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| {{DISPLAYTITLE:{{mvar|e}} (mathematical constant)}}
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| {{Redirect|Euler's number|γ, a constant in number theory|<!-- Euler's constant -->Euler–Mascheroni constant|other uses|List of things named after Leonhard Euler#Euler's numbers}}
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| [[Image:Exp derivative at 0.svg|right|frame|Functions {{math|''f''(''x'') {{=}} ''a''<sup>''x''</sup>}} are shown for several values of {{math|''a''}}.
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| {{mvar|e}} is the unique value of {{math|''a''}}, such that the derivative of {{math|''f''(''x'') {{=}} ''a''<sup>''x''</sup>}} at the point {{math|''x'' {{=}} 0}} is equal to 1. The blue curve illustrates this case, {{math|''e''<sup>''x''</sup>}}. For comparison, functions {{math|2<sup>''x''</sup>}} (dotted curve) and {{math|4<sup>''x''</sup>}} (dashed curve) are shown; they are not [[tangent]] to the line of slope 1 and y-intercept 1 (red).]]
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| The number '''{{mvar|e}}''' is an important [[mathematical constant]] that is the base of the [[natural logarithm]]. It is approximately equal to 2.71828,<ref>[[Oxford English Dictionary]], 2nd ed.: [http://oxforddictionaries.com/definition/english/natural%2Blogarithm natural logarithm]</ref> and 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 study of [[compound interest]]. It can also be calculated as the sum of the infinite [[series (mathematics)|series]]<ref>[[Encyclopedic Dictionary of Mathematics]] 142.D</ref>
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| :<math>e = \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots</math> | | Reserving ʏouг body's power іf you ɑre ill, will assist tҺe healing process, and hɑve you straight back to total durability morе rapidly. Be sure that yօu are always inhaling appropriately աith your exercise regimen. ҬҺis will lіkely make cеrtain yoս arе maximizing tɦe performance of eacɦ and еvеry inhale tɦat you simply cߋnsider. You кnow yoս mіght ƅe inhaling appropriately aѕ ѕoon aѕ your belly iѕ what is shifting and neѵeг үouг shoulders. If уou loved tɦiѕ article ɑnd ʏou would like to receive а lot mօre faϲtѕ cߋncerning [https://www.facebook.com/fitnessfreakshirts squat t shirts] kindly pay а visit tо οur own web-site. |
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| The constant can be defined in many ways; for example, {{mvar|e}} is the unique [[real number]] such that the value of the [[derivative]] (slope of the [[tangent line]]) of the function {{math|1=''f''(''x'') = ''e''<sup>''x''</sup>}} at the point {{math|1=''x'' = 0}} is equal to 1.<ref>{{cite book|title = Calculus|author = Jerrold E. Marsden, Alan Weinstein|publisher = Springer|year = 1985|isbn = 0-387-90974-5|url=http://books.google.com/?id=KVnbZ0osbAkC&printsec=frontcover}}</ref> The function {{math|''e''<sup>''x''</sup>}} so defined is called the [[exponential function]], and its [[Inverse function|inverse]] is the [[natural logarithm]], or logarithm to [[base (exponentiation)|base]] {{mvar|e}}. The natural logarithm of a positive number {{math|''k''}} can also 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 number whose natural logarithm is 1. There are also more [[#Alternative characterizations|alternative characterizations]].
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| Sometimes called '''Euler's number''' after the [[Switzerland|Swiss]] [[mathematician]] [[Leonhard Euler]], {{mvar|e}} is not to be confused with {{math|γ}}—the [[Euler–Mascheroni constant]], sometimes called simply ''Euler's constant''. The number {{mvar|e}} is also known as '''Napier's constant''', but Euler's choice of the symbol {{mvar|e}} is said to have been retained in his honor.<ref name="mathworld">{{cite web|last=Sondow|first=Jonathan|title=e|url=http://mathworld.wolfram.com/e.html|work=[[MathWorld|Wolfram Mathworld]]|publisher=[[Wolfram Research]]|accessdate=10 May 2011}}</ref> The number {{mvar|e}} is of eminent importance in mathematics,<ref>{{cite book|title = An Introduction to the History of Mathematics|author = Howard Whitley Eves|year = 1969|publisher = Holt, Rinehart & Winston|isbn =0-03-029558-0}}</ref> alongside [[0 (number)|0]], [[1 (number)|1]], [[pi|{{pi}}]] and [[imaginary unit|{{mvar|i}}]]. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of [[Euler's identity]]. Like the constant {{pi}}, {{mvar|e}} is [[irrational number|irrational]]: it is not a ratio of [[integers]]; and it is [[transcendental number|transcendental]]: it is not a root of ''any'' non-zero [[polynomial]] with rational coefficients. The numerical value of {{mvar|e}} truncated to 50 [[decimal|decimal places]] is
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| :{{gaps|2.71828|18284|59045|23536|02874|71352|66249|77572|47093|69995...}} {{OEIS|A001113}}.
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| {{E (mathematical constant)}}
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| ==History==
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| The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by [[John Napier]].<ref name="OConnor">{{cite web|url=<!-- http://www.gap-system.org/~history/PrintHT/e.html -->http://www-history.mcs.st-and.ac.uk/HistTopics/e.html|title=The number ''e''|publisher=MacTutor History of Mathematics|first1=J J|last1=O'Connor|first2=E F|last2=Robertson}}</ref> However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. It is assumed that the table was written by [[William Oughtred]]. The discovery of the constant itself is credited to [[Jacob Bernoulli]], who attempted to find the value of the following expression (which is in fact {{mvar|e}}):
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| :<math>\lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n</math>
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| The first known use of the constant, represented by the letter {{math|''b''}}, was in correspondence from [[Gottfried Leibniz]] to [[Christiaan Huygens]] in 1690 and 1691. [[Leonhard Euler]] introduced the letter {{mvar|e}} as the base for natural logarithms, writing in a letter to [[Christian Goldbach]] of 25 November 1731.<ref>{{Cite book|last=Remmert|first=Reinhold|authorlink=Reinhold Remmert|title=Theory of Complex Functions|page=136|publisher=[[Springer-Verlag]]|year=1991|isbn=0-387-97195-5|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref> 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, ''[http://www.math.dartmouth.edu/~euler/pages/E853.html Meditatio in experimenta explosione tormentorum nuper instituta]''.</ref> and the first appearance of {{mvar|e}} in a publication was [[Mechanica|Euler's ''Mechanica'']] (1736). While in the subsequent years some researchers used the letter {{math|''c''}}, {{mvar|e}} was more common and eventually became the standard.
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| ==Applications==
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| ===Compound interest===
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| [[File:Compound Interest with Varying Frequencies.svg|thumb|right|350px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies]]
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| [[Jacob Bernoulli]] discovered this constant by studying a question about [[compound interest]]:<ref name="OConnor" />
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| :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?
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| 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 $1.00×1.5<sup>2</sup> = $2.25 at the end of the year. Compounding quarterly yields $1.00×1.25<sup>4</sup> = $2.4414..., and compounding monthly yields $1.00×(1+1/12)<sup>12</sup> = $2.613035... If there are {{math|''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 + 1/''n'')<sup>''n''</sup>}}.
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| Bernoulli noticed that this sequence approaches a limit (the [[force of interest]]) with larger {{math|''n''}} and, thus, smaller compounding intervals. Compounding weekly ({{math|1=''n'' = 52}}) yields $2.692597..., while compounding daily ({{math|1=''n'' = 365}}) yields $2.714567..., just two cents more. The limit as {{math|''n''}} grows large is the number that came to be known as {{mvar|e}}; with ''continuous'' compounding, the account value will reach $2.7182818.... More generally, an account that starts at $1 and offers an annual interest rate of {{math|''R''}} will, after {{math|''t''}} years, yield {{math|''e''<sup>''Rt''</sup>}} dollars with continuous compounding. (Here {{math|''R''}} is a fraction, so for 5% interest, {{math|1=''R'' = 5/100 = 0.05}})
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| ===Bernoulli trials===
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| The number {{mvar|e}} itself also has applications to [[probability theory]], where it arises in a way not obviously related to exponential growth. Suppose that a gambler plays a slot machine that pays out with a probability of one in {{math|''n''}} and plays it {{math|''n''}} times. Then, for large {{math|''n''}} (such as a million) the [[probability]] that the gambler will lose every bet is (approximately) {{math|1/''e''}}. For {{math|1=''n'' = 20}} it is already 1/2.72.
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| This is an example of a [[Bernoulli trials]] process. Each time the gambler plays the slots, there is a one in one million chance of winning. Playing one million times is modelled by the [[binomial distribution]], which is closely related to the [[binomial theorem]]. The probability of winning {{math|''k''}} times out of a million trials is;
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| :<math>\binom{10^6}{k} \left(10^{-6}\right)^k(1-10^{-6})^{10^6-k}.</math>
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| In particular, the probability of winning zero times ({{math|1=''k'' = 0}}) is
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| :<math>\left(1-\frac{1}{10^6}\right)^{10^6}.</math>
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| This is very close to the following limit for {{math|1/''e''}}:
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| :<math>\frac{1}{e} = \lim_{n\to\infty} \left(1-\frac{1}{n}\right)^n.</math>
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| ===Derangements===
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| Another application of {{mvar|e}}, also discovered in part by Jacob Bernoulli along with [[Pierre Raymond de Montmort]] is in the problem of [[derangement]]s, also known as the ''hat check problem'':<ref>Grinstead, C.M. and Snell, J.L.''[http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html Introduction to probability theory] (published online under the [[GFDL]]), p. 85.</ref> {{math|''n''}} guests are invited to a party, and at the door each guest checks his hat with the butler who then places them into {{math|''n''}} boxes, each labelled with the name of one guest. But the butler does not know the identities of the guests, and so he 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. The answer is:
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| :<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>
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| As the number {{math|''n''}} of guests 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 is in the right box is {{math|''n''!/''e''}} rounded to the nearest integer, for every positive {{math|''n''}}.<ref>Knuth (1997) ''[[The Art of Computer Programming]]'' Volume I, Addison-Wesley, p. 183 ISBN 0-201-03801-3.</ref>
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| ===Asymptotics===
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| The number {{mvar|e}} occurs naturally in connection with many problems involving [[asymptotics]]. A prominent example is [[Stirling's formula]] for the [[Asymptotic analysis|asymptotics]] of the [[factorial function]], in which both the numbers {{mvar|e}} and [[Pi|{{pi}}]] enter:
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| :<math>n! \sim \sqrt{2\pi n}\, \left(\frac{n}{e}\right)^n.</math>
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| A particular consequence of this is
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| :<math>e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}</math>.
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| ===Standard normal distribution===
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| (from [[Normal distribution]])
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| The simplest case of a normal distribution is known as the ''standard normal distribution'', described by this [[probability density function]]:
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| :<math>\phi(x) = \frac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} x^2}.</math>
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| The factor <math style="position:relative; top:-.2em">\scriptstyle\ 1/\sqrt{2\pi}</math> in this expression ensures that the total area under the curve ''ϕ''(''x'') is equal to one<sup>[[Gaussian integral|[proof]]]</sup>. The {{frac2|1|2}} in the exponent ensures that the distribution has unit variance (and therefore also unit standard deviation). This function is symmetric around ''x''=0, where it attains its maximum value <math style="position:relative; top:-.2em">\scriptstyle\ 1/\sqrt{2\pi}</math>; and has [[inflection point]]s at +1 and −1.
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| =={{mvar|e}} in calculus==
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| [[Image:Ln+e.svg|thumb|right|The natural log at (x-axis) {{mvar|e}}, {{math|ln(''e'')}}, is equal to 1]]
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| 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>Kline, M. (1998) ''Calculus: An intuitive and physical approach'', section 12.3 [http://books.google.co.jp/books?id=YdjK_rD7BEkC&pg=PA337 "The Derived Functions of Logarithmic Functions."], pp. 337 ff, Courier Dover Publications, 1998, ISBN 0-486-40453-6</ref> A general exponential function {{math|''y'' {{=}} ''a''<sup>''x''</sup>}} has derivative given as the [[limit of a function|limit]]:
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| :<math>\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\left(\lim_{h\to 0}\frac{a^h-1}{h}\right).</math>
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| The limit on the far right is independent of the variable {{math|''x''}}: it depends only on the base {{math|''a''}}. When the base is {{mvar|e}}, this limit is equal to one, and so {{mvar|e}} is symbolically defined by the equation:
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| :<math>\frac{d}{dx}e^x = e^x.</math>
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| Consequently, the exponential function with base {{mvar|e}} is particularly suited to doing calculus. Choosing {{mvar|e}}, as opposed to some other number, as the base of the exponential function makes calculations involving the derivative much simpler.
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| Another motivation comes from considering the base-{{math|''a''}} [[logarithm]].<ref>This is the approach taken by Kline (1998).</ref> Considering the definition of the derivative of {{math|log<sub>''a''</sub> ''x''}} as the limit:
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| :<math>\frac{d}{dx}\log_a x = \lim_{h\to 0}\frac{\log_a(x+h)-\log_a(x)}{h}=\frac{1}{x}\left(\lim_{u\to 0}\frac{1}{u}\log_a(1+u)\right),</math>
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| where the substitution {{math|''u'' {{=}} ''h''/''x''}} was made in the last step. The last limit appearing in this calculation is again an undetermined limit that depends only on the base {{math|a}}, and if that base is {{mvar|e}}, the limit is one. So symbolically,
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| :<math>\frac{d}{dx}\log_e x=\frac{1}{x}.</math>
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| The logarithm in this special base is called the [[natural logarithm]] and is represented as {{math|ln}}; it behaves well under differentiation since there is no undetermined limit to carry through the calculations.
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| There are thus two ways in which to select a special number {{math|''a'' {{=}} ''e''}}. One way is to set the derivative of the exponential function {{math|''a''<sup>''x''</sup>}} to {{math|''a''<sup>''x''</sup>}}, and solve for {{math|''a''}}. The other way is to set the derivative of the base {{math|''a''}} logarithm to {{math|1/''x''}} and solve for {{math|''a''}}. In each case, one arrives at a convenient choice of base for doing calculus. In fact, these two solutions for {{math|''a''}} are actually ''the same'', the number {{mvar|e}}.
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| ===Alternative characterizations===
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| [[Image:hyperbola E.svg|thumb|right|The area between the {{math|''x''}}-axis and the graph {{math|''y'' {{=}} 1/''x''}}, between {{math|''x'' {{=}} 1}} and {{math|''x'' {{=}} ''e''}} is 1.]]
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| {{See also|Representations of e}}
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| Other characterizations of {{mvar|e}} are also possible: one is as the [[limit of a sequence]], another is as the sum of an [[infinite series]], and still others rely on [[integral calculus]]. So far, the following two (equivalent) properties have been introduced:
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| 1. The number {{mvar|e}} is the unique positive [[real number]] such that
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| :<math>\frac{d}{dt}e^t = e^t.</math>
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| 2. The number {{mvar|e}} is the unique positive real number such that
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| :<math>\frac{d}{dt} \log_e t = \frac{1}{t}.</math>
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| The following three characterizations can be [[characterizations of the exponential function#Equivalence of the characterizations|proven equivalent]]:
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| 3. The number {{mvar|e}} is the [[limit of a sequence|limit]]
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| :<math>e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n</math>
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| Similarly:
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| :<math>e = \lim_{x\to 0} \left( 1 + x \right)^{\frac{1}{x}}</math>
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| 4. The number {{mvar|e}} is the sum of the [[infinite series]]
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| :<math>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>
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| where {{math|''n''!}} is the [[factorial]] of {{math|''n''}}.
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| 5. The number {{mvar|e}} is the unique positive real number such that
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| :<math>\int_1^e \frac{1}{t} \, dt = 1.</math>
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| ==Properties==
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| ===Calculus===
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| As in the motivation, the [[exponential function]] {{math|''e''<sup>''x''</sup>}} is important in part because it is the unique nontrivial function (up to multiplication by a constant) which is its own [[derivative]]
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| :<math>\frac{d}{dx}e^x=e^x</math>
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| and therefore its own [[antiderivative]] as well:
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| :<math>\int e^x\,dx = e^x + C.</math>
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| ===Exponential-like functions===
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| {{See also|Steiner's problem}}
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| [[Image:Xth root of x.svg|thumb|right|250px|The [[global maximum]] of <math>\sqrt[x]{x}</math> occurs at {{math|''x'' {{=}} ''e''}}.]]
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| The [[global maximum]] for the function
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| :<math> f(x) = \sqrt[x]{x}</math>
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| occurs at {{math|''x'' {{=}} ''e''}}. Similarly, {{math|''x'' {{=}} 1/''e''}} is where the [[global minimum]] occurs for the function
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| :<math> f(x) = x^x\, </math>
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| defined for positive {{math|''x''}}. More generally, {{math|''x'' {{=}} ''e''<sup>−1/''n''</sup>}} is where the global minimum occurs for the function
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| :<math> \!\ f(x) = x^{x^n} </math>
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| for any {{math|n > 0}}. The infinite [[tetration]]
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| :<math> x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} </math> or <sup>∞</sup><math>x</math>
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| converges if and only if {{math|''e''<sup>−''e''</sup> ≤ ''x'' ≤ ''e''<sup>1/''e''</sup>}} (or approximately between 0.0660 and 1.4447), due to a theorem of [[Leonhard Euler]].
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| ===Number theory===
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| The real number {{mvar|e}} is [[Irrational number|irrational]]. [[Leonhard Euler|Euler]] proved this by showing that its [[simple continued fraction]] expansion is infinite.<ref>{{cite web|url=http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2028%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|accessdate=2010-06-18}}</ref> (See also [[Joseph Fourier|Fourier]]'s [[proof that e is irrational|proof that {{mvar|e}} is irrational]].)
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| Furthermore, by the [[Lindemann–Weierstrass theorem]], {{mvar|e}} is [[Transcendental number|transcendental]], meaning that it is not a solution of any non-constant 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.
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| 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).
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| ===Complex numbers===
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| The [[exponential function]] {{math|''e''<sup>''x''</sup>}} may be written as a [[Taylor series]]
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| :<math> e^{x} = 1 + {x \over 1!} + {x^{2} \over 2!} + {x^{3} \over 3!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}</math>
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| Because this series keeps many important properties for {{math|''e''<sup>''x''</sup>}} even when {{math|''x''}} is [[complex number|complex]], it is commonly used to extend the definition of {{math|''e''<sup>''x''</sup>}} to the complex numbers. This, with the Taylor series for [[trigonometric functions|sin and cos {{math|''x''}}]], allows one to derive [[Euler's formula]]:
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| :<math>e^{ix} = \cos x + i\sin x,\,\!</math>
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| which holds for all {{math|''x''}}. The special case with {{math|''x'' {{=}} [[Pi|π]]}} is [[Euler's identity]]:
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| :<math>e^{i\pi} + 1 = 0\,\!</math>
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| from which it follows that, in the [[principal branch]] of the logarithm,
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| :<math>\ln (-1) = i\pi.\,\!</math>
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| Furthermore, using the laws for exponentiation,
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| :<math>(\cos x + i\sin x)^n = \left(e^{ix}\right)^n = e^{inx} = \cos (nx) + i \sin (nx),</math>
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| which is [[de Moivre's formula]].
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| The expression
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| :<math>\cos x + i \sin x \,</math>
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| is sometimes referred to as {{math|cis(''x'')}}.
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| ===Differential equations===
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| The general function
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| :<math>y(x) = Ce^x\,</math>
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| is the solution to the differential equation:
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| :<math>y' = y.\,</math>
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| ==Representations==
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| {{Main|List of representations of e}}
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| The number {{mvar|e}} can be represented as a [[real number]] in a variety of ways: as an [[infinite series]], an [[infinite product]], a [[continued fraction]], or a [[limit of a sequence]]. The chief among these representations, particularly in introductory [[calculus]] courses is the limit
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| :<math>\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n,</math>
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| given above, as well as the series
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| :<math>e=\sum_{n=0}^\infty \frac{1}{n!}</math>
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| given by evaluating the above [[power series]] for {{math|''e''<sup>''x''</sup>}} at {{math|''x'' {{=}} 1}}.
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| Less common is the [[continued fraction]] {{OEIS|id=A003417}}.
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| <!--move to history section or say <ref>[[Leonhard Euler|Euler]] was the first showed that {{mvar|e}} can be represented as a continued fraction.</ref>-->
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| :<math>
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| e = [2;1,\mathbf 2,1,1,\mathbf 4,1,1,\mathbf 6,1,1,...,\mathbf {2n},1,1,...] = [1;\mathbf 0,1,1,\mathbf 2,1,1,\mathbf 4,1,1,...,\mathbf {2n},1,1,...],
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| </math><ref>Hofstadter, D. R., "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought" Basic Books (1995) ISBN 0-7139-9155-0</ref>
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| which written out looks like
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| :<math>e = 2+
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| \cfrac{1}
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| {1+\cfrac{1}
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| {\mathbf 2 +\cfrac{1}
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| {1+\cfrac{1}
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| {1+\cfrac{1}
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| {\mathbf 4 +\cfrac{1}
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| {1+\cfrac{1}
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| {1+\ddots}
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| }
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| }
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| }
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| }
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| }
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| }
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| = 1+
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| \cfrac{1}
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| {\mathbf 0 + \cfrac{1}
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| {1 + \cfrac{1}
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| {1 + \cfrac{1}
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| {\mathbf 2 + \cfrac{1}
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| {1 + \cfrac{1}
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| {1 + \cfrac{1}
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| {\mathbf 4 + \cfrac{1}
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| {1 + \cfrac{1}
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| {1 + \ddots}
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| }
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| }
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| }
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| }
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| }
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| }
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| }
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| }.
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| </math>
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| This continued fraction for {{mvar|e}} converges three times as quickly:
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| :<math> e = [ 1 ; 0.5 , 12 , 5 , 28 , 9 , 44 , 13 , \ldots , 4(4n-1) , (4n+1) , \ldots ],</math>
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| which written out looks like
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| :<math> e = 1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\cfrac{1}{22+\cfrac{1}{26+\ddots\,}}}}}}}.</math>
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| Many other series, sequence, continued fraction, and infinite product representations of {{mvar|e}} have been developed.
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| ===Stochastic representations===
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| 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 {{math|''V''}} be the least number {{math|''n''}} such that the sum of the first {{math|''n''}} samples exceeds 1:
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| :<math>V = \min { \left \{ n \mid X_1+X_2+\cdots+X_n > 1 \right \} }.</math>
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| Then the [[expected value]] of {{math|''V''}} is {{mvar|e}}: {{math|E(''V'') {{=}} ''e''}}.<ref>Russell, K. G. (1991) ''[http://links.jstor.org/sici?sici=0003-1305%28199102%2945%3A1%3C66%3AETVOEB%3E2.0.CO%3B2-U Estimating the Value of e by Simulation]'' The American Statistician, Vol. 45, No. 1. (Feb., 1991), pp. 66–68.</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>
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| ===Known digits===
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| The number of known digits of {{mvar|e}} has increased dramatically 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 e and its computation]</ref><ref>Gourdon, X.; [http://numbers.computation.free.fr/Constants/PiProgram/computations.html Reported large computations with PiFast]</ref>
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| | |
| {| class="wikitable" style="margin: 1em auto 1em auto"
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| |+ '''Number of known decimal digits of {{mvar|e}} '''
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| ! Date || Decimal digits || Computation performed by
| |
| |-
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| | 1748 ||align=right| 23 || [[Leonhard Euler]]<ref>''Introductio in analysin infinitorum'' [http://books.google.de/books?id=jQ1bAAAAQAAJ&pg=PA90 p. 90]</ref>
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| |-
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| | 1853 ||align=right| 137 || [[William Shanks]]
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| |-
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| | 1871 ||align=right| 205 || [[William Shanks]]
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| |-
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| | 1884 ||align=right| 346 || J. Marcus Boorman
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| |-
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| | 1949 ||align=right| 2,010 || [[John von Neumann]] (on the [[ENIAC]])
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| |-
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| | 1961 ||align=right| 100,265 || [[Daniel Shanks]] and [[John Wrench]]<ref name="We have computed e on a 7090 to 100,265D by the obvious program.">{{cite journal|author=Daniel Shanks and John W Wrench|quote=We have computed e on a 7090 to 100,265D by the obvious program|title=Calculation of Pi to 100,000 Decimals|journal =Mathematics of Computation|volume= 16 |year=1962| issue =77| pages =76–99 (78)|url=http://www.ams.org/journals/mcom/1962-16-077/S0025-5718-1962-0136051-9/S0025-5718-1962-0136051-9.pdf|doi=10.2307/2003813}}</ref>
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| |-
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| | 1978 ||align=right| 116,000 || [[Steve Wozniak]] on the [[Apple II]]<ref name="wozniak198106">{{cite news | url=http://archive.org/stream/byte-magazine-1981-06/1981_06_BYTE_06-06_Operating_Systems#page/n393/mode/2up | title=The Impossible Dream: Computing ''e'' to 116 Places with a Personal Computer | work=BYTE | date=June 1981 | accessdate=18 October 2013 | author=Wozniak, Steve | pages=392}}</ref>
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| |-
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| | 1994 April 1 ||align=right| 1,000,000 || [[Robert J. Nemiroff]] & Jerry Bonnell <ref>[http://apod.nasa.gov/htmltest/gifcity/e.1mil Email from Robert Nemiroff and Jerry Bonnell – The Number e to 1 Million Digits]. None. Retrieved on 2012-02-24.</ref>
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| |-
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| | 1999 November 21 ||align=right| 1,250,000,000 || Xavier Gourdon <ref name="Email from Xavier Gourdon to Simon Plouffe">[http://web.archive.org/web/20021223163426/http://pi.lacim.uqam.ca/piDATA/expof1.txt Email from Xavier Gourdon to Simon Plouffe – I have made a new e computation (with verification): 1,250,000,000 digits]. None. Retrieved on 2012-02-24.</ref>
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| |-
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| | 2000 July 16 ||align=right| 3,221,225,472 || Colin Martin & Xavier Gourdon <ref name="PiHacks message 177 - E to 3,221,225,472 D">[http://groups.yahoo.com/group/pi-hacks/message/177 PiHacks message 177 – E to 3,221,225,472 D]. Groups.yahoo.com. Retrieved on 2012-02-24.</ref>
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| |-
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| | 2003 September 18 ||align=right| 50,100,000,000 || Shigeru Kondo & Xavier Gourdon <ref name="PiHacks message 1071 - Two new records: 50 billions for E and 25 billions for pi">[http://groups.yahoo.com/group/pi-hacks/message/1071 PiHacks message 1071 – Two new records: 50 billions for E and 25 billions for pi]. Groups.yahoo.com. Retrieved on 2012-02-24.</ref>
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| |-
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| | 2007 April 27 ||align=right| 100,000,000,000 || Shigeru Kondo & Steve Pagliarulo <ref name="English Version of PI WORLD">[http://web.archive.org/web/20021221061853/http://ja0hxv.calico.jp/pai/eevalue.html English Version of PI WORLD]. Ja0hxv.calico.jp. Retrieved on 2012-02-24.</ref>
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| |-
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| | 2009 May 6 ||align=right| 200,000,000,000 || Rajesh Bohara & Steve Pagliarulo <ref name="English Version of PI WORLD"/>
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| |-
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| | 2010 July 5 ||align=right| 1,000,000,000,000 || Shigeru Kondo & Alexander J. Yee <ref>[http://www.numberworld.org/digits/E/ A list of notable large computations of e]. Numberworld.org. Last updated: March 7, 2011. Retrieved on 2012-02-24.</ref>
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| |}
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| ==In computer culture==
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| In contemporary [[internet culture]], individuals and organizations frequently pay homage to the number {{mvar|e}}.
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| For instance, in the [[Initial Public Offering|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, which is {{mvar|e}} billion [[United States dollar|dollars]] to the nearest dollar. Google was also responsible for a billboard<ref>[http://braintags.com/archives/2004/07/first-10digit-prime-found-in-consecutive-digits-of-e/ First 10-digit prime found in consecutive digits of {{math|e}}}]. Brain Tags. Retrieved on 2012-02-24.</ref> 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". Solving this problem and visiting the advertised web site (now defunct) led to an even more difficult problem to solve, which in turn led to [[Google Labs]] where the visitor was invited to submit a resume.<ref>{{cite news|first=Andrea|last=Shea|url=http://www.npr.org/templates/story/story.php?storyId=3916173|title=Google Entices Job-Searchers with Math Puzzle|work=NPR|accessdate=2007-06-09}}</ref> The first 10-digit prime in {{mvar|e}} is 7427466391, which starts at the 99th digit.<ref>{{cite web |first=Marcus |last=Kazmierczak |url=http://mkaz.com/math/google-billboard |title=Google Billboard |publisher=mkaz.com |date=2004-07-29 |accessdate=2007-06-09}}</ref>
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| In another instance, 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. Similarly, the version numbers of his [[TeX]] program approach {{pi}}.<ref>{{Cite journal|url=http://www.tex.ac.uk/tex-archive/digests/tex-mag/v5.n1|title=The Future of TeX and Metafont|first=Donald|last=Knuth|authorlink=Donald Knuth|journal=TeX Mag|volume=5|issue=1}}</ref>
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| ==Notes==
| |
| {{Reflist|colwidth=30em}}
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| ==Further reading==
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| * Maor, Eli; ''{{mvar|e}}: The Story of a Number'', ISBN 0-691-05854-7
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| * [http://www.johnderbyshire.com/Books/Prime/Blog/page.html#endnote10 Commentary on Endnote 10] of the book ''[[Prime Obsession]]'' for another stochastic representation
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| ==External links==
| |
| {{Commons category|E (mathematical constant)}}
| |
| *[http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/ An Intuitive Guide To Exponential Functions &{{mvar|e}}] for the non-mathematician
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| *[<!-- http://www.gutenberg.org/etext/127 -->http://gutenberg.org/ebooks/127 The number {{mvar|e}} to 1 million places] and [<!-- http://antwrp.gsfc.nasa.gov/htmltest/rjn_dig.html -->http://apod.nasa.gov/htmltest/gifcity/e.2mil 2 and 5 million places (link obsolete)]
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| *[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)
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| *[http://www.subidiom.com/e {{mvar|e}} Search Engine] 2 billion searchable digits of {{mvar|e}}, {{pi}} and √2
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| {{Good article}}
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| {{DEFAULTSORT:E (Mathematical Constant)}}
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| [[Category:Transcendental numbers]]
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| [[Category:Mathematical constants]]
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| [[Category:E (mathematical constant)|*]]
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| {{Link FA|ka}}
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| {{Link FA|mk}}
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| {{Link FA|lmo}}
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