Editing E (mathematical constant) (section)

== History ==
The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by [[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, [[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 [[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|author2-link= 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 [[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>

[[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 [[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|author-link=Reinhold Remmert|title=Theory of Complex Functions|page=136|publisher=[[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]] 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]].<ref>{{cite book|author-link=Walter Rudin |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>