Editing E (mathematical constant) (section)

{{Short description|2.71828..., base of natural logarithms}}
{{redirect|Euler's number|other Euler's numbers|List of things named after Leonhard Euler#Numbers}}
{{DISPLAYTITLE:{{mvar|e}} (mathematical constant)}}
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{{Infobox mathematical constant
| name = Euler's number
| symbol = {{mvar|e}}
| type = [[Transcendental number|Transcendental]]
| approximation = 2.71828...<ref name="OEIS decimal expansion" />
| discovery_date = 1685
| discovery_person = [[Jacob Bernoulli]]
| discovery_work = ''Quæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685''
| named_after = {{flatlist |
* [[Leonhard Euler]]
* [[John Napier]]
}}
}}
[[File:Hyperbola E.svg|thumb|237px|right|Graph of the equation {{math|1=''y'' = 1/''x''}}. Here, {{mvar|e}} is the unique number larger than 1 that makes the shaded [[area under the curve]] equal to 1.]]
{{e (mathematical constant)}}

The number '''{{mvar|e}}''' is a [[mathematical constant]] approximately equal to 2.71828 that is the [[base of a logarithm|base]] of the [[natural logarithm]] and [[exponential function]]. It is sometimes called '''Euler's number''', after the Swiss mathematician [[Leonhard Euler]], though this can invite confusion with [[Euler numbers]], or with [[Euler's constant]], a different constant typically denoted <math>\gamma</math>. Alternatively, {{mvar|e}} can be called '''Napier's constant''' after [[John Napier]].<ref name="Miller"/><ref name="Weisstein">{{Cite web|last=Weisstein|first=Eric W.|title=e|url=https://mathworld.wolfram.com/e.html|access-date=2020-08-10|website=mathworld.wolfram.com|language=en|ref=mathworld}}</ref> The Swiss mathematician [[Jacob Bernoulli]] discovered the constant while studying compound interest.<ref name="Pickover">{{cite book |title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics |edition=illustrated |first1=Clifford A. |last1=Pickover |publisher=Sterling Publishing Company |year=2009 |isbn=978-1-4027-5796-9 |page=166 |url=https://books.google.com/books?id=JrslMKTgSZwC}} [https://books.google.com/books?id=JrslMKTgSZwC&pg=PA166 Extract of page 166]</ref><ref name="OConnor">{{MacTutor|mode=cs1 |class=HistTopics |id=e |title=The number {{mvar|e}} |date=September 2001}}</ref>

The number {{mvar|e}} is of great importance in mathematics,<ref>{{Cite book |last=Sawyer |first=W. W. |title=Mathematician's Delight |publisher=Penguin |year=1961 |url=https://archive.org/details/MathemateciansDelight-W.W.Sawyer/page/n153/mode/2up|url-access = registration|pages=155 |language=en}}</ref> alongside 0, 1, [[Pi|{{pi}}]], and {{mvar|[[Imaginary unit|i]]}}. All five appear in one formulation of [[Euler's identity]] <math>e^{i\pi}+1=0</math> and play important and recurring roles across mathematics.<ref>{{cite book |title=Euler's Pioneering Equation: The most beautiful theorem in mathematics |edition=illustrated |first1=Robinn |last1=Wilson |publisher=Oxford University Press |year=2018 |isbn=978-0-19-251405-9 |page=(preface) |url=https://books.google.com/books?id=345HDwAAQBAJ}}</ref><ref>{{cite book |title=Pi: A Biography of the World's Most Mysterious Number |edition=illustrated |first1=Alfred S. |last1=Posamentier |first2=Ingmar |last2=Lehmann |publisher=Prometheus Books |year=2004 |isbn=978-1-59102-200-8 |page=68 |url=https://books.google.com/books?id=QFPvAAAAMAAJ}}</ref> Like the constant {{pi}}, {{mvar|e}} is [[Irrational number|irrational]], meaning that it cannot be represented as a ratio of integers, and moreover it is [[Transcendental number|transcendental]], meaning that it is not a root of any non-zero [[polynomial]] with rational coefficients.<ref name="Weisstein" /> To 30 decimal places, the value of {{mvar|e}} is:<ref name="OEIS decimal expansion">{{Cite OEIS|A001113|Decimal expansion of {{mvar|e}}}}</ref>
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