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{{DISPLAYTITLE:{{mvar|e}} (mathematical constant)}}
{{Short description|2.71828..., base of natural logarithms}}

{{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 = [[Biography:Jacob Bernoulli|Jacob Bernoulli]]
| discovery_work = ''Quæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685''
| named_after = {{flatlist |
* [[Biography:Leonhard Euler|Leonhard Euler]]
* [[Biography:John Napier|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|mathematical constant]] approximately equal to 2.71828 that can be characterized in many ways. It is the base of [[Natural logarithm|natural logarithm]]s. It is the [[Limit of a sequence|limit]] of {{math|(1 + 1/''n'')<sup>''n''</sup>}} as {{mvar|n}} approaches infinity, an expression that arises in the computation of [[Compound interest|compound interest]]. It can also be calculated as the sum of the infinite [[Series (mathematics)|series]]
<math display ="block">e = \sum\limits_{n = 0}^{\infty} \frac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots.</math>

It is also the unique positive number {{mvar|a}} such that the graph of the function {{math|1=''y'' = ''a''<sup>''x''</sup>}} has a [[Slope|slope]] of 1 at {{math|1=''x'' = 0}}.

The (natural) [[Exponential function|exponential function]] {{math|1=''f''(''x'') = ''e''<sup>''x''</sup>}} is the unique function {{mvar|f}} that equals its own [[Derivative|derivative]] and satisfies the equation {{math|1=''f''(0) = 1}}; hence one can also define {{mvar|e}} as {{math|''f''(1)}}. The natural logarithm, or logarithm to base {{mvar|e}}, is the [[Inverse function|inverse function]] to the natural exponential function. The natural logarithm of a number {{math|''k'' > 1}} can be defined directly as the [[Integral|area under]] the curve {{math|1=''y'' = 1/''x''}} between {{math|1=''x'' = 1}} and {{math|1=''x'' = ''k''}}, in which case {{mvar|e}} is the value of {{mvar|k}} for which this area equals {{math|1}} (see image). There are various other characterizations.

The number {{mvar|e}} is sometimes called '''Euler's number''', after the Swiss mathematician [[Biography:Leonhard Euler|Leonhard Euler]], though this can invite confusion with [[Euler's constant]], a different constant typically denoted <math>\gamma</math>. Alternatively, {{mvar|e}} can be called '''Napier's constant''' after [[Biography:John Napier|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 constant was discovered by the Swiss mathematician [[Biography:Jacob Bernoulli|Jacob Bernoulli]] 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|polynomial]] with rational coefficients.<ref name="Weisstein" /> To 40 decimal places, the value of {{mvar|e}} is:<ref name="OEIS decimal expansion">{{Cite OEIS|A001113|Decimal expansion of {{mvar|e}}}}</ref>
{{block indent
|{{gaps|2.71828|18284|59045|23536|02874|71352|66249|77572...}}
}}

== History ==
The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by [[Biography:John Napier|John Napier]]. However, this did not contain the constant itself, but simply a list of [[Natural logarithm|logarithms to the base <math>e</math>]]. It is assumed that the table was written by William Oughtred. In 1661, [[Biography:Christiaan Huygens|Christiaan Huygens]] studied how to compute logarithms by geometrical methods and calculated a quantity that, in retrospect, is the base-10 logarithm of {{mvar|e}}, but he did not recognize {{mvar|e}} itself as a quantity of interest.<ref name="OConnor" /><ref>{{cite journal|url=http://www.math.bas.bg/mathmod/Proceedings_CTF/CTF-1981/files_CTF-1981/CTF-1981-254-257.pdf |first=E. M. |last=Bruins |title=The Computation of Logarithms by Huygens |journal=Constructive Function Theory |year=1983 |pages=254–257}}</ref>

The constant itself was introduced by [[Biography:Jacob Bernoulli|Jacob Bernoulli]] in 1683, for solving the problem of continuous compounding of interest.<ref name="Bernoulli, 1690">Jacob Bernoulli considered the problem of continuous compounding of interest, which led to a series expression for {{mvar|e}}.  See: Jacob Bernoulli (1690) "Quæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685" (Some questions about interest, with a solution of a problem about games of chance, proposed in the ''Journal des Savants'' (''Ephemerides Eruditorum Gallicanæ''), in the year (anno) 1685.**), ''Acta eruditorum'', pp.&nbsp;219–23.  [https://books.google.com/books?id=s4pw4GyHTRcC&pg=PA222 On page 222], Bernoulli poses the question: ''"Alterius naturæ hoc Problema est: Quæritur, si creditor aliquis pecuniæ summam fænori exponat, ea lege, ut singulis momentis pars proportionalis usuræ annuæ sorti annumeretur; quantum ipsi finito anno debeatur?"''  (This is a problem of another kind: The question is, if some lender were to invest [a] sum of money [at] interest, let it accumulate, so that [at] every moment [it] were to receive [a] proportional part of [its] annual interest; how much would be owing [at the] end of [the] year?)  Bernoulli constructs a power series to calculate the answer, and then writes: " … quæ nostra serie [mathematical expression for a geometric series] &c. major est. … si {{math|1=''a'' = ''b''}}, debebitur plu quam {{math|2½''a''}} & minus quam {{math|3''a''}}." ( … which our series [a geometric series] is larger [than]. … if {{math|1=''a''=''b''}}, [the lender] will be owed more than {{math|2½''a''}} and less than {{math|3''a''}}.)  If {{math|1=''a'' = ''b''}}, the geometric series reduces to the series for {{math|''a'' × ''e''}}, so {{math|2.5 < ''e'' < 3}}.  (** The reference is to a problem which Jacob Bernoulli posed and which appears in the ''Journal des Sçavans'' of 1685 at the bottom of [http://gallica.bnf.fr/ark:/12148/bpt6k56536t/f307.image.langEN page 314.])</ref><ref>{{cite book|author1=Carl Boyer|author2=Uta Merzbach|title=A History of Mathematics|url=https://archive.org/details/historyofmathema00boye|url-access=registration|page=[https://archive.org/details/historyofmathema00boye/page/419 419]|publisher=Wiley|year=1991|isbn=978-0-471-54397-8|edition=2nd}}</ref>
In his solution, the constant {{mvar|e}} occurs as the [[Limit (mathematics)|limit]]
<math display="block">\lim_{n\to \infty} \left( 1 + \frac{1}{n} \right)^n,</math>
where {{mvar|n}} represents the number of intervals in a year on which the compound interest is evaluated (for example, <math>n=12</math> for monthly compounding).

The first symbol used for this constant was the letter {{mvar|b}} by Gottfried Leibniz in letters to [[Biography:Christiaan Huygens|Christiaan Huygens]] in 1690 and 1691.<ref>{{cite web |url=https://leibniz.uni-goettingen.de/files/pdf/Leibniz-Edition-III-5.pdf |title=Sämliche Schriften Und Briefe |last=Leibniz |first=Gottfried Wilhelm |date=2003 |language=de |quote=look for example letter nr. 6}}</ref>

[[Biography:Leonhard Euler|Leonhard Euler]] started to use the letter {{mvar|e}} for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons,<ref name="Meditatio">Euler, ''[https://scholarlycommons.pacific.edu/euler-works/853/ Meditatio in experimenta explosione tormentorum nuper instituta]''. {{lang|la|Scribatur pro numero cujus logarithmus est unitas, e, qui est 2,7182817…}} (English: Written for the number of which the logarithm has the unit, e, that is 2,7182817...")</ref> and in a letter to [[Biography:Christian Goldbach|Christian Goldbach]] on 25 November 1731.<ref>Lettre XV.  Euler à Goldbach, dated November 25, 1731 in: P.H. Fuss, ed., ''Correspondance Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIeme Siècle'' … (Mathematical and physical correspondence of some famous geometers of the 18th century), vol. 1, (St. Petersburg, Russia: 1843), pp.&nbsp;56–60, see especially [https://books.google.com/books?id=gf1OEXIQQgsC&pg=PA58 p. 58.]  From p. 58: ''" … ( e denotat hic numerum, cujus logarithmus hyperbolicus est = 1), … "'' ( … (e denotes that number whose hyperbolic [i.e., natural] logarithm is equal to 1) … )</ref><ref>{{Cite book|last=Remmert|first=Reinhold|title=Theory of Complex Functions|page=136|publisher=[[Physics:Springer-Verlag|Springer-Verlag]]|year=1991|isbn=978-0-387-97195-7}}</ref>  The first appearance of {{mvar|e}} in a printed publication was in Euler's ''[[Mechanica]]'' (1736).<ref>Leonhard Euler, ''Mechanica, sive Motus scientia analytice exposita'' (St. Petersburg (Petropoli), Russia: Academy of Sciences, 1736), vol. 1, Chapter 2, Corollary 11, paragraph 171, p.&nbsp;68.  [https://books.google.com/books?id=qalsP7uMiV4C&pg=PA68 From page 68:] ''Erit enim <math>\frac{dc}{c} = \frac{dy ds}{rdx}</math> seu <math>c = e^{\int\frac{dy ds}{rdx}}</math> ubi {{mvar|e}} denotat numerum, cuius logarithmus hyperbolicus est 1.''  (So it [i.e., {{mvar|c}}, the speed] will be <math>\frac{dc}{c} = \frac{dy ds}{rdx}</math> or <math>c = e^{\int\frac{dy ds}{rdx}}</math>, where {{mvar|e}} denotes the number whose hyperbolic [i.e., natural] logarithm is 1.)</ref> It is unknown why Euler chose the letter {{mvar|e}}.<ref>{{cite book |last=Calinger |first=Ronald |date=2016 |title=Leonhard Euler: Mathematical Genius in the Enlightenment |publisher=Princeton University Press |isbn=978-0-691-11927-4}} p. 124.</ref> Although some researchers used the letter {{mvar|c}} in the subsequent years, the letter {{mvar|e}} was more common and eventually became standard.<ref name="Miller">{{cite web |last1=Miller |first1=Jeff |title=Earliest Uses of Symbols for Constants |url=https://mathshistory.st-andrews.ac.uk/Miller/mathsym/constants/ |website=MacTutor |publisher=University of St. Andrews, Scotland |access-date=31 October 2023}}</ref>

Euler proved that {{mvar|e}} is the sum of the infinite series
<math display="block">e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots ,</math>
where {{math|''n''!}} is the [[Factorial|factorial]] of {{mvar|n}}.<ref name="OConnor" /> The equivalence of the two characterizations using the limit and the infinite series can be proved via the [[Binomial theorem|binomial theorem]].<ref>{{cite book|first=Walter |last=Rudin |title=Principles of Mathematical Analysis |edition=3rd |publisher=McGraw–Hill |year=1976 |pages=63–65 |isbn=0-07-054235-X}}</ref>

== Applications ==

=== Compound interest ===
[[File:Compound Interest with Varying Frequencies.svg|thumb|right|350px|The effect of earning 20% annual interest on an {{nowrap|initial $1,000}} investment at various compounding frequencies. The limiting curve on top is the graph <math>y=1000e^{0.2t}</math>, where {{mvar|y}} is in dollars, {{mvar|t}} in years, and 0.2 = 20%.]]

Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest:<ref name="OConnor" />
{{Blockquote|An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?}}

If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding {{nowrap|1=$1.00 × 1.5<sup>2</sup> = $2.25}} at the end of the year. Compounding quarterly yields {{nowrap|1=$1.00 × 1.25<sup>4</sup> = $2.44140625}}, and compounding monthly yields {{nowrap|1=$1.00 × (1 + 1/12)<sup>12</sup> = $2.613035...}}. If there are {{mvar|n}} compounding intervals, the interest for each interval will be {{math|100%/''n''}} and the value at the end of the year will be $1.00&nbsp;×&nbsp;{{math|(1 + 1/''n'')<sup>''n''</sup>}}.<ref name="Gonick"/><ref name=":0" />

Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger {{mvar|n}} and, thus, smaller compounding intervals.<ref name="OConnor" /> Compounding weekly ({{math|1=''n'' = 52}}) yields $2.692596..., while compounding daily ({{math|1=''n'' = 365}}) yields $2.714567... (approximately two cents more). The limit as {{mvar|n}} grows large is the number that came to be known as {{mvar|e}}. That is, with ''continuous'' compounding, the account value will reach $2.718281828... More generally, an account that starts at $1 and offers an annual interest rate of {{mvar|R}} will, after {{mvar|t}} years, yield {{math|''e''<sup>''Rt''</sup>}} dollars with continuous compounding. Here, {{mvar|R}} is the decimal equivalent of the rate of interest expressed as a ''percentage'', so for 5% interest, {{math|1=''R'' = 5/100 = 0.05}}.<ref name="Gonick">{{cite book
 | last = Gonick
 | first = Larry
 | year = 2012
 | title = The Cartoon Guide to Calculus
 | publisher = William Morrow
 | url = https://www.larrygonick.com/titles/science/cartoon-guide-to-calculus-2/
 | isbn = 978-0-06-168909-3
 | pages = 29–32
}}</ref><ref name=":0" />

=== Bernoulli trials ===
[[File:Bernoulli trial sequence.svg|thumb|300px|Graphs of probability {{mvar|P}} of {{em|not}} observing independent events each of probability {{math|1/''n''}} after {{mvar|n}} Bernoulli trials, and {{math|1 − ''P''&thinsp;}} vs {{mvar|n}}&thinsp;; it can be observed that as {{mvar|n}} increases, the probability of a {{math|1/''n''}}-chance event never appearing after ''n'' tries rapidly {{nowrap|converges to {{math|1/''e''}}.}}]]
The number {{mvar|e}} itself also has applications in [[Probability theory|probability theory]], in a way that is not obviously related to exponential growth. Suppose that a gambler plays a slot machine that pays out with a probability of one in {{mvar|n}} and plays it {{mvar|n}} times. As {{mvar|n}} increases, the probability that gambler will lose all {{mvar|n}} bets approaches {{math|1/''e''}}. For {{math|1=''n'' = 20}}, this is already approximately 1/2.789509....

This is an example of a [[Bernoulli trial]] process. Each time the gambler plays the slots, there is a one in {{mvar|n}} chance of winning. Playing {{mvar|n}} times is modeled by the [[Binomial distribution|binomial distribution]], which is closely related to the [[Binomial theorem|binomial theorem]] and [[Pascal's triangle]]. The probability of winning {{mvar|k}} times out of {{mvar|n}} trials is:<ref>{{cite book|first=Mehran |last=Kardar |title=Statistical Physics of Particles |title-link=Statistical Physics of Particles |year=2007 |publisher={{wipe|Cambridge University Press}} |isbn=978-0-521-87342-0 |oclc=860391091 |page=41}}</ref>
:<math>\Pr[k~\mathrm{wins~of}~n] = \binom{n}{k} \left(\frac{1}{n}\right)^k\left(1 - \frac{1}{n}\right)^{n-k}.</math>

In particular, the probability of winning zero times ({{math|1=''k'' = 0}}) is
:<math>\Pr[0~\mathrm{wins~of}~n] = \left(1 - \frac{1}{n}\right)^{n}.</math>

The limit of the above expression, as {{mvar|n}} tends to infinity, is precisely {{math|1/''e''}}.

=== Exponential growth and decay ===
{{Further|Exponential growth|Exponential decay}}
[[Exponential growth]] is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous [[Rate (mathematics)#Of change|rate of change]] (that is, the [[Derivative|derivative]]) of a quantity with respect to time is [[Proportionality (mathematics)|proportional]] to the quantity itself.<ref name=":0">{{cite book|chapter-url=https://openstax.org/books/college-algebra-2e/pages/6-1-exponential-functions |chapter=6.1 Exponential Functions |title=College Algebra 2e |publisher=OpenStax |first1=Jay |last1=Abramson |display-authors=0 |year=2023 |isbn=978-1-951693-41-1}}</ref> Described as a function, a quantity undergoing exponential growth is an [[Exponentiation#Power functions|exponential function]] of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as [[Quadratic growth|quadratic growth]]). If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing [[Exponential decay|exponential decay]] instead. The law of exponential growth can be written in different but mathematically equivalent forms, by using a different [[Exponentiation|base]], for which the number {{mvar|e}} is a common and convenient choice:
<math display="block">x(t) = x_0\cdot e^{kt} = x_0\cdot e^{t/\tau}.</math>
Here, <math>x_0</math> denotes the initial value of the quantity {{mvar|x}}, {{mvar|k}} is the growth constant, and <math>\tau</math> is the time it takes the quantity to grow by a factor of {{mvar|e}}.

=== Standard normal distribution ===
{{Main|Normal distribution}}

The normal distribution with zero mean and unit standard deviation is known as the ''standard normal distribution'',{{r|openstax}} given by the [[Physics:Probability density function|probability density function]]<!--{{r|nordistr}}-->
<math display="block"> \phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2} x^2}. </math>

The constraint of unit standard deviation (and thus also unit variance) results in the {{frac2|1|2}} in the exponent, and the constraint of unit total area under the curve <math>\phi(x)</math> results in the factor <math>\textstyle 1/\sqrt{2\pi}</math>. This function is symmetric around {{math|1=''x'' = 0}}, where it attains its maximum value <math>\textstyle 1/\sqrt{2\pi}</math>, and has [[Inflection point|inflection point]]s at {{math|1=''x'' = ±1}}.

=== Derangements ===
{{Main|Derangement}}
Another application of {{mvar|e}}, also discovered in part by Jacob Bernoulli along with Pierre Remond de Montmort, is in the problem of [[Derangement|derangement]]s, also known as the ''hat check problem'':<ref>{{cite book| last1=Grinstead|first1= Charles M. |last2= Snell|first2= James Laurie |date=1997 |url=http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html |title= Introduction to Probability |archive-url=https://web.archive.org/web/20110727200156/http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html |archive-date=2011-07-27 |type=published online under the GFDL|publisher=American Mathematical Society |page= 85 |isbn=978-0-8218-9414-9 |author2-link=J. Laurie Snell }}</ref> {{mvar|n}} guests are invited to a party and, at the door, the guests all check their hats with the butler, who in turn places the hats into {{mvar|n}} boxes, each labelled with the name of one guest. But the butler has not asked the identities of the guests, and so puts the hats into boxes selected at random. The problem of de Montmort is to find the probability that ''none'' of the hats gets put into the right box. This probability, denoted by <math>p_n\!</math>, is:

:<math>p_n = 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \cdots + \frac{(-1)^n}{n!} = \sum_{k = 0}^n \frac{(-1)^k}{k!}.</math>

As {{mvar|n}} tends to infinity, {{math|''p''<sub>''n''</sub>}} approaches {{math|1/''e''}}. Furthermore, the number of ways the hats can be placed into the boxes so that none of the hats are in the right box is {{math|''n''!/''e'',}} [[Rounding|rounded]] to the nearest integer, for every positive&nbsp;{{mvar|n}}.<ref>{{cite book | last=Knuth |first=Donald |date= 1997 | title=The Art of Computer Programming |title-link=The Art of Computer Programming | volume =I|publisher= Addison-Wesley|page= 183 |isbn=0-201-03801-3}}</ref>

=== Optimal planning problems ===
The maximum value of <math>\sqrt[x]{x}</math> occurs at <math>x = e</math>.  Equivalently, for any value of the base {{math|''b'' > 1}}, it is the case that the maximum value of <math>x^{-1}\log_b x</math> occurs at <math>x = e</math> ([[Steiner's calculus problem|Steiner's problem]], discussed below).

This is useful in the problem of a stick of length {{mvar|L}} that is broken into {{mvar|n}} equal parts.  The value of {{mvar|n}} that maximizes the product of the lengths is then either<ref name="Finch-2003-p14">{{cite book|title=Mathematical constants|url=https://archive.org/details/mathematicalcons0000finc|url-access=registration|author=Steven Finch|year=2003|publisher=Cambridge University Press|page=[https://archive.org/details/mathematicalcons0000finc/page/14 14]|isbn=978-0-521-81805-6}}</ref>
:<math>n = \left\lfloor \frac{L}{e} \right\rfloor</math> or <math>\left\lceil \frac{L}{e} \right\rceil.</math>

The quantity <math>x^{-1}\log_b x</math> is also a measure of information gleaned from an event occurring with probability <math>1/x</math>, so that essentially the same optimal division appears in optimal planning problems like the [[Secretary problem|secretary problem]].

=== Asymptotics ===
The number {{mvar|e}} occurs naturally in connection with many problems involving asymptotics. An example is Stirling's formula for the [[Asymptotic analysis|asymptotics]] of the factorial function, in which both the numbers {{mvar|e}} and [[Pi|{{pi}}]] appear:<ref name="greg">{{cite book|first=Greg |last=Gbur |year=2011 |title=Mathematical Methods for Optical Physics and Engineering |isbn=978-0-521516-10-5 |publisher=Cambridge University Press |page=779}}</ref>
<math display="block>n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n.</math>

As a consequence,<ref name="greg"/>
<math display="block>e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}} .</math>

== Properties ==

=== Calculus ===
[[File:Exp derivative at 0.svg|thumb|right|The graphs of the functions {{math|''x'' ↦ ''a''{{sup|''x''}}}} are shown for {{math|1=''a'' = 2}} (dotted), {{math|1=''a'' = ''e''}} (blue), and {{math|1=''a'' = 4}} (dashed). They all pass through the point {{math|(0,1)}}, but the red line (which has slope {{math|1}}) is tangent to only {{math|''e''{{sup|''x''}}}} there.]]

[[File:Ln+e.svg|thumb|right|The value of the natural log function for argument {{mvar|e}}, i.e. {{math|ln ''e''}}, equals {{math|1.}}]]

The principal motivation for introducing the number {{mvar|e}}, particularly in [[Calculus|calculus]], is to perform differential and integral calculus with [[Exponential function|exponential function]]s and [[Logarithm|logarithm]]s.<ref name="kline">{{cite book | last = Kline | first = M. | year = 1998 | title = Calculus: An intuitive and physical approach | page = [https://books.google.com/books?id=YdjK_rD7BEkC&pg=PA337 337 ff] | publisher = Dover Publications | isbn = 0-486-40453-6}}</ref> A general exponential {{nowrap|function {{math|''y'' {{=}} ''a''<sup>''x''</sup>}}}} has a derivative, given by a [[Limit of a function|limit]]:

:<math>\begin{align}
  \frac{d}{dx}a^x
    &= \lim_{h\to 0}\frac{a^{x+h} - a^x}{h} = \lim_{h\to 0}\frac{a^x a^h - a^x}{h} \\
    &= a^x \cdot \left(\lim_{h\to 0}\frac{a^h - 1}{h}\right).
\end{align}</math>

The parenthesized limit on the right is independent of the {{nowrap|variable {{mvar|x}}.}} Its value turns out to be the logarithm of {{mvar|a}} to base {{mvar|e}}. Thus, when the value of {{mvar|a}} is set {{nowrap|to {{mvar|e}},}} this limit is equal {{nowrap|to {{math|1}},}} and so one arrives at the following simple identity:
:<math>\frac{d}{dx}e^x = e^x.</math>

Consequently, the exponential function with base {{mvar|e}} is particularly suited to doing calculus. {{nowrap|Choosing {{mvar|e}}}} (as opposed to some other number) as the base of the exponential function makes calculations involving the derivatives much simpler.

Another motivation comes from considering the derivative of the base-{{mvar|a}} logarithm (i.e., {{math|log<sub>''a''</sub> ''x''}}),{{r|kline}} for&nbsp;{{math|''x'' > 0}}:

:<math>\begin{align}
  \frac{d}{dx}\log_a x
    &= \lim_{h\to 0}\frac{\log_a(x + h) - \log_a(x)}{h} \\
    &= \lim_{h\to 0}\frac{\log_a(1 + h/x)}{x\cdot h/x} \\
    &= \frac{1}{x}\log_a\left(\lim_{u\to 0}(1 + u)^\frac{1}{u}\right) \\
    &= \frac{1}{x}\log_a e,
\end{align}</math>

where the substitution {{math|''u'' {{=}} ''h''/''x''}} was made. The base-{{mvar|a}} logarithm of {{mvar|e}} is 1, if {{mvar|a}} equals {{mvar|e}}. So symbolically,
:<math>\frac{d}{dx}\log_e x = \frac{1}{x}.</math>

The logarithm with this special base is called the [[Natural logarithm|natural logarithm]], and is denoted as {{math|ln}}; it behaves well under differentiation since there is no undetermined limit to carry through the calculations.

Thus, there are two ways of selecting such special numbers {{mvar|a}}. One way is to set the derivative of the exponential function {{math|''a''<sup>''x''</sup>}} equal to {{math|''a''<sup>''x''</sup>}}, and solve for {{mvar|a}}. The other way is to set the derivative of the base {{mvar|a}} logarithm to {{math|1/''x''}} and solve for {{mvar|a}}. In each case, one arrives at a convenient choice of base for doing calculus. It turns out that these two solutions for {{mvar|a}} are actually ''the same'': the number {{mvar|e}}.

[[File:Area under rectangular hyperbola.svg|thumb|right|The five colored regions are of equal area, and define units of [[Hyperbolic angle|hyperbolic angle]] along the {{nowrap|[[Hyperbola|hyperbola]] <math>xy=1.</math>}}]]

The [[Taylor series]] for the exponential function can be deduced from the facts that the exponential function is its own derivative and that it equals 1 when evaluated at 0:<ref>{{cite book|first1=Gilbert |last1=Strang |first2=Edwin |last2=Herman |title=Calculus, volume 2 |publisher=OpenStax |display-authors=0 |chapter=6.3 Taylor and Maclaurin Series |chapter-url=https://openstax.org/books/calculus-volume-2/pages/6-3-taylor-and-maclaurin-series |year=2023 |isbn=978-1-947172-14-2}}</ref>
<math display="block">e^x = \sum_{n=0}^\infty \frac{x^n}{n!}.</math>
Setting <math>x = 1</math> recovers the definition of {{mvar|e}} as the sum of an infinite series.

The natural logarithm function can be defined as the integral from 1 to <math>x</math> of <math>1/t</math>, and the exponential function can then be defined as the inverse function of the natural logarithm. The number {{mvar|e}} is the value of the exponential function evaluated at <math>x = 1</math>, or equivalently, the number whose natural logarithm is 1. It follows that {{mvar|e}} is the unique positive real number such that
<math display="block">\int_1^e \frac{1}{t} \, dt = 1.</math>

Because {{math|''e''<sup>''x''</sup>}} is the unique function ([[Up to|up to]] multiplication by a constant {{mvar|K}}) that is equal to its own [[Derivative|derivative]],

<math display="block">\frac{d}{dx}Ke^x = Ke^x,</math>

it is therefore its own [[Antiderivative|antiderivative]] as well:<ref>{{cite book|first1=Gilbert |last1=Strang |first2=Edwin |last2=Herman |title=Calculus, volume 2 |publisher=OpenStax |display-authors=0 |chapter=4.10 Antiderivatives |chapter-url=https://openstax.org/books/calculus-volume-1/pages/4-10-antiderivatives |year=2023 |isbn=978-1-947172-14-2}}</ref>

<math display="block">\int Ke^x\,dx = Ke^x + C .</math>

Equivalently, the family of functions

<math display="block">y(x) = Ke^x</math>

where {{mvar|K}} is any real or complex number, is the full solution to the [[Differential equation|differential equation]]

<math display="block">y' = y .</math>

=== Inequalities ===
[[File:Exponentials vs x+1.pdf|thumb|right|Exponential functions {{math|''y'' {{=}} 2<sup>''x''</sup>}} and {{math|''y'' {{=}} 4<sup>''x''</sup>}} intersect the graph of {{math|''y'' {{=}} ''x'' + 1}}, respectively, at {{math|''x'' {{=}} 1}} and {{math|''x'' {{=}} -1/2}}.  The number {{mvar|e}} is the unique base such that {{math|''y'' {{=}} ''e''<sup>''x''</sup>}} intersects only at {{math|''x'' {{=}} 0}}.  We may infer that {{mvar|e}} lies between 2 and 4.]]
The number {{mvar|e}} is the unique real number such that
<math display="block">\left(1 + \frac{1}{x}\right)^x < e < \left(1 + \frac{1}{x}\right)^{x+1}</math>
for all positive {{mvar|x}}.<ref>{{cite book|last1=Dorrie|first1=Heinrich|title=100 Great Problems of Elementary Mathematics|date=1965|publisher=Dover|pages=44–48}}</ref>

Also, we have the inequality
<math display="block">e^x \ge x + 1</math>
for all real {{mvar|x}}, with equality if and only if {{math|''x'' {{=}} 0}}.  Furthermore, {{mvar|e}} is the unique base of the exponential for which the inequality {{math|''a''<sup>''x''</sup> ≥ ''x'' + 1}} holds for all {{mvar|x}}.<ref>A standard calculus exercise using the [[Mean value theorem|mean value theorem]]; see for example Apostol (1967) ''Calculus'', §&nbsp;6.17.41.</ref>  This is a limiting case of [[Bernoulli's inequality]].

=== Exponential-like functions ===
[[File:Xth root of x.svg|thumb|right|250px|The global maximum of {{math|{{sqrt|''x''|''x''}}}} {{nowrap|occurs at {{math|''x'' {{=}} ''e''}}.}}]]

[[Steiner's calculus problem|Steiner's problem]] asks to find the global maximum for the function

<math display="block"> f(x) = x^\frac{1}{x} .</math>

This maximum occurs precisely at {{math|''x'' {{=}} ''e''}}.  (One can check that the derivative of {{math|ln ''f''(''x'')}} is zero only for this value of&nbsp;{{mvar|x}}.)

Similarly, {{math|''x'' {{=}} 1/''e''}} is where the global minimum occurs for the function

<math display="block"> f(x) = x^x .</math>

The infinite [[Tetration|tetration]]

:<math> x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} </math> or <math>{^\infty}x</math>

converges if and only if {{math|''x'' ∈ [(1/''e'')<sup>''e''</sup>, ''e''<sup>1/''e''</sup>] ≈ [0.06599, 1.4447] }},<ref>{{Cite OEIS|A073230|Decimal expansion of (1/e)^e}}</ref><ref>{{Cite OEIS|A073229|Decimal expansion of e^(1/e)}}</ref> shown by a theorem of [[Biography:Leonhard Euler|Leonhard Euler]].<ref>Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." ''Acta Acad. Scient. Petropol. 2'', 29–51, 1783. Reprinted in Euler, L. ''Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae''. Leipzig, Germany: Teubner, pp. 350–369, 1921. ([http://math.dartmouth.edu/~euler/docs/originals/E532.pdf facsimile])</ref><ref>{{Cite journal |last=Knoebel |first=R. Arthur |date=1981 |title=Exponentials Reiterated |url=https://www.jstor.org/stable/2320546 |journal=The American Mathematical Monthly |volume=88 |issue=4 |pages=235–252 |doi=10.2307/2320546 |jstor=2320546 |issn=0002-9890}}</ref><ref>{{Cite journal |last=Anderson |first=Joel |date=2004 |title=Iterated Exponentials |url=https://www.jstor.org/stable/4145040 |journal=The American Mathematical Monthly |volume=111 |issue=8 |pages=668–679 |doi=10.2307/4145040 |jstor=4145040 |issn=0002-9890}}</ref>

=== Number theory ===
The real number {{mvar|e}} is [[Irrational number|irrational]]. [[Biography:Leonhard Euler|Euler]] proved this by showing that its simple continued fraction expansion does not terminate.<ref>{{cite web|url=http://vanilla47.com/PDFs/Leonhard%20Euler/How%20Euler%20Did%20It%20by%20Ed%20Sandifer/Who%20proved%20e%20is%20irrational.pdf|title=How Euler Did It: Who proved {{mvar|e}} is Irrational?|last=Sandifer|first=Ed|date=Feb 2006|publisher=MAA Online|access-date=2010-06-18|url-status=dead|archive-url=https://web.archive.org/web/20140223072640/http://vanilla47.com/PDFs/Leonhard%20Euler/How%20Euler%20Did%20It%20by%20Ed%20Sandifer/Who%20proved%20e%20is%20irrational.pdf|archive-date=2014-02-23}}</ref> (See also [[Biography:Joseph Fourier|Fourier]]'s [[Proof that e is irrational|proof that {{mvar|e}} is irrational]].)

Furthermore, by the [[Lindemann–Weierstrass theorem]], {{mvar|e}} is [[Transcendental number|transcendental]], meaning that it is not a solution of any non-zero polynomial equation with rational coefficients. It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with [[Liouville number]]); the proof was given by [[Biography:Charles Hermite|Charles Hermite]] in 1873.<ref>{{cite book | last=Gelfond | first=A. O. |  translator-last=Boron | translator-first=Leo F. | translator-link=Leo F. Boron | orig-year=1960 | year=2015 | title=Transcendental and Algebraic Numbers | publisher=[[Social:Dover Publications|Dover Publications]] |location=New York |series=Dover Books on Mathematics | isbn=978-0-486-49526-2 |mr=0057921 | url={{Google books|408wBgAAQBAJ|Transcendental and Algebraic Numbers|plainurl=yes}} |page=41}}</ref>

It is conjectured that {{mvar|e}} is [[Normal number|normal]], meaning that when {{mvar|e}} is expressed in any [[Radix|base]] the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length).<ref>{{cite book|author-link=Davar Khoshnevisan |last=Khoshnevisan |first=Davar |chapter=Normal numbers are normal |year=2006 |title=Clay Mathematics Institute Annual Report 2006 |publisher=Clay Mathematics Institute |pages=15, 27–31 |chapter-url=http://www.claymath.org/library/annual_report/ar2006/06report_normalnumbers.pdf
}}</ref>

In [[Algebraic geometry|algebraic geometry]], a ''[[Period (algebraic geometry)|period]]'' is a number that can be expressed as an integral of an [[Algebraic function|algebraic function]] over an algebraic [[Domain of a function|domain]]. The constant {{pi}} is a period, but it is conjectured that {{mvar|e}} is not.<ref>{{Cite web |last1=Kontsevich |first1=Maxim |last2=Zagier |first2=Don |author-link2=Don Zagier |year=2001 |title=Periods |url=https://www.ihes.fr/~maxim/TEXTS/Periods.pdf}}</ref>

=== Complex numbers ===
The [[Exponential function|exponential function]] {{math|''e''<sup>''x''</sup>}} may be written as a [[Taylor series]]

<math display="block"> e^{x} = 1 + {x \over 1!} + {x^{2} \over 2!} + {x^{3} \over 3!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}.</math>

Because this series is [[Convergent series|convergent]] for every [[Complex number|complex]] value of {{mvar|x}}, it is commonly used to extend the definition of {{math|''e''<sup>''x''</sup>}} to the complex numbers.<ref name="Dennery">{{cite book |first1=P. |last1=Dennery |first2=A. |last2=Krzywicki |title=Mathematics for Physicists |year=1995 |orig-year=1967 |publisher=Dover |isbn=0-486-69193-4 |pages=23–25}}</ref> This, with the Taylor series for [[Trigonometric functions|{{math|sin}} and {{math|cos ''x''}}]], allows one to derive [[Euler's formula]]:

<math display="block">e^{ix} = \cos x + i\sin x ,</math>

which holds for every complex {{mvar|x}}.<ref name="Dennery"/> The special case with {{math|''x'' {{=}} [[Pi|{{pi}}]]}} is [[Euler's identity]]:

<math display="block">e^{i\pi} + 1 = 0 ,</math>
which is considered to be an exemplar of [[Mathematical beauty|mathematical beauty]] as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in [[Lindemann–Weierstrass theorem#Transcendence of e and π|a proof]] that {{pi}} is [[Transcendental number|transcendental]], which implies the impossibility of [[Squaring the circle|squaring the circle]].<ref>{{cite arXiv|title=The Transcendence of π and the Squaring of the Circle|last1=Milla|first1=Lorenz|eprint=2003.14035|year=2020|class=math.HO }}</ref><ref>{{Cite web|url=https://math.colorado.edu/~rohi1040/expository/eistranscendental.pdf |archive-url=https://web.archive.org/web/20210623215444/https://math.colorado.edu/~rohi1040/expository/eistranscendental.pdf |archive-date=2021-06-23 |url-status=live|title=e is transcendental|last=Hines|first=Robert|website=University of Colorado}}</ref> Moreover, the identity implies that, in the [[Principal branch|principal branch]] of the logarithm,<ref name="Dennery"/>

<math display="block">\ln (-1) = i\pi .</math>

Furthermore, using the laws for exponentiation,

<math display="block">(\cos x + i\sin x)^n = \left(e^{ix}\right)^n = e^{inx} = \cos (nx) + i \sin (nx) ,</math>

which is [[De Moivre's formula|de Moivre's formula]].<ref name="Sultan"/>

The expressions of {{math|cos ''x''}} and {{math|sin ''x''}} in terms of the [[Exponential function|exponential function]] can be deduced from the Taylor series:<ref name="Dennery"/>
<math display="block">
  \cos x = \frac{e^{ix} + e^{-ix}}{2} , \qquad
  \sin x = \frac{e^{ix} - e^{-ix}}{2i}.
</math>

The expression
<math display=inline>\cos x + i \sin x</math>
is sometimes abbreviated as {{math|cis(''x'')}}.<ref name="Sultan">{{cite book|title=The Mathematics That Every Secondary School Math Teacher Needs to Know |first1=Alan |last1=Sultan |first2=Alice F. |last2=Artzt |year=2010 |pages=326–328 |publisher=Routledge |isbn=978-0-203-85753-3}}</ref>

== Representations ==

The number {{mvar|e}} can be represented in a variety of ways: as an infinite series, an [[Infinite product|infinite product]], a [[Continued fraction|continued fraction]], or a [[Limit of a sequence|limit of a sequence]]. In addition to the limit and the series given above, there is also the [[Continued fraction|continued fraction]]
<!--move to history section or say <ref>[[Biography:Leonhard Euler|Euler]] was the first showed that {{mvar|e}} can be represented as a continued fraction.</ref>-->

:<math>
  e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, ..., 1, 2n, 1, ...],
</math><ref>{{cite book |last=Hofstadter|first=D.R.|title= Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought|publisher= Basic Books |date=1995|isbn=0-7139-9155-0}}</ref><ref name="OEIS continued fraction">{{Cite OEIS|A003417|Continued fraction for e}}</ref>

which written out looks like

:<math>e = 2 +
\cfrac{1}
   {1 + \cfrac{1}
      {2 + \cfrac{1}
         {1 + \cfrac{1}
            {1 + \cfrac{1}
               {4 + \cfrac{1}
            {1 + \cfrac{1}
               {1 + \ddots}
                  }
               }
            }
         }
      }
   }
.
</math>

The following infinite product evaluates to {{mvar|e}}:<ref name="Finch-2003-p14"/>
<math display="block">e = \frac{2}{1} \left(\frac{4}{3}\right)^{1/2} \left(\frac{6 \cdot 8}{5 \cdot 7}\right)^{1/4} \left(\frac{10 \cdot 12 \cdot 14 \cdot 16}{9 \cdot 11 \cdot 13 \cdot 15}\right)^{1/8} \cdots.</math>

Many other series, sequence, continued fraction, and infinite product representations of {{mvar|e}} have been proved.

=== Stochastic representations ===
In addition to exact analytical expressions for representation of {{mvar|e}}, there are stochastic techniques for estimating {{mvar|e}}.  One such approach begins with an infinite sequence of independent random variables {{math|''X''<sub>1</sub>}}, {{math|''X''<sub>2</sub>}}..., drawn from the [[Uniform distribution (continuous)|uniform distribution]] on [0, 1]. Let {{mvar|V}} be the least number {{mvar|n}} such that the sum of the first {{mvar|n}} observations exceeds 1:

:<math>V = \min\left\{ n \mid X_1 + X_2 + \cdots + X_n > 1 \right\}.</math>

Then the [[Expected value|expected value]] of {{mvar|V}} is {{mvar|e}}: {{math|E(''V'') {{=}} ''e''}}.<ref>{{cite journal|last=Russell|first= K.G. |jstor=2685243 |title=Estimating the Value of e by Simulation |journal= The American Statistician|volume= 45|issue= 1|date= February 1991|pages= 66–68 |doi=10.1080/00031305.1991.10475769}}</ref><ref>Dinov, ID (2007) ''[http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_LawOfLargeNumbers#Estimating_e_using_SOCR_simulation Estimating e using SOCR simulation]'', SOCR Hands-on Activities (retrieved December 26, 2007).</ref>

=== Known digits ===
The number of known digits of {{mvar|e}} has increased substantially during the last decades. This is due both to the increased performance of computers and to algorithmic improvements.<ref>Sebah, P. and Gourdon, X.; [http://numbers.computation.free.fr/Constants/E/e.html The constant {{mvar|e}} and its computation]</ref><ref>Gourdon, X.; [http://numbers.computation.free.fr/Constants/PiProgram/computations.html Reported large computations with PiFast]</ref>

{| class="wikitable" style="margin: 1em auto 1em auto"
|+ Number of known decimal digits of {{mvar|e}}
! Date || Decimal digits || Computation performed by
|-
| 1690 ||align=right| 1 || [[Biography:Jacob Bernoulli|Jacob Bernoulli]]<ref name="Bernoulli, 1690" />
|-
| 1714 ||align=right| 13 || [[Biography:Roger Cotes|Roger Cotes]]<ref>Roger Cotes (1714) "Logometria," ''Philosophical Transactions of the Royal Society of London'', '''29''' (338) : 5–45; [https://archive.today/20140410203227/http://babel.hathitrust.org/cgi/pt?id=ucm.5324351035;view=2up;seq=16 see especially the bottom of page 10.]  From page 10: ''"Porro eadem ratio est inter 2,718281828459 &c et 1, … "'' (Furthermore, by the same means, the ratio is between 2.718281828459… and 1, … )</ref>
|-
| 1748 ||align=right| 23 || [[Biography:Leonhard Euler|Leonhard Euler]]<ref>Leonhard Euler, ''Introductio in Analysin Infinitorum'' (Lausanne, Switzerland: Marc Michel Bousquet & Co., 1748), volume 1, [https://archive.org/details/bub_gb_jQ1bAAAAQAAJ/page/n115 page 90.]</ref>
|-
| 1853 ||align=right| 137 || William Shanks<ref>William Shanks, ''Contributions to Mathematics'', ... (London, England: G. Bell, 1853), [https://books.google.com/books?id=d-9ZAAAAcAAJ&pg=PA89 page 89.]</ref>
|-
| 1871 ||align=right| 205 || William Shanks<ref>William Shanks (1871) [https://books.google.com/books?id=sclTAAAAcAAJ&pg=PA27 "On the numerical values of {{mvar|e}}, {{math|log<sub>''e''</sub> 2}}, {{math|log<sub>''e''</sub> 3}}, {{math|log<sub>''e''</sub> 5}}, and {{math|log<sub>''e''</sub> 10}}, also on the numerical value of {{mvar|M}} the modulus of the common system of logarithms, all to 205 decimals,"] ''Proceedings of the Royal Society of London'', '''20''' : 27–29.</ref>
|-
| 1884 ||align=right| 346 || J. Marcus Boorman<ref>J. Marcus Boorman (October 1884) [https://books.google.com/books?id=mG8yAQAAMAAJ&pg=PA204 "Computation of the Naperian base,"] ''Mathematical Magazine'', '''1''' (12) : 204–205.</ref>
|-
| 1949 ||align=right| 2,010 || [[Biography:John von Neumann|John von Neumann]] (on the [[Engineering:ENIAC|ENIAC]])
|-
|-
|}

Since around 2010, the proliferation of modern high-speed [[Engineering:Desktop computer|desktop computer]]s has made it feasible for amateurs to compute trillions of digits of {{mvar|e}} within acceptable amounts of time. On Dec 5, 2020, a record-setting calculation was made, giving {{mvar|e}} to 31,415,926,535,897 (approximately {{mvar|π}}{{x10^|13}}) digits.<ref>{{cite web
 | title= e
 | editor= Alexander Yee
 | work= Numberworld
 | date= 5 December 2020
 | url=http://www.numberworld.org/digits/E/
}}</ref>

== Computing the digits ==
One way to compute the digits of {{mvar|e}} is with the series<ref name="Finch-2005">{{Cite book |last=Finch |first= Steven R. |url=http://worldcat.org/oclc/180072364 |title=Mathematical constants |date=2005 |publisher=Cambridge Univ. Press |isbn=978-0-521-81805-6 |oclc=180072364}}</ref>
<math display=block>e=\sum_{k=0}^\infty \frac{1}{k!}.</math>

A faster method involves two recursive functions <math>p(a,b)</math> and <math>q(a,b)</math>. The functions are defined as <math display=block>\binom{p(a,b)}{q(a,b)}= \begin{cases} \binom{1}{b}, & \text{if }b=a+1\text{,} \\ \binom{p(a,m)q(m,b)+p(m,b)}{q(a,m)q(m,b)}, & \text{otherwise, where }m=\lfloor(a+b)/2\rfloor .\end{cases}</math>

The expression <math display=block>1+\frac{p(0,n)}{q(0,n)}</math> produces the {{mvar|n}}th partial sum of the series above. This method uses [[Binary splitting|binary splitting]] to compute {{mvar|e}} with fewer single-digit arithmetic operations and thus reduced bit complexity. Combining this with [[Fast Fourier transform|fast Fourier transform]]-based methods of multiplying integers makes computing the digits very fast.<ref name="Finch-2005" />

== In computer culture ==

During the emergence of [[Social:Internet culture|internet culture]], individuals and organizations sometimes paid homage to the number {{mvar|e}}.

In an early example, the [[Computer scientist|computer scientist]] [[Biography:Donald Knuth|Donald Knuth]] let the version numbers of his program [[Software:Metafont|Metafont]] approach {{mvar|e}}. The versions are 2, 2.7, 2.71, 2.718, and so forth.<ref>{{Cite journal
 | title= The Future of TeX and Metafont
 | first= Donald
 | last= Knuth
 | journal= TeX Mag
 | volume= 5
 | issue= 1
 | page= 145
 | date= 1990-10-03
 | url= http://www.ntg.nl/maps/05/34.pdf
 | access-date= 2017-02-17
}}</ref>

In another instance, the [[Finance:Initial public offering|IPO]] filing for [[Company:Google|Google]] in 2004, rather than a typical round-number amount of money, the company announced its intention to raise 2,718,281,828 USD, which is {{mvar|e}} billion {{wipe|United States dollar}} rounded to the nearest dollar.<ref>{{Cite journal |last1=Roberge |first1=Jonathan |last2=Melançon |first2=Louis |date=June 2017 |title=Being the King Kong of algorithmic culture is a tough job after all: Google's regimes of justification and the meanings of Glass |url=http://journals.sagepub.com/doi/10.1177/1354856515592506 |journal=Convergence: The International Journal of Research into New Media Technologies |language=en |volume=23 |issue=3 |pages=306–324 |doi=10.1177/1354856515592506 |issn=1354-8565}}</ref>

Google was also responsible for a billboard<ref>{{cite web|url=http://braintags.com/archives/2004/07/first-10digit-prime-found-in-consecutive-digits-of-e/|title=First 10-digit prime found in consecutive digits of {{math|e}}|website=Brain Tags|access-date=2012-02-24|archive-url=https://web.archive.org/web/20131203102744/http://braintags.com/archives/2004/07/first-10digit-prime-found-in-consecutive-digits-of-e/|archive-date=2013-12-03}}</ref>
that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts; Seattle, Washington; and {{wipe|Austin, Texas}}. It read "{first 10-digit prime found in consecutive digits of {{mvar|e}}}.com". The first 10-digit prime in {{mvar|e}} is 7427466391, which starts at the 99th digit.<ref>{{cite web
 | first= Marcus
 | last= Kazmierczak
 | title= Google Billboard
 | publisher= mkaz.com
 | date= 2004-07-29
 | url= http://mkaz.com/math/google-billboard
 | access-date= 2007-06-09
}}</ref> Solving this problem and visiting the advertised (now defunct) website led to an even more difficult problem to solve, which consisted in finding the fifth term in the sequence 7182818284, 8182845904, 8747135266, 7427466391. It turned out that the sequence consisted of 10-digit numbers found in consecutive digits of {{mvar|e}} whose digits summed to 49. The fifth term in the sequence is 5966290435, which starts at the 127th digit.<ref>[http://explorepdx.com/firsten.html The first 10-digit prime in {{math|e}}] {{Webarchive|url=https://web.archive.org/web/20210411010312/http://explorepdx.com/firsten.html |date=2021-04-11 }}. Explore Portland Community. Retrieved on 2020-12-09.</ref>
Solving this second problem finally led to a [[Organization:Google Labs|Google Labs]] webpage where the visitor was invited to submit a résumé.<ref>{{cite news
 | first= Andrea
 | last= Shea
 | title= Google Entices Job-Searchers with Math Puzzle
 | work= NPR
 | url= https://www.npr.org/templates/story/story.php?storyId=3916173
 | access-date= 2007-06-09
}}</ref>

== References ==
{{Reflist|30em|refs=

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 | last = Bryc | first = Wlodzimierz
 | year = 1995
 | title = The Normal Distribution: Characterizations with Applications 
 | url = https://books.google.com/books?id=tyXjBwAAQBAJ&pg=PA23
 | page = 23
 | publisher = Springer-Varleg
}}</ref>-->

<ref name="openstax">{{cite book
 | title = Statistics
 | chapter-url = https://openstax.org/books/statistics/pages/6-1-the-standard-normal-distribution
 | chapter = 6.1 The Standard Normal Distribution |first1=Barbara
 | last1 = Illowsky
 | first2 = Susan | last2 = Dean
 | display-authors = etal
 | publisher = OpenStax
 | isbn = 978-1-951693-22-0
 | year = 2023
}}</ref> 

}}

== Further reading ==
* Maor, Eli; ''{{mvar|e}}: The Story of a Number'', {{isbn|0-691-05854-7}}
* [http://www.johnderbyshire.com/Books/Prime/Blog/page.html#endnote10 Commentary on Endnote 10] of the book ''[[Prime Obsession]]'' for another stochastic representation
* {{cite journal|author=McCartin, Brian J.|title=e: The Master of All|journal=The Mathematical Intelligencer|volume=28|year=2006|issue= 2|pages=10–21|url=http://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/mccartin.pdf|doi=10.1007/bf02987150|s2cid=123033482}}

== External links ==
* <!-- https://www.gutenberg.org/ebooks/127 -->[https://web.archive.org/web/20070210095028/http://www.gutenberg.org/ebooks/127 The number {{mvar|e}} to 1 million places] and [https://apod.nasa.gov/htmltest/rjn_dig.html NASA.gov] 2 and 5 million places
* [http://mathworld.wolfram.com/eApproximations.html {{mvar|e}} Approximations]&nbsp;– Wolfram MathWorld
* [http://jeff560.tripod.com/constants.html Earliest Uses of Symbols for Constants] Jan. 13, 2008
* [http://www.gresham.ac.uk/lectures-and-events/the-story-of-e "The story of {{mvar|e}}"], by Robin Wilson at Gresham College, 28 February 2007 (available for audio and video download)
* [http://pisearch.org/e {{mvar|e}} Search Engine] 2 billion searchable digits of {{mvar|e}}, {{pi}} and {{radic|2}}

{{Irrational number}}

[[Category:E (mathematical constant)| ]]
[[Category:Mathematical constants]]
[[Category:Real transcendental numbers]]

{{Sourceattribution|E (mathematical constant)}}