Editing E (mathematical constant) (section)

== Properties ==

=== Calculus ===
{{See also|Characterizations of the exponential function}}
[[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]], is to perform [[derivative (mathematics)|differential]] and [[integral calculus]] with [[exponential function]]s and [[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]], 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''<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]] along the {{nowrap|[[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 name="strangherman">{{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}}</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]] multiplication by a constant {{mvar|K}}) that is equal to its own [[derivative]],

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

it is therefore its own [[antiderivative]] as well:<ref>{{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}}</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]]

<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]]; 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]]

:<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 [[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]]. [[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 [[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 [[Charles Hermite]] in 1873.<ref>{{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}}</ref> The number {{mvar|e}} is one of only a few transcendental numbers for which the exact [[Irrationality measure#Irrationality exponent|irrationality exponent]] is known (given by <math>\mu(e)=2</math>).<ref>{{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}}</ref>

An [[List of unsolved problems in mathematics|unsolved problem]] thus far is the question of whether or not the numbers {{mvar|e}} and {{mvar|π}} are [[Algebraic independence|algebraically independent]]. This would be resolved by [[Schanuel's conjecture]] – a currently unproven generalization of the Lindemann–Weierstrass theorem.<ref>{{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 }}</ref><ref>{{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}}</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]], a ''[[period (algebraic geometry)|period]]'' is a number that can be expressed as an integral of an [[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-link1=Maxim Kontsevich |author-link2=Don Zagier |year=2001 |title=Periods |url=https://www.ihes.fr/~maxim/TEXTS/Periods.pdf}}</ref>

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

<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]] 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]].<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]] 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>

for any integer {{mvar|n}}, which is [[de Moivre's formula]].<ref name="Sultan"/>

The expressions of {{math|cos ''x''}} and {{math|sin ''x''}} in terms of the [[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>