E (mathematical constant)#Definitions

The number {{mvar|e}} is the limit

$$\backslash lim\_\{n\backslash to\; \backslash infty\}\backslash left(1+\backslash frac\; 1n\backslash right)^n,$$

an expression that arises in the computation of compound interest.

It is the sum of the infinite series

$$e\; =\; \backslash sum\backslash limits\_\{n\; =\; 0\}^\{\backslash infty\}\; \backslash frac\{1\}\{n!\}\; =\; 1\; +\; \backslash frac\{1\}\{1\}\; +\; \backslash frac\{1\}\{1\backslash cdot\; 2\}\; +\; \backslash frac\{1\}\{1\backslash cdot\; 2\backslash cdot\; 3\}\; +\; \backslash cdots.$$

It is the unique positive number {{mvar|a}} such that the graph of the function {{math|1=y = a^x}} has a slope of 1 at {{math|1=x = 0}}.

One has $$e=\backslash exp(1),$$ where $\backslash exp$ is the (natural) exponential function, the unique function that equals its own derivative and satisfies the equation $\backslash exp(0)=1.$ Since the exponential function is commonly denoted as $x\backslash mapsto\; e^x,$ one has also

$$e=e^1.$$

The logarithm of base {{mvar|b}} can be defined as the inverse function of the function $x\backslash mapsto\; b^x.$ Since $b=b^1,$ one has $\backslash log\_b\; b=\; 1.$ The equation $e=e^1$ implies therefore that {{mvar|e}} is the base of the natural logarithm.

The number {{mvar|e}} can also be characterized in terms of an integral:{{dlmf}}

$$\backslash int\_1^e\; \backslash frac\; \{dx\}x\; =1.$$

For other characterizations, see {{slink||Representations}}.

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 logarithms to the base $e$. 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.{{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}}

The constant itself was introduced by Jacob Bernoulli in 1683, for solving the problem of continuous compounding of interest.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. 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|2½a}} & minus quam {{math|3a}}." ( … which our series [a geometric series] is larger [than]. … if {{math|1=a=b}}, [the lender] will be owed more than {{math|2½a}} and less than {{math|3a}}.) 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.]){{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}}

In his solution, the constant {{mvar|e}} occurs as the limit

$$\backslash lim\_\{n\backslash to\; \backslash infty\}\; \backslash left(\; 1\; +\; \backslash frac\{1\}\{n\}\; \backslash right)^n,$$

where {{mvar|n}} represents the number of intervals in a year on which the compound interest is evaluated (for example, $n=12$ 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.{{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}}

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,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...") and in a letter to Christian Goldbach on 25 November 1731.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. 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) … ){{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}} The first appearance of {{mvar|e}} in a printed publication was in Euler's Mechanica (1736).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. 68. [https://books.google.com/books?id=qalsP7uMiV4C&pg=PA68 From page 68:] Erit enim $\backslash frac\{dc\}\{c\}\; =\; \backslash frac\{dy\; ds\}\{rdx\}$ seu $c\; =\; e^\{\backslash int\backslash frac\{dy\; ds\}\{rdx\}\}$ ubi {{mvar|e}} denotat numerum, cuius logarithmus hyperbolicus est 1. (So it [i.e., {{mvar|c}}, the speed] will be $\backslash frac\{dc\}\{c\}\; =\; \backslash frac\{dy\; ds\}\{rdx\}$ or $c\; =\; e^\{\backslash int\backslash frac\{dy\; ds\}\{rdx\}\}$, where {{mvar|e}} denotes the number whose hyperbolic [i.e., natural] logarithm is 1.) It is unknown why Euler chose the letter {{mvar|e}}.{{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. Although some researchers used the letter {{mvar|c}} in the subsequent years, the letter {{mvar|e}} was more common and eventually became standard.{{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}}

Euler proved that {{mvar|e}} is the sum of the infinite series

$$e\; =\; \backslash sum\_\{n\; =\; 0\}^\backslash infty\; \backslash frac\{1\}\{n!\}\; =\; \backslash frac\{1\}\{0!\}\; +\; \backslash frac\{1\}\{1!\}\; +\; \backslash frac\{1\}\{2!\}\; +\; \backslash frac\{1\}\{3!\}\; +\; \backslash frac\{1\}\{4!\}\; +\; \backslash cdots\; ,$$

where {{math|n!}} is the factorial of {{mvar|n}}. The equivalence of the two characterizations using the limit and the infinite series can be proved via the binomial theorem.{{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}}

File:Compound Interest with Varying Frequencies.svg

Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest:

{{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 × 1.5² = $2.25}} at the end of the year. Compounding quarterly yields {{nowrap|1=$1.00 × 1.25⁴ = $2.44140625}}, and compounding monthly yields {{nowrap|1=$1.00 × (1 + 1/12)¹² = $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 × {{math|(1 + 1/n)ⁿ}}.

Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger {{mvar|n}} and, thus, smaller compounding intervals. 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^Rt}} 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}}.{{cite book

| last = Gonick

| first = Larry

| author-link = Larry Gonick

| 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

}}

File:Bernoulli trial sequence.svg

The number {{mvar|e}} itself also has applications in 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}}, which is approximately 36.79%. For {{math|1=n = 20}}, this is already 1/2.789509... (approximately 35.85%).

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, which is closely related to the binomial theorem and Pascal's triangle. The probability of winning {{mvar|k}} times out of {{mvar|n}} trials is:{{cite book|first=Mehran |last=Kardar |author-link=Mehran Kardar |title=Statistical Physics of Particles |title-link=Statistical Physics of Particles |year=2007 |publisher=Cambridge University Press |isbn=978-0-521-87342-0 |oclc=860391091 |page=41}}

:$\backslash Pr[k~\backslash mathrm\{wins~of\}~n]\; =\; \backslash binom\{n\}\{k\}\; \backslash left(\backslash frac\{1\}\{n\}\backslash right)^k\backslash left(1\; -\; \backslash frac\{1\}\{n\}\backslash right)^\{n-k\}.$

In particular, the probability of winning zero times ({{math|1=k = 0}}) is

:$\backslash Pr[0~\backslash mathrm\{wins~of\}~n]\; =\; \backslash left(1\; -\; \backslash frac\{1\}\{n\}\backslash right)^\{n\}.$

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

{{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 of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself.{{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=etal |year=2023 |isbn=978-1-951693-41-1}} Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing exponential decay instead. The law of exponential growth can be written in different but mathematically equivalent forms, by using a different base, for which the number {{mvar|e}} is a common and convenient choice:

$$x(t)\; =\; x\_0\backslash cdot\; e^\{kt\}\; =\; x\_0\backslash cdot\; e^\{t/\backslash tau\}.$$

Here, $x\_0$ denotes the initial value of the quantity {{mvar|x}}, {{mvar|k}} is the growth constant, and $\backslash tau$ is the time it takes the quantity to grow by a factor of {{mvar|e}}.

{{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 probability density function

$$\backslash phi(x)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; e^\{-\backslash frac\{1\}\{2\}\; x^2\}.$$

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 $\backslash phi(x)$ results in the factor $\backslash textstyle\; 1/\backslash sqrt\{2\backslash pi\}$. This function is symmetric around {{math|1=x = 0}}, where it attains its maximum value $\backslash textstyle\; 1/\backslash sqrt\{2\backslash pi\}$, and has inflection points at {{math|1=x = ±1}}.

{{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 derangements, also known as the hat check problem:{{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 }} {{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 $p\_n\backslash !$, is:

:$p\_n\; =\; 1\; -\; \backslash frac\{1\}\{1!\}\; +\; \backslash frac\{1\}\{2!\}\; -\; \backslash frac\{1\}\{3!\}\; +\; \backslash cdots\; +\; \backslash frac\{(-1)^n\}\{n!\}\; =\; \backslash sum\_\{k\; =\; 0\}^n\; \backslash frac\{(-1)^k\}\{k!\}.$

As {{mvar|n}} tends to infinity, {{math|p_n}} 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,}} rounded to the nearest integer, for every positive {{mvar|n}}.{{cite book | last=Knuth |first=Donald |author-link=Donald Knuth |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}}

The maximum value of $\backslash sqrt[x]\{x\}$ occurs at $x\; =\; e$. Equivalently, for any value of the base {{math|b > 1}}, it is the case that the maximum value of $x^\{-1\}\backslash log\_b\; x$ occurs at $x\; =\; e$ (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{{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}}

:$n\; =\; \backslash left\backslash lfloor\; \backslash frac\{L\}\{e\}\; \backslash right\backslash rfloor$ or $\backslash left\backslash lceil\; \backslash frac\{L\}\{e\}\; \backslash right\backslash rceil.$

The quantity $x^\{-1\}\backslash log\_b\; x$ is also a measure of information gleaned from an event occurring with probability $1/x$ (approximately $36.8\backslash \%$ when $x=e$), so that essentially the same optimal division appears in optimal planning problems like the secretary problem.

The number {{mvar|e}} occurs naturally in connection with many problems involving asymptotics. An example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers {{mvar|e}} and {{pi}} appear:{{cite book|first=Greg |last=Gbur |author-link=Greg Gbur |year=2011 |title=Mathematical Methods for Optical Physics and Engineering |isbn=978-0-521516-10-5 |publisher=Cambridge University Press |page=779}}

y {{=}} a^x}}}} has a derivative, given by a limit:

:$\backslash begin\{align\}$

\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}

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:

:$\backslash frac\{d\}\{dx\}e^x\; =\; e^x.$

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_a x}}),{{r|kline}} for {{math|x > 0}}:

:$\backslash begin\{align\}$

\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}

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,

:$\backslash frac\{d\}\{dx\}\backslash log\_e\; x\; =\; \backslash frac\{1\}\{x\}.$

The logarithm with this special base is called the natural logarithm, and is usually 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^x}} equal to {{math|a^x}}, 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 along the {{nowrap|hyperbola $xy=1.$}}]]

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:{{cite book|first1=Gilbert |last1=Strang |first2=Edwin |last2=Herman |title=Calculus, volume 2 |publisher=OpenStax |display-authors=etal |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}}

$$e^x\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{x^n\}\{n!\}.$$

Setting $x\; =\; 1$ 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 $x$ of $1/t$, 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 $x\; =\; 1$, or equivalently, the number whose natural logarithm is 1. It follows that {{mvar|e}} is the unique positive real number such that

$$\backslash int\_1^e\; \backslash frac\{1\}\{t\}\; \backslash ,\; dt\; =\; 1.$$

Because {{math|e^x}} is the unique function (up to multiplication by a constant {{mvar|K}}) that is equal to its own derivative,

$$\backslash frac\{d\}\{dx\}Ke^x\; =\; Ke^x,$$

it is therefore its own antiderivative as well:{{cite book|first1=Gilbert |last1=Strang |first2=Edwin |last2=Herman |title=Calculus, volume 2 |publisher=OpenStax |display-authors=etal |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}}

$$\backslash int\; Ke^x\backslash ,dx\; =\; Ke^x\; +\; C\; .$$

Equivalently, the family of functions

$$y(x)\; =\; Ke^x$$

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

$$y\text{'}\; =\; y\; .$$

File:Exponentials vs x+1.pdf

The number {{mvar|e}} is the unique real number such that

for all positive {{mvar|x}}.{{cite book|last1=Dorrie|first1=Heinrich|title=100 Great Problems of Elementary Mathematics|date=1965|publisher=Dover|pages=44–48}}

Also, we have the inequality

$$e^x\; \backslash ge\; x\; +\; 1$$

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^x ≥ x + 1}} holds for all {{mvar|x}}.A standard calculus exercise using the mean value theorem; see for example Apostol (1967) Calculus, § 6.17.41. This is a limiting case of Bernoulli's inequality.

File:Xth root of x.svg of {{math|{{sqrt|x|x}}}} {{nowrap|occurs at {{math|x {{=}} e}}.}}]]

Steiner's problem asks to find the global maximum for the function

$$f(x)\; =\; x^\backslash frac\{1\}\{x\}\; .$$

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 {{mvar|x}}.)

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

$$f(x)\; =\; x^x\; .$$

The infinite tetration

:$x^\{x^\{x^\{\backslash cdot^\{\backslash cdot^\{\backslash cdot\}\}\}\}\}$ or $\{^\backslash infty\}x$

converges if and only if {{math|x ∈ [(1/e)^e, e^1/e] ≈ [0.06599, 1.4447] }},{{Cite OEIS|A073230|Decimal expansion of (1/e)^e}}{{Cite OEIS|A073229|Decimal expansion of e^(1/e)}} shown by a theorem of Leonhard Euler.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]){{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}}{{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}}

The real number {{mvar|e}} is irrational. Euler proved this by showing that its simple continued fraction expansion does not terminate.{{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}} (See also Fourier's proof that e is irrational.)

Furthermore, by the Lindemann–Weierstrass theorem, {{mvar|e}} is 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 Charles Hermite in 1873.{{cite book | last=Gelfond | first=A. O. | author-link=Alexander Gelfond | translator-last=Boron | translator-first=Leo F. | translator-link=Leo F. Boron | orig-year=1960 | year=2015 | title=Transcendental and Algebraic Numbers | publisher=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}} The number {{mvar|e}} is one of only a few transcendental numbers for which the exact irrationality exponent is known (given by $\backslash mu(e)=2$).{{Cite web |last=Weisstein |first=Eric W. |title=Irrationality Measure |url=https://mathworld.wolfram.com/IrrationalityMeasure.html |access-date=2024-09-14 |website=mathworld.wolfram.com |language=en}}

An unsolved problem thus far is the question of whether or not the numbers {{mvar|e}} and {{mvar|π}} are algebraically independent. This would be resolved by Schanuel's conjecture – a currently unproven generalization of the Lindemann–Weierstrass theorem.{{Cite book |last1=Murty |first1=M. Ram |url=https://link.springer.com/book/10.1007/978-1-4939-0832-5 |title=Transcendental Numbers |last2=Rath |first2=Purusottam |date=2014 |publisher=Springer |language=en |doi=10.1007/978-1-4939-0832-5|isbn=978-1-4939-0831-8 }}{{Cite web |last=Waldschmidt |first=Michel |date=2021 |title=Schanuel's Conjecture: algebraic independence of transcendental numbers |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SchanuelEn.pdf}}

It is conjectured that {{mvar|e}} is normal, meaning that when {{mvar|e}} is expressed in any base the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length).{{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

}}

In algebraic geometry, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. The constant {{pi}} is a period, but it is conjectured that {{mvar|e}} is not.{{Cite web |last1=Kontsevich |first1=Maxim |last2=Zagier |first2=Don |author-link1=Maxim Kontsevich |author-link2=Don Zagier |year=2001 |title=Periods |url=https://www.ihes.fr/~maxim/TEXTS/Periods.pdf}}

The exponential function {{math|e^x}} may be written as a Taylor series{{cite book |author-last1=Whittaker |author-first1=Edmund Taylor |author-link1=Edmund Taylor Whittaker |title=A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions |title-link=A Course of Modern Analysis |author-last2=Watson |author-first2=George Neville |author-link2=George Neville Watson |date=1927-01-02 |publisher=Cambridge University Press |isbn= |edition=4th |publication-place=Cambridge, UK |page=581}}

$$e^\{x\}\; =\; 1\; +\; \{x\; \backslash over\; 1!\}\; +\; \{x^\{2\}\; \backslash over\; 2!\}\; +\; \{x^\{3\}\; \backslash over\; 3!\}\; +\; \backslash cdots\; =\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash frac\{x^n\}\{n!\}.$$

Because this series is convergent for every complex value of {{mvar|x}}, it is commonly used to extend the definition of {{math|e^x}} to the complex numbers.{{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}} This, with the Taylor series for trigonometric functions, allows one to derive Euler's formula:

$$e^\{ix\}\; =\; \backslash cos\; x\; +\; i\backslash sin\; x\; ,$$

which holds for every complex {{mvar|x}}. The special case with {{math|x {{=}} {{pi}}}} is Euler's identity:

$$e^\{i\backslash pi\}\; +\; 1\; =\; 0\; ,$$

which is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in a proof that {{pi}} is transcendental, which implies the impossibility of squaring the circle.{{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 }}{{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}} Moreover, the identity implies that, in the principal branch of the logarithm,

$$\backslash ln\; (-1)\; =\; i\backslash pi\; .$$

Furthermore, using the laws for exponentiation,

$$(\backslash cos\; x\; +\; i\backslash sin\; x)^n\; =\; \backslash left(e^\{ix\}\backslash right)^n\; =\; e^\{inx\}\; =\; \backslash cos\; nx\; +\; i\; \backslash sin\; nx$$

for any integer {{mvar|n}}, which is de Moivre's formula.

The expressions of {{math|cos x}} and {{math|sin x}} in terms of the exponential function can be deduced from the Taylor series:

\cos x = \frac{e^{ix} + e^{-ix}}{2} , \qquad

\sin x = \frac{e^{ix} - e^{-ix}}{2i}.

The expression

$\backslash cos\; x\; +\; i\; \backslash sin\; x$

is sometimes abbreviated as {{math|cis(x)}}.{{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}}

{{Main|List of representations of e|l1=List of representations of {{mvar|e}}}}

The number {{mvar|e}} can be represented in a variety of ways: as an infinite series, an infinite product, a continued fraction, or a limit of a sequence. In addition to the limit and the series given above, there is also the simple continued fraction

:$$

e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, ..., 1, 2n, 1, ...],

{{cite book |last=Hofstadter|first=D.R.|author-link=Douglas Hofstadter |title= Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought|publisher= Basic Books |date=1995|isbn=0-7139-9155-0}}{{Cite OEIS|A003417|Continued fraction for e}}

which written out looks like

:$e\; =\; 2\; +$

\cfrac{1}

{1 + \cfrac{1}

{2 + \cfrac{1}

{1 + \cfrac{1}

{1 + \cfrac{1}

{4 + \cfrac{1}

{1 + \cfrac{1}

{1 + \ddots}

}

}

}

}

}

}

.

The following infinite product evaluates to {{mvar|e}}:

$$e\; =\; \backslash frac\{2\}\{1\}\; \backslash left(\backslash frac\{4\}\{3\}\backslash right)^\{1/2\}\; \backslash left(\backslash frac\{6\; \backslash cdot\; 8\}\{5\; \backslash cdot\; 7\}\backslash right)^\{1/4\}\; \backslash left(\backslash frac\{10\; \backslash cdot\; 12\; \backslash cdot\; 14\; \backslash cdot\; 16\}\{9\; \backslash cdot\; 11\; \backslash cdot\; 13\; \backslash cdot\; 15\}\backslash right)^\{1/8\}\; \backslash cdots.$$

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

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|X₁}}, {{math|X₂}}..., drawn from the 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:

:$V\; =\; \backslash min\backslash left\backslash \{\; n\; \backslash mid\; X\_1\; +\; X\_2\; +\; \backslash cdots\; +\; X\_n\; >\; 1\; \backslash right\backslash \}.$

Then the expected value of {{mvar|V}} is {{mvar|e}}: {{math|E(V) {{=}} e}}.{{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}}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).

The number of known digits of {{mvar|e}} has increased substantially since the introduction of the computer, due both to increasing performance of computers and to algorithmic improvements.Sebah, P. and Gourdon, X.; [http://numbers.computation.free.fr/Constants/E/e.html The constant {{mvar|e}} and its computation]Gourdon, X.; [http://numbers.computation.free.fr/Constants/PiProgram/computations.html Reported large computations with PiFast]

Since around 2010, the proliferation of modern high-speed desktop computers has made it feasible for amateurs to compute trillions of digits of {{mvar|e}} within acceptable amounts of time. On Dec 24, 2023, a record-setting calculation was made by Jordan Ranous, giving {{mvar|e}} to 35,000,000,000,000 digits.{{cite web

| title= y-cruncher - A Multi-Threaded Pi Program

| editor= Alexander Yee

| work= Numberworld

| date= 15 March 2025

| url=http://www.numberworld.org/y-cruncher/#Records

}}

One way to compute the digits of {{mvar|e}} is with the series{{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}}

$$e=\backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{1\}\{k!\}.$$

A faster method involves two recursive functions $p(a,b)$ and $q(a,b)$. The functions are defined as

The expression $$1+\backslash frac\{p(0,n)\}\{q(0,n)\}$$ produces the {{mvar|n}}th partial sum of the series above. This method uses binary splitting to compute {{mvar|e}} with fewer single-digit arithmetic operations and thus reduced bit complexity. Combining this with fast Fourier transform-based methods of multiplying integers makes computing the digits very fast.

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

In an early example, the computer scientist Donald Knuth let the version numbers of his program Metafont approach {{mvar|e}}. The versions are 2, 2.7, 2.71, 2.718, and so forth.{{Cite journal

| title= The Future of TeX and Metafont

| first= Donald

| last= Knuth

| author-link= Donald 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

}}

In another instance, the IPO filing for 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 dollars rounded to the nearest dollar.{{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}}

Google was also responsible for a billboard{{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}}

that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts; Seattle, Washington; and 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.{{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

| archive-date= 2010-09-23

| archive-url= https://web.archive.org/web/20100923111259/http://mkaz.com/math/google-billboard/

| url-status= dead

}} Solving this problem and visiting the advertised (now defunct) website led to an even more difficult problem to solve, which consisted of 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.[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.

Solving this second problem finally led to a Google Labs webpage where the visitor was invited to submit a résumé.{{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

}}

The last release of the official Python 2 interpreter has version number 2.7.18, a reference to e.{{Cite web |last=Peterson |first=Benjamin |date=20 April 2020 |title=Python 2.7.18, the end of an era |url=https://lwn.net/Articles/818000/ |website=LWN.net}}

{{Reflist|30em|refs=

{{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

}}

}}

• {{Cite book |title={{mvar|e}}: The Story of a Number |date=2009 |publisher=Princeton University Press |isbn=978-0-691-05854-2 |editor-last=Maor |editor-first=Eli |series=Princeton science library |location=Princeton, N.J}}
• [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 |last=McCartin |first=Brian J. |date=March 2006 |title={{mvar|e}}: The Master of All |url=http://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/mccartin.pdf |journal=The Mathematical Intelligencer |volume=28 |issue=2 |pages=10–21 |doi=10.1007/BF02987150 |issn=0343-6993 |s2cid=123033482}}

{{Commons category||{{mvar|e}} (mathematical constant)}}

{{Wikiquote|E (mathematical constant)|{{mvar|e}} (mathematical constant)}}

• [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] – 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}}

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Category:Leonhard Euler

Category:Mathematical constants

Category:Real transcendental numbers