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:''e'' ≈ 2,71828 18284 59045 23536 02874 71352
==Definisi==
:<math>\lim_{x \rightarrow 0} (1 + x)^{\frac{1}{x}}</math>
:<math>\lim_{x \rightarrow \infty} (1 + \frac{1}{x})^x</math>
==Lihat pula==
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==History==
The first references to the constant were published in [[1618]] in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of natural logarithms calculated from the constant. It is assumed that the table was written by [[William Oughtred]]. The first indication of ''e'' as a constant was discovered by [[Jacob Bernoulli]], trying to find the value of the following expression:
: <math>\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n</math>
The first known use of the constant, represented by the letter b, was in correspondence from [[Gottfried Leibniz]] to [[Christiaan Huygens]] in [[1690]] and [[1691]]. [[Leonhard Euler]] started to use the letter e for the constant in [[1727]], and the first use of ''e'' in a publication was Euler's ''Mechanica'' ([[1736]]). While in the subsequent years some researchers used the letter c, e was more common and eventually became the standard.
The exact reasons for the use of the letter e are unknown, but it may be because it is the first letter of the word ''[[exponential]]''. Another possibility is that Euler used it because it was the first [[vowel]] after [[a]], which he was already using for another number, but his reason for using vowels is unknown. It is unlikely that Euler chose the letter because it is his first initial, since he was a very modest man, and tried to give proper credit to the work of others.{{rf|1|OConnor}}
==Definitions==
The three most common definitions of ''e'' are listed below.
# The [[limit (mathematics)|limit]]
#:<math>e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n</math>
#:
# The sum of the [[infinite series]]
#:<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>
#::where ''n''! is the [[factorial]] of ''n''.
#:
# The unique [[real number]] ''e'' > 0 such that
#:<math>\int_{1}^{e} \frac{1}{t} \, dt = {1}</math>
#::(that is, the number ''e'' such that area under the [[hyperbola]] <math> f(t)=1/t </math> from 1 to ''e'' is equal to 1).
These different definitions can be [[characterizations of the exponential function|proven]] to be equivalent.
==Properties==
The [[exponential function]] ''f(x)=e''<sup>''x''</sup> is important because it is the unique function (up to multiplication by a constant) which is its own [[derivative]], and therefore, its own [[antiderivative|primitive]]:
:<math>\frac{d}{dx}e^x=e^x</math> and
:<math>\int e^x\,dx=e^x + C</math>, where ''C'' is the [[arbitrary constant of integration]].
It is known that ''e'' is [[Irrational number|irrational]] ([[Proof that e is irrational|proof]]) and even more, [[Transcendental number|transcendental]] ([[Lindemann-Weierstrass theorem|proof]]). It was the first number to be proved transcendental without having been specifically constructed for this purpose (cf. [[Liouville number]]); the proof was given by [[Charles Hermite]] in [[1873]]. It is conjectured to be [[normal number|normal]]. It features in [[Euler's formula]], one of the most important formulas in mathematics:
:<math>e^{ix} = \cos(x) + i\sin(x),\,\!</math>
described by [[Richard Feynman]] (p. I-22-10) as "Euler's jewel".
The special case with ''x'' = π is known as [[Euler's identity]]:
:<math>e^{i\pi}+1 =0 .\,\!</math>
The following is an infinite [[simple continued fraction]] expansion of ''e'' (sequence [[OEIS:A005131|A005131]] in [[Online Encyclopedia of Integer Sequences|OEIS]]):
:<math>e = [1; 0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, \ldots,1, 2n, 1,\ldots] \,</math>
The following is an infinite [[generalized continued fraction]] expansion of ''e'':
:<math>e= 2+\frac{1}{1+\frac{1}{2+\frac{2}{3+\frac{3}{\ddots}}}}.</math>
The number ''e'' is also equal to the sum of the following [[infinite series]]:
:<math>e = \left [ \sum_{k=0}^\infty \frac{(-1)^k}{k!} \right ]^{-1}</math>
:<math>e = \left [ \sum_{k=0}^\infty \frac{1-2k}{(2k)!} \right ]^{-1}</math>
:<math>e = \frac{1}{2} \sum_{k=0}^\infty \frac{k+1}{k!}</math>
:<math>e = 2 \sum_{k=0}^\infty \frac{k+1}{(2k+1)!}</math>
:<math>e = \sum_{k=0}^\infty \frac{3-4k^2}{(2k+1)!}</math>
:<math>e = \sum_{k=0}^\infty \frac{3k^2+1}{(3k)!}</math>
:<math>e = \left [ \sum_{k=0}^\infty \frac{4k+3}{2^{2k+1}\,(2k+1)!} \right ]^2</math>
:<math>e = \frac{-12}{\pi^2} \left [ \sum_{k=1}^\infty \frac{1}{k^2} \ \cos \left ( \frac{9}{k\pi+\sqrt{k^2\pi^2-9}} \right ) \right ]^{-1/3} </math>
:<math>e = \sum_{k=1}^\infty \frac{k^2}{2(k!)}</math>
The number ''e'' is also given by several [[infinite product]] forms including the [[Pippenger product]]
:<math> e= 2 \left ( \frac{2}{1} \right )^{1/2} \left ( \frac{2}{3}\; \frac{4}{3} \right )^{1/4} \left ( \frac{4}{5}\; \frac{6}{5}\; \frac{6}{7}\; \frac{8}{7} \right )^{1/8} \cdots </math>
as well as,
:<math> \frac{2\cdot 2^{(\ln(2)-1)^2} \cdots}{2^{\ln(2)-1}\cdot 2^{(\ln(2)-1)^3}\cdots }</math>
The number ''e'' is equal to the [[limit of a sequence|limit]] of several [[infinite sequences]]:
:<math> e= \lim_{n \to \infty} n\cdot\left ( \frac{\sqrt{2 \pi n}}{n!} \right )^{1/n} </math> and
:<math> e=\lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}} </math> (both by [[Stirling's formula]]).
The symmetric limit,
:<math>e=\lim_{n \to \infty} \left [ \frac{(n+1)^{n+1}}{n^n}- \frac{n^n}{(n-1)^{n-1}} \right ]</math>
may be obtained by manipulation of the basic limit definition of ''e''. Another limit is
:<math>e= \lim_{n \to \infty}(p_n \#)^{1/p_n} </math>
where <math> p_n </math> is the ''n''th [[prime number|prime]] and <math> p_n \# </math> is the [[primorial]] of the ''n''th prime.
It was shown by Euler that the infinite [[tetration]]
:<math> x^{x^{\cdot^{\cdot^{\cdot}}}}, </math>
converges only if <math>e^{-e} \le x \le e^{1/e}. </math>
==Non-mathematical uses of ''e''==
One of the most famous mathematical constants, ''e'' is also frequently referenced outside of mathematics. Some examples are:
* In 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, which is, of course, ''e'' billion [[United States dollar|dollars]] to the nearest dollar.
* Google was also responsible for a mysterious billboard [http://mattwalsh.com/twiki/pub/Main/GoogleBillboardContestFindingPrimesInE/IMG_0742.JPG] that appeared in the heart of [[Silicon Valley]], and later in [[Cambridge, Massachusetts]], which read ''{first 10-digit prime found in consecutive digits of ''e''}.com''. Solving this problem and visiting the web site advertised led to an even more difficult problem to solve, which in turn leads to [[Google Labs]] where the visitor is invited to submit a resume. The first 10-digit prime in ''e'' is 7427466391, which surprisingly starts as late as at the 101st digit. [http://www.mkaz.com/math/google/]
* The famous [[Computer Science|computer scientist]] [[Donald Knuth]] let the version numbers of his book [[METAFONT]] approach ''e'' (the versions are 2, 2.7, 2.71, 2.718, etc.).
==References==
* Maor, Eli; ''e: The Story of a Number'', ISBN 0691058547
* O'Connor, J.J., and Roberson, E.F.; ''The MacTutor History of Mathematics archive'': [http://www-history.mcs.st-andrews.ac.uk/HistTopics/e.html "The number ''e''"]; University of St Andrews Scotland (2001)
==Notes==
{{ent|1|OConnor}} O'Connor, "The number ''e''"
==External links==
* [http://www.gutenberg.org/etext/127 The number ''e'' to 1 million places] and [http://antwrp.gsfc.nasa.gov/htmltest/rjn_dig.html 2, 5] or [http://67.49.215.31/constants.htm 10 million places]
* [http://members.aol.com/jeff570/constants.html Earliest Uses of Symbols for Constants]
* [http://www.austms.org.au/Modules/Exp/ e the EXPONENTIAL - the Magic Number of GROWTH] - Keith Tognetti, University of Wollongong, NSW, Australia
* [http://members.optusnet.com.au/exponentialist/The_Scales_Of_e.htm 'The Scales Of e' demonstrates that fixed rate and variable rate compound growth are both exponential in nature.]
[[Category:Transcendental numbers]]
[[Category:Mathematical constants]]
[[Category:Exponentials]]
[[Category:Logarithms]]
[[Category:Famous numbers|2.71828]]
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[[Kategori:Matematika]]
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