691-11822-2 \"Special:BookSources\/978-0-691-11822-2\")\n    \n    .\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-13)**\n    Stipp, David (2017). *A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics* (First ed.). Basic Books. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-465-09377-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-465-09377-9 \"Special:BookSources\/978-0-465-09377-9\")\n    \n    .\n13. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-14)**\n    Wilson, Robin (2018). *Euler's pioneering equation: the most beautiful theorem in mathematics*. Oxford: Oxford University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-19-879493-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-19-879493-6 \"Special:BookSources\/978-0-19-879493-6\")\n    \n    .\n14. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-15)** Conway & Guy, p. [254–255](https:\/\/books.google.com\/books?id=0--3rcO7dMYC&pg=PA254).\n15. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-Sandifer2007_16-0)** Sandifer, p. 4.\n16. **[^](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_identity#cite_ref-17)** Wilson, p. 151-152.\n\n- [Conway, John H.](https:\/\/en.wikipedia.org\/wiki\/John_Horton_Conway \"John Horton Conway\"), and [Guy, Richard K.](https:\/\/en.wikipedia.org\/wiki\/Richard_K._Guy \"Richard K. Guy\") (1996), *[The Book of Numbers](https:\/\/en.wikipedia.org\/wiki\/The_Book_of_Numbers_\\(math_book\\) \"The Book of Numbers (math book)\")*, Springer [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [978-0-387-97993-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-387-97993-9 \"Special:BookSources\/978-0-387-97993-9\")\n- [Crease, Robert P.](https:\/\/en.wikipedia.org\/wiki\/Robert_P._Crease \"Robert P. Crease\") (10 May 2004), \"[The greatest equations ever](http:\/\/physicsworld.com\/cws\/article\/print\/2004\/may\/10\/the-greatest-equations-ever)\", *[Physics World](https:\/\/en.wikipedia.org\/wiki\/Physics_World \"Physics World\")* \\[registration required\\]\n- [Dunham, William](https:\/\/en.wikipedia.org\/wiki\/William_Dunham_\\(mathematician\\) \"William Dunham (mathematician)\") (1999), *Euler: The Master of Us All*, [Mathematical Association of America](https:\/\/en.wikipedia.org\/wiki\/Mathematical_Association_of_America \"Mathematical Association of America\") [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [978-0-88385-328-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-88385-328-3 \"Special:BookSources\/978-0-88385-328-3\")\n- [Euler, Leonhard](https:\/\/en.wikipedia.org\/wiki\/Euler,_Leonhard \"Euler, Leonhard\"), *Complete work.* *[Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus](http:\/\/gallica.bnf.fr\/ark:\/12148\/bpt6k69587.image.r=%22has+celeberrimas+formulas%22.f169.langEN)*, Leipzig: B. G. Teubneri\n- [Kasner, E.](https:\/\/en.wikipedia.org\/wiki\/Edward_Kasner \"Edward Kasner\"), and [Newman, J.](https:\/\/en.wikipedia.org\/wiki\/James_R._Newman \"James R. Newman\") (1940), *[Mathematics and the Imagination](https:\/\/en.wikipedia.org\/wiki\/Mathematics_and_the_Imagination \"Mathematics and the Imagination\")*, [Simon & Schuster](https:\/\/en.wikipedia.org\/wiki\/Simon_%26_Schuster \"Simon & Schuster\")\n- [Maor, Eli](https:\/\/en.wikipedia.org\/wiki\/Eli_Maor \"Eli Maor\") (1998), *e: The Story of a number*, [Princeton University Press](https:\/\/en.wikipedia.org\/wiki\/Princeton_University_Press \"Princeton University Press\") [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-691-05854-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-691-05854-7 \"Special:BookSources\/0-691-05854-7\")\n- Nahin, Paul J. (2006), *Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills*, [Princeton University Press](https:\/\/en.wikipedia.org\/wiki\/Princeton_University_Press \"Princeton University Press\") [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [978-0-691-11822-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-691-11822-2 \"Special:BookSources\/978-0-691-11822-2\")\n- [Paulos, John Allen](https:\/\/en.wikipedia.org\/wiki\/John_Allen_Paulos \"John Allen Paulos\") (1992), *Beyond Numeracy: An Uncommon Dictionary of Mathematics*, [Penguin Books](https:\/\/en.wikipedia.org\/wiki\/Penguin_Books \"Penguin Books\") [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-14-014574-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-14-014574-5 \"Special:BookSources\/0-14-014574-5\")\n- Reid, Constance (various editions), *[From Zero to Infinity](https:\/\/en.wikipedia.org\/wiki\/From_Zero_to_Infinity \"From Zero to Infinity\")*, [Mathematical Association of America](https:\/\/en.wikipedia.org\/wiki\/Mathematical_Association_of_America \"Mathematical Association of America\")\n- Sandifer, C. Edward (2007), *[Euler's Greatest Hits](https:\/\/books.google.com\/books?id=sohHs7ExOsYC&pg=PA4)*, [Mathematical Association of America](https:\/\/en.wikipedia.org\/wiki\/Mathematical_Association_of_America \"Mathematical Association of America\") [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [978-0-88385-563-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-88385-563-8 \"Special:BookSources\/978-0-88385-563-8\")\n- Stipp, David (2017). *A Most Elegant Equation: Euler's formula and the beauty of mathematics*. [Basic Books](https:\/\/en.wikipedia.org\/wiki\/Basic_Books \"Basic Books\").\n- [Wells, David](https:\/\/en.wikipedia.org\/w\/index.php?title=David_G._Wells&action=edit&redlink=1 \"David G. Wells (page does not exist)\") (1990). \"Are these the most beautiful?\". *[The Mathematical Intelligencer](https:\/\/en.wikipedia.org\/wiki\/The_Mathematical_Intelligencer \"The Mathematical Intelligencer\")*. **12** (3): 37–41\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/BF03024015](https:\/\/doi.org\/10.1007%2FBF03024015). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [121503263](https:\/\/api.semanticscholar.org\/CorpusID:121503263).\n- [Wilson, Robin](https:\/\/en.wikipedia.org\/wiki\/Robin_Wilson_\\(mathematician\\) \"Robin Wilson (mathematician)\") (2018). *Euler's Pioneering Equation: The most beautiful theorem in mathematics*. [Oxford University Press](https:\/\/en.wikipedia.org\/wiki\/Oxford_University_Press \"Oxford University Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-192-51406-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-192-51406-6 \"Special:BookSources\/978-0-192-51406-6\")\n  \n  .\n- [Zeki, S.](https:\/\/en.wikipedia.org\/wiki\/Semir_Zeki \"Semir Zeki\"); Romaya, J. P.; Benincasa, D. M. T.; [Atiyah, M. F.](https:\/\/en.wikipedia.org\/wiki\/Michael_Atiyah \"Michael Atiyah\") (2014). \"The experience of mathematical beauty and its neural correlates\". *Frontiers in Human Neuroscience*. **8**: 68. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.3389\/fnhum.2014.00068](https:\/\/doi.org\/10.3389%2Ffnhum.2014.00068). [PMC](https:\/\/en.wikipedia.org\/wiki\/PMC_\\(identifier\\) \"PMC (identifier)\") [3923150](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3923150). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [24592230](https:\/\/pubmed.ncbi.nlm.nih.gov\/24592230).\n\n- [Intuitive understanding of Euler's formula](http:\/\/betterexplained.com\/articles\/intuitive-understanding-of-eulers-formula\/)","meta_canonical":null}

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Boilerpipe Text	From Wikipedia, the free encyclopedia In mathematics , Euler's identity [ note 1 ] (also known as Euler's equation ) is the equality where Euler's identity is named after the Swiss mathematician Leonhard Euler . It is a special case of Euler's formula when evaluated for . Euler's identity is considered an exemplar of mathematical beauty , as it shows a profound connection between the most fundamental numbers in mathematics. Mathematical beauty [ edit ] Euler's identity is often cited as an example of deep mathematical beauty . [ 3 ] Three of the basic arithmetic operations occur exactly once each: addition , multiplication , and exponentiation . The identity also links five fundamental mathematical constants : [ 4 ] The number 0 , the additive identity The number 1 , the multiplicative identity The number π ( π = 3.14159... ), the fundamental circle constant The number e ( e = 2.71828... ), also known as Euler's number, which occurs widely in mathematical analysis The number i , the imaginary unit such that i 2 = −1 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". [ 5 ] 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". [ 6 ] Mathematics writer Constance Reid has said that Euler's identity is "the most famous formula in all mathematics". [ 7 ] 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". [ 8 ] A 1990 poll of readers by The Mathematical Intelligencer named Euler's identity the "most beautiful theorem in mathematics". [ 9 ] In a 2004 poll of readers by Physics World , Euler's identity tied with Maxwell's equations (of electromagnetism ) as the "greatest equation ever". [ 10 ] 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) [ 11 ] A Most Elegant Equation: Euler's formula and the beauty of mathematics , by David Stipp (2017) [ 12 ] Euler's Pioneering Equation: The most beautiful theorem in mathematics , by Robin Wilson (2018) [ 13 ] Imaginary exponents [ edit ] In this animation N takes various increasing values from 1 to 100. The computation of (1 + ⁠ iπ / N ⁠ ) N is displayed as the combined effect of N repeated multiplications in the complex plane , with the final point being the actual value of (1 + ⁠ iπ / N ⁠ ) N . It can be seen that as N gets larger (1 + ⁠ iπ / N ⁠ ) N approaches a limit of −1. Euler's identity asserts that is equal to −1. The expression is a special case of the expression , where z is any complex number . In general, is defined for complex z by extending one of the definitions of the exponential function from real exponents to complex exponents. For example, one common definition is: Euler's identity therefore states that the limit, as n approaches infinity, of is equal to −1. This limit is illustrated in the animation to the 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 x , where the inputs of the trigonometric functions sine and cosine are given in radians . In particular, when x = π , Since and it follows that which yields Euler's identity: Geometric interpretation [ edit ] Any complex number can be represented by the point on the complex plane . This point can also be represented in polar coordinates as , where r is the absolute value of z (distance from the origin), and 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 , implying that . According to Euler's formula, this is equivalent to saying . Euler's identity says that . Since is for r = 1 and , 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 radians. Additionally, when any complex number z is multiplied by , it has the effect of rotating counterclockwise by an angle of 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 radians around the origin has the same effect as reflecting the point across the origin. Similarly, setting equal to yields the related equation , which can be interpreted as saying that rotating any point by one turn around the origin returns it to its original position. 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: Euler's identity is the case where n = 2 . A similar identity also applies to quaternion exponential : let { i , j , k } be the basis quaternions ; then, More generally, let q be a quaternion with a zero real part and a norm equal to 1; that is, with Then one has The same formula applies to octonions , with a zero real part and a norm equal to 1. These formulas are a direct generalization of Euler's identity, since and are the only complex numbers with a zero real part and a norm (absolute value) equal to 1. Euler's identity is a direct result of Euler's formula , first published in his monumental 1748 work of mathematical analysis, Introductio in analysin infinitorum , [ 14 ] 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. [ 15 ] Robin Wilson writes: [ 16 ] 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 ], e ix = cos x + i sin x . Moreover, it seems to be unknown who first stated the result explicitly De Moivre's formula Exponential function Gelfond's constant ^ The term "Euler's identity" (or "Euler identity") is also used elsewhere to refer to other concepts, including the related general formula e ix = cos x + i sin x , [ 1 ] and the Euler product formula . [ 2 ] See also List of topics named after Leonhard Euler . ^ Dunham, 1999, p. xxiv . ^ Stepanov, S. A. (2001) [1994]. "Euler identity" . Encyclopedia of Mathematics . EMS Press . ^ Gallagher, James (13 February 2014). "Mathematics: Why the brain sees maths as beauty" . BBC News . Retrieved 26 December 2017 . ^ Paulos, 1992, p. 117. ^ Nahin, 2006, p. 1 . ^ Nahin, 2006, p. xxxii. ^ Reid, chapter e . ^ Maor, p. 160 , and Kasner & Newman, p. 103–104 . ^ Wells, 1990. ^ Crease, 2004. ^ Nahin, Paul (2011). Dr. Euler's fabulous formula: cures many mathematical ills . Princeton University Press. ISBN 978-0-691-11822-2 . ^ Stipp, David (2017). A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics (First ed.). Basic Books. ISBN 978-0-465-09377-9 . ^ Wilson, Robin (2018). Euler's pioneering equation: the most beautiful theorem in mathematics . Oxford: Oxford University Press. ISBN 978-0-19-879493-6 . ^ Conway & Guy, p. 254–255 . ^ Sandifer, p. 4. ^ Wilson, p. 151-152. Conway, John H. , and Guy, Richard K. (1996), The Book of Numbers , Springer ISBN 978-0-387-97993-9 Crease, Robert P. (10 May 2004), " The greatest equations ever ", Physics World [registration required] Dunham, William (1999), Euler: The Master of Us All , Mathematical Association of America ISBN 978-0-88385-328-3 Euler, Leonhard , Complete work. Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus , Leipzig: B. G. Teubneri Kasner, E. , and Newman, J. (1940), Mathematics and the Imagination , Simon & Schuster Maor, Eli (1998), 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 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), Euler's Greatest Hits , Mathematical Association of America ISBN 978-0-88385-563-8 Stipp, David (2017). A Most Elegant Equation: Euler's formula and the beauty of mathematics . Basic Books . Wells, David (1990). "Are these the most beautiful?". The Mathematical Intelligencer . 12 (3): 37– 41. doi : 10.1007/BF03024015 . S2CID 121503263 . Wilson, Robin (2018). Euler's Pioneering Equation: The most beautiful theorem in mathematics . Oxford University Press . ISBN 978-0-192-51406-6 . Zeki, S. ; Romaya, J. P.; Benincasa, D. M. T.; Atiyah, M. F. (2014). "The experience of mathematical beauty and its neural correlates". Frontiers in Human Neuroscience . 8 : 68. doi : 10.3389/fnhum.2014.00068 . PMC 3923150 . PMID 24592230 . Intuitive understanding of Euler's formula
Markdown	[Jump to content](https://en.wikipedia.org/wiki/Euler%27s_identity#bodyContent) Main menu Main menu move to sidebar hide Navigation - [Main page](https://en.wikipedia.org/wiki/Main_Page "Visit the main page [z]") - [Contents](https://en.wikipedia.org/wiki/Wikipedia:Contents "Guides to browsing Wikipedia") - [Current events](https://en.wikipedia.org/wiki/Portal:Current_events "Articles related to current events") - [Random article](https://en.wikipedia.org/wiki/Special:Random "Visit a randomly selected article [x]") - [About Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:About "Learn about Wikipedia and how it works") - [Contact us](https://en.wikipedia.org/wiki/Wikipedia:Contact_us "How to contact Wikipedia") Contribute - [Help](https://en.wikipedia.org/wiki/Help:Contents "Guidance on how to use and edit Wikipedia") - [Learn to edit](https://en.wikipedia.org/wiki/Help:Introduction "Learn how to edit Wikipedia") - [Community portal](https://en.wikipedia.org/wiki/Wikipedia:Community_portal "The hub for editors") - [Recent changes](https://en.wikipedia.org/wiki/Special:RecentChanges "A list of recent changes to Wikipedia [r]") - [Upload file](https://en.wikipedia.org/wiki/Wikipedia:File_upload_wizard "Add images or other media for use on Wikipedia") - [Special pages](https://en.wikipedia.org/wiki/Special:SpecialPages "A list of all special pages [q]") [![](https://en.wikipedia.org/static/images/icons/enwiki-25.svg) ![Wikipedia](https://en.wikipedia.org/static/images/mobile/copyright/wikipedia-wordmark-en-25.svg) ![The Free Encyclopedia](https://en.wikipedia.org/static/images/mobile/copyright/wikipedia-tagline-en-25.svg)](https://en.wikipedia.org/wiki/Main_Page) [Search](https://en.wikipedia.org/wiki/Special:Search "Search Wikipedia [f]") Appearance - [Donate](https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en) - [Create account](https://en.wikipedia.org/w/index.php?title=Special:CreateAccount&returnto=Euler%27s+identity "You are encouraged to create an account and log in; however, it is not mandatory") - [Log in](https://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Euler%27s+identity "You're encouraged to log in; however, it's not mandatory. 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[o]") ## Contents move to sidebar hide - [(Top)](https://en.wikipedia.org/wiki/Euler%27s_identity) - [1 Mathematical beauty](https://en.wikipedia.org/wiki/Euler%27s_identity#Mathematical_beauty) - [2 Explanations](https://en.wikipedia.org/wiki/Euler%27s_identity#Explanations) Toggle Explanations subsection - [2\.1 Imaginary exponents](https://en.wikipedia.org/wiki/Euler%27s_identity#Imaginary_exponents) - [2\.2 Geometric interpretation](https://en.wikipedia.org/wiki/Euler%27s_identity#Geometric_interpretation) - [3 Generalizations](https://en.wikipedia.org/wiki/Euler%27s_identity#Generalizations) - [4 History](https://en.wikipedia.org/wiki/Euler%27s_identity#History) - [5 See also](https://en.wikipedia.org/wiki/Euler%27s_identity#See_also) - [6 Notes](https://en.wikipedia.org/wiki/Euler%27s_identity#Notes) - [7 References](https://en.wikipedia.org/wiki/Euler%27s_identity#References) Toggle References subsection - [7\.1 Sources](https://en.wikipedia.org/wiki/Euler%27s_identity#Sources) - [8 External links](https://en.wikipedia.org/wiki/Euler%27s_identity#External_links) Toggle the table of contents # Euler's identity 54 languages - [العربية](https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%B7%D8%A7%D8%A8%D9%82%D8%A9_%D8%A3%D9%88%D9%8A%D9%84%D8%B1 "متطابقة أويلر – Arabic") - [Asturianu](https://ast.wikipedia.org/wiki/Identid%C3%A1_d%27Euler "Identidá d'Euler – Asturian") - [Башҡортса](https://ba.wikipedia.org/wiki/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80_%D1%82%D0%BE%D0%B6%D0%B4%D0%B5%D1%81%D1%82%D0%B2%D0%BE%D2%BB%D1%8B_\(%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BB%D1%8B_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7\) "Эйлер тождествоһы (комплекслы анализ) – Bashkir") - [Български](https://bg.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%BD%D0%B0_%D0%9E%D0%B9%D0%BB%D0%B5%D1%80 "Равенство на Ойлер – Bulgarian") - [বাংলা](https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%AF%E0%A6%BC%E0%A6%B2%E0%A6%BE%E0%A6%B0%E0%A7%87%E0%A6%B0_%E0%A6%85%E0%A6%AD%E0%A7%87%E0%A6%A6 "অয়লারের অভেদ – Bangla") - [Català](https://ca.wikipedia.org/wiki/Identitat_d%27Euler "Identitat d'Euler – Catalan") - [کوردی](https://ckb.wikipedia.org/wiki/%DA%BE%D8%A7%D9%88%D8%A6%DB%95%D9%86%D8%AC%D8%A7%D9%85%DB%8C_%D8%A6%DB%86%DB%8C%D9%84%DB%95%D8%B1 "ھاوئەنجامی ئۆیلەر – Central Kurdish") - [Čeština](https://cs.wikipedia.org/wiki/Eulerova_rovnost "Eulerova rovnost – Czech") - [Чӑвашла](https://cv.wikipedia.org/wiki/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80_%C3%A7%D0%B0%D0%B2%D0%B0%D1%85%D0%BB%C4%83%D1%85%C4%95_\(%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BB%C4%83_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7\) "Эйлер çавахлăхĕ (комплекслă анализ) – Chuvash") - [Deutsch](https://de.wikipedia.org/wiki/Eulersche_Formel#Eulersche_Identit.C3.A4t "Eulersche Formel – German") - [Ελληνικά](https://el.wikipedia.org/wiki/%CE%A4%CE%B1%CF%85%CF%84%CF%8C%CF%84%CE%B7%CF%84%CE%B1_%CF%84%CE%BF%CF%85_%CE%8C%CE%B9%CE%BB%CE%B5%CF%81 "Ταυτότητα του Όιλερ – Greek") - [Esperanto](https://eo.wikipedia.org/wiki/E%C5%ADlera_idento "Eŭlera idento – Esperanto") - [Español](https://es.wikipedia.org/wiki/Identidad_de_Euler "Identidad de Euler – Spanish") - [Euskara](https://eu.wikipedia.org/wiki/Eulerren_identitatea "Eulerren identitatea – Basque") - [فارسی](https://fa.wikipedia.org/wiki/%D8%A7%D8%AA%D8%AD%D8%A7%D8%AF_%D8%A7%D9%88%DB%8C%D9%84%D8%B1 "اتحاد اویلر – Persian") - [Suomi](https://fi.wikipedia.org/wiki/Eulerin_identiteetti "Eulerin identiteetti – Finnish") - [Français](https://fr.wikipedia.org/wiki/Identit%C3%A9_d%27Euler "Identité d'Euler – French") - [Arpetan](https://frp.wikipedia.org/wiki/Identit%C3%A2t_d%27Euler "Identitât d'Euler – Arpitan") - [Nordfriisk](https://frr.wikipedia.org/wiki/Euler_sin_identiteet "Euler sin identiteet – Northern Frisian") - [Gaeilge](https://ga.wikipedia.org/wiki/C%C3%A9annacht_Euler "Céannacht Euler – Irish") - [Galego](https://gl.wikipedia.org/wiki/Identidade_de_Euler "Identidade de Euler – Galician") - [עברית](https://he.wikipedia.org/wiki/%D7%96%D7%94%D7%95%D7%AA_%D7%90%D7%95%D7%99%D7%9C%D7%A8 "זהות אוילר – Hebrew") - [हिन्दी](https://hi.wikipedia.org/wiki/%E0%A4%86%E0%A4%AF%E0%A4%B2%E0%A4%B0_%E0%A4%B8%E0%A4%B0%E0%A5%8D%E0%A4%B5%E0%A4%B8%E0%A4%AE%E0%A4%BF%E0%A4%95%E0%A4%BE "आयलर सर्वसमिका – Hindi") - [Hrvatski](https://hr.wikipedia.org/wiki/Eulerov_identitet "Eulerov identitet – Croatian") - [Magyar](https://hu.wikipedia.org/wiki/Euler-%C3%B6sszef%C3%BCgg%C3%A9s "Euler-összefüggés – Hungarian") - [Հայերեն](https://hy.wikipedia.org/wiki/%D4%B7%D5%B5%D5%AC%D5%A5%D6%80%D5%AB_%D5%B6%D5%B8%D6%82%D5%B5%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6 "Էյլերի նույնություն – Armenian") - [Bahasa Indonesia](https://id.wikipedia.org/wiki/Identitas_Euler "Identitas Euler – Indonesian") - [Italiano](https://it.wikipedia.org/wiki/Identit%C3%A0_di_Eulero "Identità di Eulero – Italian") - [日本語](https://ja.wikipedia.org/wiki/%E3%82%AA%E3%82%A4%E3%83%A9%E3%83%BC%E3%81%AE%E7%AD%89%E5%BC%8F "オイラーの等式 – Japanese") - [한국어](https://ko.wikipedia.org/wiki/%EC%98%A4%EC%9D%BC%EB%9F%AC_%ED%95%AD%EB%93%B1%EC%8B%9D "오일러 항등식 – Korean") - [Latina](https://la.wikipedia.org/wiki/Euleri_identitas "Euleri identitas – Latin") - [Lombard](https://lmo.wikipedia.org/wiki/Identit%C3%A0_de_Euler "Identità de Euler – Lombard") - [Bahasa Melayu](https://ms.wikipedia.org/wiki/Identiti_Euler "Identiti Euler – Malay") - [Nederlands](https://nl.wikipedia.org/wiki/Identiteit_van_Euler "Identiteit van Euler – Dutch") - [Norsk bokmål](https://no.wikipedia.org/wiki/Eulers_likhet "Eulers likhet – Norwegian Bokmål") - [Occitan](https://oc.wikipedia.org/wiki/Identitat_d%27Euler "Identitat d'Euler – Occitan") - [Polski](https://pl.wikipedia.org/wiki/Wz%C3%B3r_Eulera#To%C5%BCsamo%C5%9B%C4%87_Eulera "Wzór Eulera – Polish") - [Piemontèis](https://pms.wikipedia.org/wiki/Identit%C3%A0_d%27Euler "Identità d'Euler – Piedmontese") - [Português](https://pt.wikipedia.org/wiki/Identidade_de_Euler "Identidade de Euler – Portuguese") - [Русский](https://ru.wikipedia.org/wiki/%D0%A2%D0%BE%D0%B6%D0%B4%D0%B5%D1%81%D1%82%D0%B2%D0%BE_%D0%AD%D0%B9%D0%BB%D0%B5%D1%80%D0%B0_\(%D0%BA%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D1%8B%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7\) "Тождество Эйлера (комплексный анализ) – Russian") - [Simple English](https://simple.wikipedia.org/wiki/Euler%27s_identity "Euler's identity – Simple English") - [Slovenščina](https://sl.wikipedia.org/wiki/Eulerjeva_ena%C4%8Dba "Eulerjeva enačba – Slovenian") - [Српски / srpski](https://sr.wikipedia.org/wiki/%D0%9E%D1%98%D0%BB%D0%B5%D1%80%D0%BE%D0%B2_%D0%B8%D0%B4%D0%B5%D0%BD%D1%82%D0%B8%D1%82%D0%B5%D1%82 "Ојлеров идентитет – Serbian") - [Svenska](https://sv.wikipedia.org/wiki/Eulers_identitet "Eulers identitet – Swedish") - [தமிழ்](https://ta.wikipedia.org/wiki/%E0%AE%86%E0%AE%AF%E0%AF%8D%E0%AE%B2%E0%AE%B0%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%8A%E0%AE%B0%E0%AF%81%E0%AE%AE%E0%AF%88 "ஆய்லரின் முற்றொருமை – Tamil") - [ไทย](https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%AD%E0%B8%81%E0%B8%A5%E0%B8%B1%E0%B8%81%E0%B8%A9%E0%B8%93%E0%B9%8C%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%AD%E0%B9%87%E0%B8%AD%E0%B8%A2%E0%B9%80%E0%B8%A5%E0%B8%AD%E0%B8%A3%E0%B9%8C "เอกลักษณ์ของอ็อยเลอร์ – Thai") - [Türkçe](https://tr.wikipedia.org/wiki/Euler_%C3%B6zde%C5%9Fli%C4%9Fi "Euler özdeşliği – Turkish") - [Татарча / tatarça](https://tt.wikipedia.org/wiki/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80_%D0%B1%D0%B5%D1%80%D0%B4%D3%99%D0%B9%D0%BB%D0%B5%D0%B3%D0%B5 "Эйлер бердәйлеге – Tatar") - [Українська](https://uk.wikipedia.org/wiki/%D0%A2%D0%BE%D1%82%D0%BE%D0%B6%D0%BD%D1%96%D1%81%D1%82%D1%8C_%D0%95%D0%B9%D0%BB%D0%B5%D1%80%D0%B0 "Тотожність Ейлера – Ukrainian") - [Tiếng Việt](https://vi.wikipedia.org/wiki/%C4%90%E1%BB%93ng_nh%E1%BA%A5t_th%E1%BB%A9c_Euler "Đồng nhất thức Euler – Vietnamese") - [吴语](https://wuu.wikipedia.org/wiki/%E6%AC%A7%E6%8B%89%E6%81%92%E7%AD%89%E5%BC%8F "欧拉恒等式 – Wu") - [ייִדיש](https://yi.wikipedia.org/wiki/%D7%90%D7%95%D7%99%D7%9C%D7%A2%D7%A8-%D7%90%D7%99%D7%93%D7%A2%D7%A0%D7%98%D7%99%D7%98%D7%A2%D7%98 "אוילער-אידענטיטעט – Yiddish") - [粵語](https://zh-yue.wikipedia.org/wiki/%E6%AD%90%E6%8B%89%E6%81%86%E7%AD%89%E5%BC%8F "歐拉恆等式 – Cantonese") - [中文](https://zh.wikipedia.org/wiki/%E6%AD%90%E6%8B%89%E6%81%86%E7%AD%89%E5%BC%8F "歐拉恆等式 – Chinese") [Edit links](https://www.wikidata.org/wiki/Special:EntityPage/Q204819#sitelinks-wikipedia "Edit interlanguage links") - [Article](https://en.wikipedia.org/wiki/Euler%27s_identity "View the content page [c]") - [Talk](https://en.wikipedia.org/wiki/Talk:Euler%27s_identity "Discuss improvements to the content page [t]") English - [Read](https://en.wikipedia.org/wiki/Euler%27s_identity) - [Edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit "Edit this page [e]") - [View history](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=history "Past revisions of this page [h]") Tools Tools move to sidebar hide Actions - [Read](https://en.wikipedia.org/wiki/Euler%27s_identity) - [Edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit "Edit this page [e]") - [View history](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=history) General - [What links here](https://en.wikipedia.org/wiki/Special:WhatLinksHere/Euler%27s_identity "List of all English Wikipedia pages containing links to this page [j]") - [Related changes](https://en.wikipedia.org/wiki/Special:RecentChangesLinked/Euler%27s_identity "Recent changes in pages linked from this page [k]") - 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[Wikimedia Commons](https://commons.wikimedia.org/wiki/Category:Euler%27s_formula) - [Wikiquote](https://en.wikiquote.org/wiki/Euler%27s_identity) - [Wikidata item](https://www.wikidata.org/wiki/Special:EntityPage/Q204819 "Structured data on this page hosted by Wikidata [g]") Appearance move to sidebar hide From Wikipedia, the free encyclopedia Mathematical equation linking e, i and π For other uses, see [List of topics named after Leonhard Euler § Identities](https://en.wikipedia.org/wiki/List_of_topics_named_after_Leonhard_Euler#Identities "List of topics named after Leonhard Euler"). \| \| \|---\| \| Part of [a series of articles](https://en.wikipedia.org/wiki/Category:E_\(mathematical_constant\) "Category:E (mathematical constant)") on the \| \| mathematical constant [e](https://en.wikipedia.org/wiki/E_\(mathematical_constant\) "E (mathematical constant)") \| \| [![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/250px-Euler%27s_formula.svg.png)](https://en.wikipedia.org/wiki/File:Euler%27s_formula.svg) \| \| Properties \| \| [Natural logarithm](https://en.wikipedia.org/wiki/Natural_logarithm "Natural logarithm") [Exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function") \| \| Applications \| \| [Compound interest](https://en.wikipedia.org/wiki/Compound_interest "Compound interest") [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") [Euler's identity]() [Half-lifes](https://en.wikipedia.org/wiki/Half-life "Half-life") Exponential [growth](https://en.wikipedia.org/wiki/Exponential_growth "Exponential growth") and [decay](https://en.wikipedia.org/wiki/Exponential_decay "Exponential decay") [Probability theory](https://en.wikipedia.org/wiki/Probability_theory "Probability theory") [Bernoulli trial](https://en.wikipedia.org/wiki/Bernoulli_trial "Bernoulli trial") [Derangement](https://en.wikipedia.org/wiki/Derangement "Derangement") [Probability density function](https://en.wikipedia.org/wiki/Probability_density_function "Probability density function") \| \| Defining e \| \| [proof that e is irrational](https://en.wikipedia.org/wiki/Proof_that_e_is_irrational "Proof that e is irrational") [representations of e](https://en.wikipedia.org/wiki/List_of_representations_of_e "List of representations of e") [Lindemann–Weierstrass theorem](https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem "Lindemann–Weierstrass theorem") \| \| People \| \| [Jakob Bernoulli](https://en.wikipedia.org/wiki/Jakob_Bernoulli "Jakob Bernoulli") [John Napier](https://en.wikipedia.org/wiki/John_Napier "John Napier") [Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") \| \| Related topics \| \| [Schanuel's conjecture](https://en.wikipedia.org/wiki/Schanuel%27s_conjecture "Schanuel's conjecture") \| \| [v](https://en.wikipedia.org/wiki/Template:E_\(mathematical_constant\) "Template:E (mathematical constant)") [t](https://en.wikipedia.org/wiki/Template_talk:E_\(mathematical_constant\) "Template talk:E (mathematical constant)") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:E_\(mathematical_constant\) "Special:EditPage/Template:E (mathematical constant)") \| In [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), Euler's identity[\[note 1\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-3) (also known as Euler's equation) is the [equality](https://en.wikipedia.org/wiki/Equality_\(mathematics\) "Equality (mathematics)") e i π \+ 1 \= 0 {\\displaystyle e^{i\\pi }+1=0} ![{\\displaystyle e^{i\\pi }+1=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7464809a40f9e486de3a454745f572fbf8bb256) where - e {\\displaystyle e} ![{\\displaystyle e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467) is [Euler's number](https://en.wikipedia.org/wiki/E_\(mathematical_constant\) "E (mathematical constant)"), the base of [natural logarithms](https://en.wikipedia.org/wiki/Natural_logarithm "Natural logarithm"), - i {\\displaystyle i} ![{\\displaystyle i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20) is the [imaginary unit](https://en.wikipedia.org/wiki/Imaginary_unit "Imaginary unit"), which by definition satisfies i 2 \= − 1 {\\displaystyle i^{2}=-1} ![{\\displaystyle i^{2}=-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88e98a401d352e5037d5043028e2d7f449e83fa6) , and - π {\\displaystyle \\pi } ![{\\displaystyle \\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a) is [pi](https://en.wikipedia.org/wiki/Pi "Pi"), the ratio of the [circumference](https://en.wikipedia.org/wiki/Circumference "Circumference") of a circle to its [diameter](https://en.wikipedia.org/wiki/Diameter "Diameter"). Euler's identity is named after the Swiss mathematician [Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler"). It is a special case of [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") e i x \= cos ⁡ x \+ i sin ⁡ x {\\displaystyle e^{ix}=\\cos x+i\\sin x} ![{\\displaystyle e^{ix}=\\cos x+i\\sin x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4907c0489ab08ce550c7700a1587d4634801dff8) when evaluated for x \= π {\\displaystyle x=\\pi } ![{\\displaystyle x=\\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b4512a97fa6b7772825e2c887e010a99e217005). Euler's identity is considered an exemplar of [mathematical beauty](https://en.wikipedia.org/wiki/Mathematical_beauty "Mathematical beauty"), as it shows a profound connection between the most fundamental numbers in mathematics. ## Mathematical beauty \[[edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit&section=1 "Edit section: Mathematical beauty")\] Euler's identity is often cited as an example of deep [mathematical beauty](https://en.wikipedia.org/wiki/Mathematical_beauty "Mathematical beauty").[\[3\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-Gallagher2014-4) Three of the basic [arithmetic](https://en.wikipedia.org/wiki/Arithmetic "Arithmetic") operations occur exactly once each: [addition](https://en.wikipedia.org/wiki/Addition "Addition"), [multiplication](https://en.wikipedia.org/wiki/Multiplication "Multiplication"), and [exponentiation](https://en.wikipedia.org/wiki/Exponentiation "Exponentiation"). The identity also links five fundamental [mathematical constants](https://en.wikipedia.org/wiki/Mathematical_constant "Mathematical constant"):[\[4\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-5) - The [number 0](https://en.wikipedia.org/wiki/0 "0"), the [additive identity](https://en.wikipedia.org/wiki/Additive_identity "Additive identity") - The [number 1](https://en.wikipedia.org/wiki/1 "1"), the [multiplicative identity](https://en.wikipedia.org/wiki/Multiplicative_identity "Multiplicative identity") - The [number π](https://en.wikipedia.org/wiki/Pi "Pi") (π = 3.14159...), the fundamental [circle](https://en.wikipedia.org/wiki/Circle "Circle") constant - The [number e](https://en.wikipedia.org/wiki/E_\(mathematical_constant\) "E (mathematical constant)") (e = 2.71828...), also known as Euler's number, which occurs widely in [mathematical analysis](https://en.wikipedia.org/wiki/Mathematical_analysis "Mathematical analysis") - The [number i](https://en.wikipedia.org/wiki/Imaginary_unit "Imaginary unit"), the [imaginary unit](https://en.wikipedia.org/wiki/Imaginary_unit "Imaginary unit") such that i2 = −1 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](https://en.wikipedia.org/wiki/Stanford_University "Stanford University") mathematics professor [Keith Devlin](https://en.wikipedia.org/wiki/Keith_Devlin "Keith Devlin") has said, "like a Shakespearean [sonnet](https://en.wikipedia.org/wiki/Sonnet "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".[\[5\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-6) [Paul Nahin](https://en.wikipedia.org/wiki/Paul_Nahin "Paul Nahin"), a professor emeritus at the [University of New Hampshire](https://en.wikipedia.org/wiki/University_of_New_Hampshire "University of New Hampshire") who wrote a book dedicated to [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") and its applications in [Fourier analysis](https://en.wikipedia.org/wiki/Fourier_analysis "Fourier analysis"), said Euler's identity is "of exquisite beauty".[\[6\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-7) Mathematics writer [Constance Reid](https://en.wikipedia.org/wiki/Constance_Reid "Constance Reid") has said that Euler's identity is "the most famous formula in all mathematics".[\[7\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-8) [Benjamin Peirce](https://en.wikipedia.org/wiki/Benjamin_Peirce "Benjamin Peirce"), a 19th-century American philosopher, mathematician, and professor at [Harvard University](https://en.wikipedia.org/wiki/Harvard_University "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".[\[8\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-9) A 1990 poll of readers by [The Mathematical Intelligencer](https://en.wikipedia.org/wiki/The_Mathematical_Intelligencer "The Mathematical Intelligencer") named Euler's identity the "most beautiful theorem in mathematics".[\[9\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-10) In a 2004 poll of readers by [Physics World](https://en.wikipedia.org/wiki/Physics_World "Physics World"), Euler's identity tied with [Maxwell's equations](https://en.wikipedia.org/wiki/Maxwell%27s_equations "Maxwell's equations") (of [electromagnetism](https://en.wikipedia.org/wiki/Electromagnetism "Electromagnetism")) as the "greatest equation ever".[\[10\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-11) At least three books in [popular mathematics](https://en.wikipedia.org/wiki/Popular_mathematics "Popular mathematics") have been published about Euler's identity: - Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, by [Paul Nahin](https://en.wikipedia.org/wiki/Paul_Nahin "Paul Nahin") (2011)[\[11\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-12) - A Most Elegant Equation: Euler's formula and the beauty of mathematics, by David Stipp (2017)[\[12\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-13) - Euler's Pioneering Equation: The most beautiful theorem in mathematics, by [Robin Wilson](https://en.wikipedia.org/wiki/Robin_Wilson_\(mathematician\) "Robin Wilson (mathematician)") (2018)[\[13\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-14) ## Explanations \[[edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit&section=2 "Edit section: Explanations")\] ### Imaginary exponents \[[edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit&section=3 "Edit section: Imaginary exponents")\] Main article: [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") See also: [Complex exponents with a positive real base](https://en.wikipedia.org/wiki/Exponentiation#Complex_exponents_with_a_positive_real_base "Exponentiation") [![](https://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/ExpIPi.gif/250px-ExpIPi.gif)](https://en.wikipedia.org/wiki/File:ExpIPi.gif) In this animation N takes various increasing values from 1 to 100. The computation of (1 + ⁠iπ/N⁠)N is displayed as the combined effect of N repeated multiplications in the [complex plane](https://en.wikipedia.org/wiki/Complex_plane "Complex plane"), with the final point being the actual value of (1 + ⁠iπ/N⁠)N . It can be seen that as N gets larger (1 + ⁠iπ/N⁠)N approaches a limit of −1. Euler's identity asserts that e i π {\\displaystyle e^{i\\pi }} ![{\\displaystyle e^{i\\pi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14a02b8464db88c495fd187a3e181865822b21de) is equal to −1. The expression e i π {\\displaystyle e^{i\\pi }} ![{\\displaystyle e^{i\\pi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14a02b8464db88c495fd187a3e181865822b21de) is a special case of the expression e z {\\displaystyle e^{z}} ![{\\displaystyle e^{z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4772def31b56e642df3e4d1160cadff3d80ba45), where z is any [complex number](https://en.wikipedia.org/wiki/Complex_number "Complex number"). In general, e z {\\displaystyle e^{z}} ![{\\displaystyle e^{z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4772def31b56e642df3e4d1160cadff3d80ba45) is defined for complex z by extending one of the [definitions of the exponential function](https://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function "Characterizations of the exponential function") from real exponents to complex exponents. For example, one common definition is: e z \= lim n → ∞ ( 1 \+ z n ) n . {\\displaystyle e^{z}=\\lim \_{n\\to \\infty }\\left(1+{\\frac {z}{n}}\\right)^{n}.} ![{\\displaystyle e^{z}=\\lim \_{n\\to \\infty }\\left(1+{\\frac {z}{n}}\\right)^{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40ce3c19f0325e83628f164efe1da5a60bc80b49) Euler's identity therefore states that the limit, as n approaches infinity, of ( 1 \+ i π n ) n {\\displaystyle (1+{\\tfrac {i\\pi }{n}})^{n}} ![{\\displaystyle (1+{\\tfrac {i\\pi }{n}})^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25296a2d06e81ecdc74113c1c10cd0b2649aa59a) is equal to −1. This limit is illustrated in the animation to the right. [![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/250px-Euler%27s_formula.svg.png)](https://en.wikipedia.org/wiki/File:Euler%27s_formula.svg) Euler's formula for a general angle Euler's identity is a [special case](https://en.wikipedia.org/wiki/Special_case "Special case") of [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula"), which states that for any [real number](https://en.wikipedia.org/wiki/Real_number "Real number") x, e i x \= cos ⁡ x \+ i sin ⁡ x {\\displaystyle e^{ix}=\\cos x+i\\sin x} ![{\\displaystyle e^{ix}=\\cos x+i\\sin x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4907c0489ab08ce550c7700a1587d4634801dff8) where the inputs of the [trigonometric functions](https://en.wikipedia.org/wiki/Trigonometry "Trigonometry") sine and cosine are given in [radians](https://en.wikipedia.org/wiki/Radian "Radian"). In particular, when x = π, e i π \= cos ⁡ π \+ i sin ⁡ π . {\\displaystyle e^{i\\pi }=\\cos \\pi +i\\sin \\pi .} ![{\\displaystyle e^{i\\pi }=\\cos \\pi +i\\sin \\pi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/809c56fce05afb471ebb280a3588c6f8e14198fd) Since cos ⁡ π \= − 1 {\\displaystyle \\cos \\pi =-1} ![{\\displaystyle \\cos \\pi =-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e009a140f4ee530848ede1ead96aeac3fc54a23) and sin ⁡ π \= 0 , {\\displaystyle \\sin \\pi =0,} ![{\\displaystyle \\sin \\pi =0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ca27f5c501370bbc23e7203d7ca1fe4764b1fa) it follows that e i π \= − 1 \+ 0 i , {\\displaystyle e^{i\\pi }=-1+0i,} ![{\\displaystyle e^{i\\pi }=-1+0i,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24995de3ca85e21e47689f00348ae2661e091317) which yields Euler's identity: e i π \+ 1 \= 0\. {\\displaystyle e^{i\\pi }+1=0.} ![{\\displaystyle e^{i\\pi }+1=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e76caf050d8bc37cd2350c40517face26de5ecb7) ### Geometric interpretation \[[edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit&section=4 "Edit section: Geometric interpretation")\] Any complex number z \= x \+ i y {\\displaystyle z=x+iy} ![{\\displaystyle z=x+iy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08e90bb6b36fef59c6113eed2a08f10d77240741) can be represented by the point ( x , y ) {\\displaystyle (x,y)} ![{\\displaystyle (x,y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386) on the [complex plane](https://en.wikipedia.org/wiki/Complex_plane "Complex plane"). This point can also be represented in [polar coordinates](https://en.wikipedia.org/wiki/Complex_number#Polar_complex_plane "Complex number") as ( r , θ ) {\\displaystyle (r,\\theta )} ![{\\displaystyle (r,\\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8396fdc359fb06c93722137c959e7496e47ed6) , where r is the absolute value of z (distance from the origin), and θ {\\displaystyle \\theta } ![{\\displaystyle \\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af) 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 ( r cos ⁡ θ , r sin ⁡ θ ) {\\displaystyle (r\\cos \\theta ,r\\sin \\theta )} ![{\\displaystyle (r\\cos \\theta ,r\\sin \\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/560ed9ef39aed01ca74a50d7b5cae2cced5de2e1) , implying that z \= r ( cos ⁡ θ \+ i sin ⁡ θ ) {\\displaystyle z=r(\\cos \\theta +i\\sin \\theta )} ![{\\displaystyle z=r(\\cos \\theta +i\\sin \\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/350014e34fca54c3ea03a63825d6afa7a743ffa7) . According to Euler's formula, this is equivalent to saying z \= r e i θ {\\displaystyle z=re^{i\\theta }} ![{\\displaystyle z=re^{i\\theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b36ddd965193c2b7d6ea24a7c3678814d0dc8d) . Euler's identity says that − 1 \= e i π {\\displaystyle -1=e^{i\\pi }} ![{\\displaystyle -1=e^{i\\pi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/127dc145108d2d2b72f37efba2ce9e924be890a1) . Since e i π {\\displaystyle e^{i\\pi }} ![{\\displaystyle e^{i\\pi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14a02b8464db88c495fd187a3e181865822b21de) is r e i θ {\\displaystyle re^{i\\theta }} ![{\\displaystyle re^{i\\theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/304e85daab31cb0b8918d182576f69764bf0b3f3) for r = 1 and θ \= π {\\displaystyle \\theta =\\pi } ![{\\displaystyle \\theta =\\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab4db588619489e27efb50a1d0d5aa016c49ce15) , 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 π {\\displaystyle \\pi } ![{\\displaystyle \\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a) radians. Additionally, when any complex number z is [multiplied](https://en.wikipedia.org/wiki/Complex_number#Multiplication_and_division_in_polar_form "Complex number") by e i θ {\\displaystyle e^{i\\theta }} ![{\\displaystyle e^{i\\theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b4b8e67ee479d68e0e5040aaf87eff99214c90f) , it has the effect of rotating z {\\displaystyle z} ![{\\displaystyle z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98) counterclockwise by an angle of θ {\\displaystyle \\theta } ![{\\displaystyle \\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af) 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 π {\\displaystyle \\pi } ![{\\displaystyle \\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a) radians around the origin has the same effect as reflecting the point across the origin. Similarly, setting θ {\\displaystyle \\theta } ![{\\displaystyle \\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af) equal to 2 π {\\displaystyle 2\\pi } ![{\\displaystyle 2\\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06) yields the related equation e 2 π i \= 1 {\\displaystyle e^{2\\pi i}=1} ![{\\displaystyle e^{2\\pi i}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6da4586a614d29da6632bc76b2b4417996d62e88) , which can be interpreted as saying that rotating any point by one [turn](https://en.wikipedia.org/wiki/Turn_\(angle\) "Turn (angle)") around the origin returns it to its original position. ## Generalizations \[[edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit&section=5 "Edit section: Generalizations")\] Euler's identity is also a special case of the more general identity that the nth [roots of unity](https://en.wikipedia.org/wiki/Roots_of_unity "Roots of unity"), for n \> 1, add up to 0: ∑ k \= 0 n − 1 e 2 π i k n \= 0\. {\\displaystyle \\sum \_{k=0}^{n-1}e^{2\\pi i{\\frac {k}{n}}}=0.} ![{\\displaystyle \\sum \_{k=0}^{n-1}e^{2\\pi i{\\frac {k}{n}}}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f6bd22aa5cd424ee8298ca5090892d6ac900082) Euler's identity is the case where n = 2. A similar identity also applies to [quaternion exponential](https://en.wikipedia.org/wiki/Quaternion#Exponential,_logarithm,_and_power_functions "Quaternion"): let {i, j, k} be the basis [quaternions](https://en.wikipedia.org/wiki/Quaternion "Quaternion"); then, e 1 3 ( i ± j ± k ) π \+ 1 \= 0\. {\\displaystyle e^{{\\frac {1}{\\sqrt {3}}}(i\\pm j\\pm k)\\pi }+1=0.} ![{\\displaystyle e^{{\\frac {1}{\\sqrt {3}}}(i\\pm j\\pm k)\\pi }+1=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7425d865b67e05665dfa275bd0515129138f460) More generally, let q be a quaternion with a zero real part and a norm equal to 1; that is, q \= a i \+ b j \+ c k , {\\displaystyle q=ai+bj+ck,} ![{\\displaystyle q=ai+bj+ck,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f29e3114ba4e82b5c0f75a8c1141e2843715cdb7) with a 2 \+ b 2 \+ c 2 \= 1\. {\\displaystyle a^{2}+b^{2}+c^{2}=1.} ![{\\displaystyle a^{2}+b^{2}+c^{2}=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11f2994d705edbe9d0ad3ffcb854ca63d3fb6745) Then one has e q π \+ 1 \= 0\. {\\displaystyle e^{q\\pi }+1=0.} ![{\\displaystyle e^{q\\pi }+1=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1df35eeca09cd9e9557ac9218c992857a00fb3b7) The same formula applies to [octonions](https://en.wikipedia.org/wiki/Octonion "Octonion"), with a zero real part and a norm equal to 1. These formulas are a direct generalization of Euler's identity, since i {\\displaystyle i} ![{\\displaystyle i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20) and − i {\\displaystyle -i} ![{\\displaystyle -i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91fddb9f89a520937db3a8821575068cdcc76f60) are the only complex numbers with a zero real part and a norm (absolute value) equal to 1. ## History \[[edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit&section=6 "Edit section: History")\] Euler's identity is a direct result of [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula"), first published in his monumental 1748 work of mathematical analysis, [Introductio in analysin infinitorum](https://en.wikipedia.org/wiki/Introductio_in_analysin_infinitorum "Introductio in analysin infinitorum"),[\[14\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-15) 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.[\[15\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-Sandifer2007-16) [Robin Wilson](https://en.wikipedia.org/wiki/Robin_Wilson_\(mathematician\) "Robin Wilson (mathematician)") writes:[\[16\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-17) > We've seen how \[Euler's identity\] can easily be deduced from results of [Johann Bernoulli](https://en.wikipedia.org/wiki/Johann_Bernoulli "Johann Bernoulli") and [Roger Cotes](https://en.wikipedia.org/wiki/Roger_Cotes "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 \[i.e. [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula")\], eix = cos x + i sin x. Moreover, it seems to be unknown who first stated the result explicitly ## See also \[[edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit&section=7 "Edit section: See also")\] - [![icon](https://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/40px-Nuvola_apps_edu_mathematics_blue-p.svg.png)](https://en.wikipedia.org/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg)[Mathematics portal](https://en.wikipedia.org/wiki/Portal:Mathematics "Portal:Mathematics") - [De Moivre's formula](https://en.wikipedia.org/wiki/De_Moivre%27s_formula "De Moivre's formula") - [Exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function") - [Gelfond's constant](https://en.wikipedia.org/wiki/Gelfond%27s_constant "Gelfond's constant") ## Notes \[[edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit&section=8 "Edit section: Notes")\] 1. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-3) The term "Euler's identity" (or "Euler identity") is also used elsewhere to refer to other concepts, including the related general formula eix = cos x + i sin x,[\[1\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-1) and the [Euler product formula](https://en.wikipedia.org/wiki/Riemann_zeta_function#Euler's_product_formula "Riemann zeta function").[\[2\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-EOM-2) See also [List of topics named after Leonhard Euler](https://en.wikipedia.org/wiki/List_of_topics_named_after_Leonhard_Euler "List of topics named after Leonhard Euler"). ## References \[[edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit&section=9 "Edit section: References")\] 1. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-1) Dunham, 1999, [p. xxiv](https://books.google.com/books?id=uKOVNvGOkhQC&pg=PR24). 2. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-EOM_2-0) Stepanov, S. A. (2001) \[1994\]. ["Euler identity"](https://www.encyclopediaofmath.org/index.php?title=Euler_identity&oldid=33574). [Encyclopedia of Mathematics](https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics "Encyclopedia of Mathematics"). [EMS Press](https://en.wikipedia.org/wiki/European_Mathematical_Society "European Mathematical Society"). 3. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-Gallagher2014_4-0) Gallagher, James (13 February 2014). ["Mathematics: Why the brain sees maths as beauty"](https://www.bbc.co.uk/news/science-environment-26151062). [BBC News](https://en.wikipedia.org/wiki/BBC_News "BBC News"). Retrieved 26 December 2017. 4. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-5) Paulos, 1992, p. 117. 5. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-6) Nahin, 2006, [p. 1](https://books.google.com/books?id=GvSg5HQ7WPcC&pg=PA1). 6. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-7) Nahin, 2006, p. xxxii. 7. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-8) Reid, chapter e. 8. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-9) Maor, [p. 160](https://books.google.com/books?id=eIsyLD_bDKkC&pg=PA160), and Kasner & Newman, [p. 103–104](https://books.google.com/books?id=Ad8hAx-6m9oC&pg=PA103). 9. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-10) Wells, 1990. 10. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-11) Crease, 2004. 11. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-12) Nahin, Paul (2011). Dr. Euler's fabulous formula: cures many mathematical ills. Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-691-11822-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-11822-2 "Special:BookSources/978-0-691-11822-2") . 12. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-13) Stipp, David (2017). A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics (First ed.). Basic Books. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-465-09377-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-465-09377-9 "Special:BookSources/978-0-465-09377-9") . 13. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-14) Wilson, Robin (2018). Euler's pioneering equation: the most beautiful theorem in mathematics. Oxford: Oxford University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-19-879493-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-879493-6 "Special:BookSources/978-0-19-879493-6") . 14. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-15) Conway & Guy, p. [254–255](https://books.google.com/books?id=0--3rcO7dMYC&pg=PA254). 15. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-Sandifer2007_16-0) Sandifer, p. 4. 16. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-17) Wilson, p. 151-152. ### Sources \[[edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit&section=10 "Edit section: Sources")\] - [Conway, John H.](https://en.wikipedia.org/wiki/John_Horton_Conway "John Horton Conway"), and [Guy, Richard K.](https://en.wikipedia.org/wiki/Richard_K._Guy "Richard K. Guy") (1996), [The Book of Numbers](https://en.wikipedia.org/wiki/The_Book_of_Numbers_\(math_book\) "The Book of Numbers (math book)"), Springer [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-387-97993-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-97993-9 "Special:BookSources/978-0-387-97993-9") - [Crease, Robert P.](https://en.wikipedia.org/wiki/Robert_P._Crease "Robert P. Crease") (10 May 2004), "[The greatest equations ever](http://physicsworld.com/cws/article/print/2004/may/10/the-greatest-equations-ever)", [Physics World](https://en.wikipedia.org/wiki/Physics_World "Physics World") \[registration required\] - [Dunham, William](https://en.wikipedia.org/wiki/William_Dunham_\(mathematician\) "William Dunham (mathematician)") (1999), Euler: The Master of Us All, [Mathematical Association of America](https://en.wikipedia.org/wiki/Mathematical_Association_of_America "Mathematical Association of America") [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-88385-328-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-88385-328-3 "Special:BookSources/978-0-88385-328-3") - [Euler, Leonhard](https://en.wikipedia.org/wiki/Euler,_Leonhard "Euler, Leonhard"), Complete work. [Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus](http://gallica.bnf.fr/ark:/12148/bpt6k69587.image.r=%22has+celeberrimas+formulas%22.f169.langEN), Leipzig: B. G. Teubneri - [Kasner, E.](https://en.wikipedia.org/wiki/Edward_Kasner "Edward Kasner"), and [Newman, J.](https://en.wikipedia.org/wiki/James_R._Newman "James R. Newman") (1940), [Mathematics and the Imagination](https://en.wikipedia.org/wiki/Mathematics_and_the_Imagination "Mathematics and the Imagination"), [Simon & Schuster](https://en.wikipedia.org/wiki/Simon_%26_Schuster "Simon & Schuster") - [Maor, Eli](https://en.wikipedia.org/wiki/Eli_Maor "Eli Maor") (1998), e: The Story of a number, [Princeton University Press](https://en.wikipedia.org/wiki/Princeton_University_Press "Princeton University Press") [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-691-05854-7](https://en.wikipedia.org/wiki/Special:BookSources/0-691-05854-7 "Special:BookSources/0-691-05854-7") - Nahin, Paul J. (2006), Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, [Princeton University Press](https://en.wikipedia.org/wiki/Princeton_University_Press "Princeton University Press") [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-691-11822-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-11822-2 "Special:BookSources/978-0-691-11822-2") - [Paulos, John Allen](https://en.wikipedia.org/wiki/John_Allen_Paulos "John Allen Paulos") (1992), Beyond Numeracy: An Uncommon Dictionary of Mathematics, [Penguin Books](https://en.wikipedia.org/wiki/Penguin_Books "Penguin Books") [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-14-014574-5](https://en.wikipedia.org/wiki/Special:BookSources/0-14-014574-5 "Special:BookSources/0-14-014574-5") - Reid, Constance (various editions), [From Zero to Infinity](https://en.wikipedia.org/wiki/From_Zero_to_Infinity "From Zero to Infinity"), [Mathematical Association of America](https://en.wikipedia.org/wiki/Mathematical_Association_of_America "Mathematical Association of America") - Sandifer, C. Edward (2007), [Euler's Greatest Hits](https://books.google.com/books?id=sohHs7ExOsYC&pg=PA4), [Mathematical Association of America](https://en.wikipedia.org/wiki/Mathematical_Association_of_America "Mathematical Association of America") [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-88385-563-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-88385-563-8 "Special:BookSources/978-0-88385-563-8") - Stipp, David (2017). A Most Elegant Equation: Euler's formula and the beauty of mathematics. [Basic Books](https://en.wikipedia.org/wiki/Basic_Books "Basic Books"). - [Wells, David](https://en.wikipedia.org/w/index.php?title=David_G._Wells&action=edit&redlink=1 "David G. Wells (page does not exist)") (1990). "Are these the most beautiful?". [The Mathematical Intelligencer](https://en.wikipedia.org/wiki/The_Mathematical_Intelligencer "The Mathematical Intelligencer"). 12 (3): 37–41\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF03024015](https://doi.org/10.1007%2FBF03024015). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [121503263](https://api.semanticscholar.org/CorpusID:121503263). - [Wilson, Robin](https://en.wikipedia.org/wiki/Robin_Wilson_\(mathematician\) "Robin Wilson (mathematician)") (2018). Euler's Pioneering Equation: The most beautiful theorem in mathematics. [Oxford University Press](https://en.wikipedia.org/wiki/Oxford_University_Press "Oxford University Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-192-51406-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-192-51406-6 "Special:BookSources/978-0-192-51406-6") . - [Zeki, S.](https://en.wikipedia.org/wiki/Semir_Zeki "Semir Zeki"); Romaya, J. P.; Benincasa, D. M. T.; [Atiyah, M. F.](https://en.wikipedia.org/wiki/Michael_Atiyah "Michael Atiyah") (2014). "The experience of mathematical beauty and its neural correlates". Frontiers in Human Neuroscience. 8: 68. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3389/fnhum.2014.00068](https://doi.org/10.3389%2Ffnhum.2014.00068). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [3923150](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3923150). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [24592230](https://pubmed.ncbi.nlm.nih.gov/24592230). ## External links \[[edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit&section=11 "Edit section: External links")\] ![](https://upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/40px-Wikiquote-logo.svg.png) Wikiquote has quotations related to *[Euler's identity](https://en.wikiquote.org/wiki/Euler%27s_identity "q:Euler's identity"). - [Intuitive understanding of Euler's formula](http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/) \| [v](https://en.wikipedia.org/wiki/Template:Leonhard_Euler "Template:Leonhard Euler") [t](https://en.wikipedia.org/wiki/Template_talk:Leonhard_Euler "Template talk:Leonhard Euler") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Leonhard_Euler "Special:EditPage/Template:Leonhard Euler")[Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") \| \|---\| \| [Euler–Lagrange equation](https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation "Euler–Lagrange equation") [Euler–Lotka equation](https://en.wikipedia.org/wiki/Euler%E2%80%93Lotka_equation "Euler–Lotka equation") [Euler–Maclaurin formula](https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula "Euler–Maclaurin formula") [Euler–Maruyama method](https://en.wikipedia.org/wiki/Euler%E2%80%93Maruyama_method "Euler–Maruyama method") [Euler–Mascheroni constant](https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant "Euler–Mascheroni constant") [Euler–Poisson–Darboux equation](https://en.wikipedia.org/wiki/Euler%E2%80%93Poisson%E2%80%93Darboux_equation "Euler–Poisson–Darboux equation") [Euler–Rodrigues formula](https://en.wikipedia.org/wiki/Euler%E2%80%93Rodrigues_formula "Euler–Rodrigues formula") [Euler–Tricomi equation](https://en.wikipedia.org/wiki/Euler%E2%80%93Tricomi_equation "Euler–Tricomi equation") [Euler's continued fraction formula](https://en.wikipedia.org/wiki/Euler%27s_continued_fraction_formula "Euler's continued fraction formula") [Euler's critical load](https://en.wikipedia.org/wiki/Euler%27s_critical_load "Euler's critical load") [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") [Euler's four-square identity](https://en.wikipedia.org/wiki/Euler%27s_four-square_identity "Euler's four-square identity") [Euler's identity]() [Euler's pump and turbine equation](https://en.wikipedia.org/wiki/Euler%27s_pump_and_turbine_equation "Euler's pump and turbine equation") [Euler's rotation theorem](https://en.wikipedia.org/wiki/Euler%27s_rotation_theorem "Euler's rotation theorem") [Euler's sum of powers conjecture](https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture "Euler's sum of powers conjecture") [Euler's theorem](https://en.wikipedia.org/wiki/Euler%27s_theorem "Euler's theorem") [Euler equations (fluid dynamics)](https://en.wikipedia.org/wiki/Euler_equations_\(fluid_dynamics\) "Euler equations (fluid dynamics)") [Euler function](https://en.wikipedia.org/wiki/Euler_function "Euler function") [Euler method](https://en.wikipedia.org/wiki/Euler_method "Euler method") [Euler numbers](https://en.wikipedia.org/wiki/Euler_numbers "Euler numbers") [Euler number (physics)](https://en.wikipedia.org/wiki/Euler_number_\(physics\) "Euler number (physics)") [Euler–Bernoulli beam theory](https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory "Euler–Bernoulli beam theory") [Namesakes](https://en.wikipedia.org/wiki/List_of_things_named_after_Leonhard_Euler "List of things named after Leonhard Euler") \| \| ![](https://upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/20px-Symbol_category_class.svg.png) [Category](https://en.wikipedia.org/wiki/Category:Leonhard_Euler "Category:Leonhard Euler")* \| ![](https://en.wikipedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1) Retrieved from "[https://en.wikipedia.org/w/index.php?title=Euler%27s\_identity\&oldid=1341920695](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&oldid=1341920695)" [Categories](https://en.wikipedia.org/wiki/Help:Category "Help:Category"): - 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Readable Markdown	From Wikipedia, the free encyclopedia In [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), Euler's identity[\[note 1\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-3) (also known as Euler's equation) is the [equality](https://en.wikipedia.org/wiki/Equality_\(mathematics\) "Equality (mathematics)") ![{\\displaystyle e^{i\\pi }+1=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7464809a40f9e486de3a454745f572fbf8bb256) where Euler's identity is named after the Swiss mathematician [Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler"). It is a special case of [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") ![{\\displaystyle e^{ix}=\\cos x+i\\sin x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4907c0489ab08ce550c7700a1587d4634801dff8) when evaluated for ![{\\displaystyle x=\\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b4512a97fa6b7772825e2c887e010a99e217005). Euler's identity is considered an exemplar of [mathematical beauty](https://en.wikipedia.org/wiki/Mathematical_beauty "Mathematical beauty"), as it shows a profound connection between the most fundamental numbers in mathematics. ## Mathematical beauty \[[edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit&section=1 "Edit section: Mathematical beauty")\] Euler's identity is often cited as an example of deep [mathematical beauty](https://en.wikipedia.org/wiki/Mathematical_beauty "Mathematical beauty").[\[3\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-Gallagher2014-4) Three of the basic [arithmetic](https://en.wikipedia.org/wiki/Arithmetic "Arithmetic") operations occur exactly once each: [addition](https://en.wikipedia.org/wiki/Addition "Addition"), [multiplication](https://en.wikipedia.org/wiki/Multiplication "Multiplication"), and [exponentiation](https://en.wikipedia.org/wiki/Exponentiation "Exponentiation"). The identity also links five fundamental [mathematical constants](https://en.wikipedia.org/wiki/Mathematical_constant "Mathematical constant"):[\[4\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-5) - The [number 0](https://en.wikipedia.org/wiki/0 "0"), the [additive identity](https://en.wikipedia.org/wiki/Additive_identity "Additive identity") - The [number 1](https://en.wikipedia.org/wiki/1 "1"), the [multiplicative identity](https://en.wikipedia.org/wiki/Multiplicative_identity "Multiplicative identity") - The [number π](https://en.wikipedia.org/wiki/Pi "Pi") (π = 3.14159...), the fundamental [circle](https://en.wikipedia.org/wiki/Circle "Circle") constant - The [number e](https://en.wikipedia.org/wiki/E_\(mathematical_constant\) "E (mathematical constant)") (e = 2.71828...), also known as Euler's number, which occurs widely in [mathematical analysis](https://en.wikipedia.org/wiki/Mathematical_analysis "Mathematical analysis") - The [number i](https://en.wikipedia.org/wiki/Imaginary_unit "Imaginary unit"), the [imaginary unit](https://en.wikipedia.org/wiki/Imaginary_unit "Imaginary unit") such that i2 = −1 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](https://en.wikipedia.org/wiki/Stanford_University "Stanford University") mathematics professor [Keith Devlin](https://en.wikipedia.org/wiki/Keith_Devlin "Keith Devlin") has said, "like a Shakespearean [sonnet](https://en.wikipedia.org/wiki/Sonnet "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".[\[5\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-6) [Paul Nahin](https://en.wikipedia.org/wiki/Paul_Nahin "Paul Nahin"), a professor emeritus at the [University of New Hampshire](https://en.wikipedia.org/wiki/University_of_New_Hampshire "University of New Hampshire") who wrote a book dedicated to [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") and its applications in [Fourier analysis](https://en.wikipedia.org/wiki/Fourier_analysis "Fourier analysis"), said Euler's identity is "of exquisite beauty".[\[6\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-7) Mathematics writer [Constance Reid](https://en.wikipedia.org/wiki/Constance_Reid "Constance Reid") has said that Euler's identity is "the most famous formula in all mathematics".[\[7\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-8) [Benjamin Peirce](https://en.wikipedia.org/wiki/Benjamin_Peirce "Benjamin Peirce"), a 19th-century American philosopher, mathematician, and professor at [Harvard University](https://en.wikipedia.org/wiki/Harvard_University "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".[\[8\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-9) A 1990 poll of readers by [The Mathematical Intelligencer](https://en.wikipedia.org/wiki/The_Mathematical_Intelligencer "The Mathematical Intelligencer") named Euler's identity the "most beautiful theorem in mathematics".[\[9\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-10) In a 2004 poll of readers by [Physics World](https://en.wikipedia.org/wiki/Physics_World "Physics World"), Euler's identity tied with [Maxwell's equations](https://en.wikipedia.org/wiki/Maxwell%27s_equations "Maxwell's equations") (of [electromagnetism](https://en.wikipedia.org/wiki/Electromagnetism "Electromagnetism")) as the "greatest equation ever".[\[10\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-11) At least three books in [popular mathematics](https://en.wikipedia.org/wiki/Popular_mathematics "Popular mathematics") have been published about Euler's identity: - Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, by [Paul Nahin](https://en.wikipedia.org/wiki/Paul_Nahin "Paul Nahin") (2011)[\[11\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-12) - A Most Elegant Equation: Euler's formula and the beauty of mathematics, by David Stipp (2017)[\[12\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-13) - Euler's Pioneering Equation: The most beautiful theorem in mathematics, by [Robin Wilson](https://en.wikipedia.org/wiki/Robin_Wilson_\(mathematician\) "Robin Wilson (mathematician)") (2018)[\[13\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-14) ### Imaginary exponents \[[edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit&section=3 "Edit section: Imaginary exponents")\] [![](https://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/ExpIPi.gif/250px-ExpIPi.gif)](https://en.wikipedia.org/wiki/File:ExpIPi.gif) In this animation N takes various increasing values from 1 to 100. The computation of (1 + ⁠iπ/N⁠)N is displayed as the combined effect of N repeated multiplications in the [complex plane](https://en.wikipedia.org/wiki/Complex_plane "Complex plane"), with the final point being the actual value of (1 + ⁠iπ/N⁠)N. It can be seen that as N gets larger (1 + ⁠iπ/N⁠)N approaches a limit of −1. Euler's identity asserts that ![{\\displaystyle e^{i\\pi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14a02b8464db88c495fd187a3e181865822b21de) is equal to −1. The expression ![{\\displaystyle e^{i\\pi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14a02b8464db88c495fd187a3e181865822b21de) is a special case of the expression ![{\\displaystyle e^{z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4772def31b56e642df3e4d1160cadff3d80ba45), where z is any [complex number](https://en.wikipedia.org/wiki/Complex_number "Complex number"). In general, ![{\\displaystyle e^{z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4772def31b56e642df3e4d1160cadff3d80ba45) is defined for complex z by extending one of the [definitions of the exponential function](https://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function "Characterizations of the exponential function") from real exponents to complex exponents. For example, one common definition is: ![{\\displaystyle e^{z}=\\lim \_{n\\to \\infty }\\left(1+{\\frac {z}{n}}\\right)^{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40ce3c19f0325e83628f164efe1da5a60bc80b49) Euler's identity therefore states that the limit, as n approaches infinity, of ![{\\displaystyle (1+{\\tfrac {i\\pi }{n}})^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25296a2d06e81ecdc74113c1c10cd0b2649aa59a) is equal to −1. This limit is illustrated in the animation to the right. [![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/250px-Euler%27s_formula.svg.png)](https://en.wikipedia.org/wiki/File:Euler%27s_formula.svg) Euler's formula for a general angle Euler's identity is a [special case](https://en.wikipedia.org/wiki/Special_case "Special case") of [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula"), which states that for any [real number](https://en.wikipedia.org/wiki/Real_number "Real number") x, ![{\\displaystyle e^{ix}=\\cos x+i\\sin x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4907c0489ab08ce550c7700a1587d4634801dff8) where the inputs of the [trigonometric functions](https://en.wikipedia.org/wiki/Trigonometry "Trigonometry") sine and cosine are given in [radians](https://en.wikipedia.org/wiki/Radian "Radian"). In particular, when x = π, ![{\\displaystyle e^{i\\pi }=\\cos \\pi +i\\sin \\pi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/809c56fce05afb471ebb280a3588c6f8e14198fd) Since ![{\\displaystyle \\cos \\pi =-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e009a140f4ee530848ede1ead96aeac3fc54a23) and ![{\\displaystyle \\sin \\pi =0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ca27f5c501370bbc23e7203d7ca1fe4764b1fa) it follows that ![{\\displaystyle e^{i\\pi }=-1+0i,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24995de3ca85e21e47689f00348ae2661e091317) which yields Euler's identity: ![{\\displaystyle e^{i\\pi }+1=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e76caf050d8bc37cd2350c40517face26de5ecb7) ### Geometric interpretation \[[edit](https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&action=edit&section=4 "Edit section: Geometric interpretation")\] Any complex number ![{\\displaystyle z=x+iy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08e90bb6b36fef59c6113eed2a08f10d77240741) can be represented by the point ![{\\displaystyle (x,y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386) on the [complex plane](https://en.wikipedia.org/wiki/Complex_plane "Complex plane"). This point can also be represented in [polar coordinates](https://en.wikipedia.org/wiki/Complex_number#Polar_complex_plane "Complex number") as ![{\\displaystyle (r,\\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8396fdc359fb06c93722137c959e7496e47ed6), where r is the absolute value of z (distance from the origin), and ![{\\displaystyle \\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af) 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 ![{\\displaystyle (r\\cos \\theta ,r\\sin \\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/560ed9ef39aed01ca74a50d7b5cae2cced5de2e1), implying that ![{\\displaystyle z=r(\\cos \\theta +i\\sin \\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/350014e34fca54c3ea03a63825d6afa7a743ffa7). According to Euler's formula, this is equivalent to saying ![{\\displaystyle z=re^{i\\theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b36ddd965193c2b7d6ea24a7c3678814d0dc8d). Euler's identity says that ![{\\displaystyle -1=e^{i\\pi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/127dc145108d2d2b72f37efba2ce9e924be890a1). Since ![{\\displaystyle e^{i\\pi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14a02b8464db88c495fd187a3e181865822b21de) is ![{\\displaystyle re^{i\\theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/304e85daab31cb0b8918d182576f69764bf0b3f3) for r = 1 and ![{\\displaystyle \\theta =\\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab4db588619489e27efb50a1d0d5aa016c49ce15), 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 ![{\\displaystyle \\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a) radians. Additionally, when any complex number z is [multiplied](https://en.wikipedia.org/wiki/Complex_number#Multiplication_and_division_in_polar_form "Complex number") by ![{\\displaystyle e^{i\\theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b4b8e67ee479d68e0e5040aaf87eff99214c90f), it has the effect of rotating ![{\\displaystyle z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98) counterclockwise by an angle of ![{\\displaystyle \\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af) 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 ![{\\displaystyle \\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a) radians around the origin has the same effect as reflecting the point across the origin. Similarly, setting ![{\\displaystyle \\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af) equal to ![{\\displaystyle 2\\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06) yields the related equation ![{\\displaystyle e^{2\\pi i}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6da4586a614d29da6632bc76b2b4417996d62e88), which can be interpreted as saying that rotating any point by one [turn](https://en.wikipedia.org/wiki/Turn_\(angle\) "Turn (angle)") around the origin returns it to its original position. Euler's identity is also a special case of the more general identity that the nth [roots of unity](https://en.wikipedia.org/wiki/Roots_of_unity "Roots of unity"), for n \> 1, add up to 0: ![{\\displaystyle \\sum \_{k=0}^{n-1}e^{2\\pi i{\\frac {k}{n}}}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f6bd22aa5cd424ee8298ca5090892d6ac900082) Euler's identity is the case where n = 2. A similar identity also applies to [quaternion exponential](https://en.wikipedia.org/wiki/Quaternion#Exponential,_logarithm,_and_power_functions "Quaternion"): let {i, j, k} be the basis [quaternions](https://en.wikipedia.org/wiki/Quaternion "Quaternion"); then, ![{\\displaystyle e^{{\\frac {1}{\\sqrt {3}}}(i\\pm j\\pm k)\\pi }+1=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7425d865b67e05665dfa275bd0515129138f460) More generally, let q be a quaternion with a zero real part and a norm equal to 1; that is, ![{\\displaystyle q=ai+bj+ck,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f29e3114ba4e82b5c0f75a8c1141e2843715cdb7) with ![{\\displaystyle a^{2}+b^{2}+c^{2}=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11f2994d705edbe9d0ad3ffcb854ca63d3fb6745) Then one has ![{\\displaystyle e^{q\\pi }+1=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1df35eeca09cd9e9557ac9218c992857a00fb3b7) The same formula applies to [octonions](https://en.wikipedia.org/wiki/Octonion "Octonion"), with a zero real part and a norm equal to 1. These formulas are a direct generalization of Euler's identity, since ![{\\displaystyle i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20) and ![{\\displaystyle -i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91fddb9f89a520937db3a8821575068cdcc76f60) are the only complex numbers with a zero real part and a norm (absolute value) equal to 1. Euler's identity is a direct result of [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula"), first published in his monumental 1748 work of mathematical analysis, [Introductio in analysin infinitorum](https://en.wikipedia.org/wiki/Introductio_in_analysin_infinitorum "Introductio in analysin infinitorum"),[\[14\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-15) 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.[\[15\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-Sandifer2007-16) [Robin Wilson](https://en.wikipedia.org/wiki/Robin_Wilson_\(mathematician\) "Robin Wilson (mathematician)") writes:[\[16\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-17) > We've seen how \[Euler's identity\] can easily be deduced from results of [Johann Bernoulli](https://en.wikipedia.org/wiki/Johann_Bernoulli "Johann Bernoulli") and [Roger Cotes](https://en.wikipedia.org/wiki/Roger_Cotes "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 \[i.e. [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula")\], eix = cos x + i sin x. Moreover, it seems to be unknown who first stated the result explicitly - [De Moivre's formula](https://en.wikipedia.org/wiki/De_Moivre%27s_formula "De Moivre's formula") - [Exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function") - [Gelfond's constant](https://en.wikipedia.org/wiki/Gelfond%27s_constant "Gelfond's constant") 1. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-3) The term "Euler's identity" (or "Euler identity") is also used elsewhere to refer to other concepts, including the related general formula eix = cos x + i sin x,[\[1\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-1) and the [Euler product formula](https://en.wikipedia.org/wiki/Riemann_zeta_function#Euler's_product_formula "Riemann zeta function").[\[2\]](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-EOM-2) See also [List of topics named after Leonhard Euler](https://en.wikipedia.org/wiki/List_of_topics_named_after_Leonhard_Euler "List of topics named after Leonhard Euler"). 1. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-1) Dunham, 1999, [p. xxiv](https://books.google.com/books?id=uKOVNvGOkhQC&pg=PR24). 2. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-EOM_2-0) Stepanov, S. A. (2001) \[1994\]. ["Euler identity"](https://www.encyclopediaofmath.org/index.php?title=Euler_identity&oldid=33574). [Encyclopedia of Mathematics](https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics "Encyclopedia of Mathematics"). [EMS Press](https://en.wikipedia.org/wiki/European_Mathematical_Society "European Mathematical Society"). 3. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-Gallagher2014_4-0) Gallagher, James (13 February 2014). ["Mathematics: Why the brain sees maths as beauty"](https://www.bbc.co.uk/news/science-environment-26151062). [BBC News](https://en.wikipedia.org/wiki/BBC_News "BBC News"). Retrieved 26 December 2017. 4. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-5) Paulos, 1992, p. 117. 5. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-6) Nahin, 2006, [p. 1](https://books.google.com/books?id=GvSg5HQ7WPcC&pg=PA1). 6. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-7) Nahin, 2006, p. xxxii. 7. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-8) Reid, chapter e. 8. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-9) Maor, [p. 160](https://books.google.com/books?id=eIsyLD_bDKkC&pg=PA160), and Kasner & Newman, [p. 103–104](https://books.google.com/books?id=Ad8hAx-6m9oC&pg=PA103). 9. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-10) Wells, 1990. 10. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-11) Crease, 2004. 11. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-12) Nahin, Paul (2011). Dr. Euler's fabulous formula: cures many mathematical ills. Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-691-11822-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-11822-2 "Special:BookSources/978-0-691-11822-2") . 12. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-13) Stipp, David (2017). A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics (First ed.). Basic Books. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-465-09377-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-465-09377-9 "Special:BookSources/978-0-465-09377-9") . 13. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-14) Wilson, Robin (2018). Euler's pioneering equation: the most beautiful theorem in mathematics. Oxford: Oxford University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-19-879493-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-879493-6 "Special:BookSources/978-0-19-879493-6") . 14. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-15) Conway & Guy, p. [254–255](https://books.google.com/books?id=0--3rcO7dMYC&pg=PA254). 15. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-Sandifer2007_16-0) Sandifer, p. 4. 16. [^](https://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-17) Wilson, p. 151-152. - [Conway, John H.](https://en.wikipedia.org/wiki/John_Horton_Conway "John Horton Conway"), and [Guy, Richard K.](https://en.wikipedia.org/wiki/Richard_K._Guy "Richard K. Guy") (1996), [The Book of Numbers](https://en.wikipedia.org/wiki/The_Book_of_Numbers_\(math_book\) "The Book of Numbers (math book)"), Springer [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-387-97993-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-97993-9 "Special:BookSources/978-0-387-97993-9") - [Crease, Robert P.](https://en.wikipedia.org/wiki/Robert_P._Crease "Robert P. Crease") (10 May 2004), "[The greatest equations ever](http://physicsworld.com/cws/article/print/2004/may/10/the-greatest-equations-ever)", [Physics World](https://en.wikipedia.org/wiki/Physics_World "Physics World") \[registration required\] - [Dunham, William](https://en.wikipedia.org/wiki/William_Dunham_\(mathematician\) "William Dunham (mathematician)") (1999), Euler: The Master of Us All, [Mathematical Association of America](https://en.wikipedia.org/wiki/Mathematical_Association_of_America "Mathematical Association of America") [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-88385-328-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-88385-328-3 "Special:BookSources/978-0-88385-328-3") - [Euler, Leonhard](https://en.wikipedia.org/wiki/Euler,_Leonhard "Euler, Leonhard"), Complete work. [Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus](http://gallica.bnf.fr/ark:/12148/bpt6k69587.image.r=%22has+celeberrimas+formulas%22.f169.langEN), Leipzig: B. G. Teubneri - [Kasner, E.](https://en.wikipedia.org/wiki/Edward_Kasner "Edward Kasner"), and [Newman, J.](https://en.wikipedia.org/wiki/James_R._Newman "James R. Newman") (1940), [Mathematics and the Imagination](https://en.wikipedia.org/wiki/Mathematics_and_the_Imagination "Mathematics and the Imagination"), [Simon & Schuster](https://en.wikipedia.org/wiki/Simon_%26_Schuster "Simon & Schuster") - [Maor, Eli](https://en.wikipedia.org/wiki/Eli_Maor "Eli Maor") (1998), e: The Story of a number, [Princeton University Press](https://en.wikipedia.org/wiki/Princeton_University_Press "Princeton University Press") [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-691-05854-7](https://en.wikipedia.org/wiki/Special:BookSources/0-691-05854-7 "Special:BookSources/0-691-05854-7") - Nahin, Paul J. (2006), Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, [Princeton University Press](https://en.wikipedia.org/wiki/Princeton_University_Press "Princeton University Press") [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-691-11822-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-11822-2 "Special:BookSources/978-0-691-11822-2") - [Paulos, John Allen](https://en.wikipedia.org/wiki/John_Allen_Paulos "John Allen Paulos") (1992), Beyond Numeracy: An Uncommon Dictionary of Mathematics, [Penguin Books](https://en.wikipedia.org/wiki/Penguin_Books "Penguin Books") [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-14-014574-5](https://en.wikipedia.org/wiki/Special:BookSources/0-14-014574-5 "Special:BookSources/0-14-014574-5") - Reid, Constance (various editions), [From Zero to Infinity](https://en.wikipedia.org/wiki/From_Zero_to_Infinity "From Zero to Infinity"), [Mathematical Association of America](https://en.wikipedia.org/wiki/Mathematical_Association_of_America "Mathematical Association of America") - Sandifer, C. Edward (2007), [Euler's Greatest Hits](https://books.google.com/books?id=sohHs7ExOsYC&pg=PA4), [Mathematical Association of America](https://en.wikipedia.org/wiki/Mathematical_Association_of_America "Mathematical Association of America") [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-88385-563-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-88385-563-8 "Special:BookSources/978-0-88385-563-8") - Stipp, David (2017). A Most Elegant Equation: Euler's formula and the beauty of mathematics. [Basic Books](https://en.wikipedia.org/wiki/Basic_Books "Basic Books"). - [Wells, David](https://en.wikipedia.org/w/index.php?title=David_G._Wells&action=edit&redlink=1 "David G. Wells (page does not exist)") (1990). "Are these the most beautiful?". [The Mathematical Intelligencer](https://en.wikipedia.org/wiki/The_Mathematical_Intelligencer "The Mathematical Intelligencer"). 12 (3): 37–41\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF03024015](https://doi.org/10.1007%2FBF03024015). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [121503263](https://api.semanticscholar.org/CorpusID:121503263). - [Wilson, Robin](https://en.wikipedia.org/wiki/Robin_Wilson_\(mathematician\) "Robin Wilson (mathematician)") (2018). Euler's Pioneering Equation: The most beautiful theorem in mathematics. [Oxford University Press](https://en.wikipedia.org/wiki/Oxford_University_Press "Oxford University Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-192-51406-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-192-51406-6 "Special:BookSources/978-0-192-51406-6") . - [Zeki, S.](https://en.wikipedia.org/wiki/Semir_Zeki "Semir Zeki"); Romaya, J. P.; Benincasa, D. M. T.; [Atiyah, M. F.](https://en.wikipedia.org/wiki/Michael_Atiyah "Michael Atiyah") (2014). "The experience of mathematical beauty and its neural correlates". Frontiers in Human Neuroscience. 8: 68. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3389/fnhum.2014.00068](https://doi.org/10.3389%2Ffnhum.2014.00068). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [3923150](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3923150). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [24592230](https://pubmed.ncbi.nlm.nih.gov/24592230). - [Intuitive understanding of Euler's formula](http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/)
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