e (mathematical constant)

The mathematical constant e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = ex at the point x = 0 is equal to 1. The function ex so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e. The number e is also commonly defined as the base of the natural logarithm (using an integral to define the latter), as the limit of a certain sequence, or as the sum of a certain series (see the alternative characterizations, below).The number e is sometimes called Euler's number after the Swiss mathematician Leonhard Euler. (e is not to be confused with γ—the Euler–Mascheroni constant, sometimes called simply Euler's constant.) It is also sometimes known as Napier's constant, although the symbol e is in honor of Euler.The number e is of eminent importance in mathematics,alongside 0, 1, π and 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.The number e is irrational; it is not a ratio of integers. Furthermore, it is transcendental; it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places.
History:
he first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier.[4] 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 e):
    e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n
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 or 1728,
 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.
Bernoulli trials:
The number e itself also has applications to probability theory, where it arises in a way not obviously related to exponential growth.
\binom{10^6}{k} \left(10^{-6}\right)^k(1-10^{-6})^{10^6-k}.
In particular, the probability of winning zero times (k = 0) is
\left(1-\frac{1}{10^6}\right)^{10^6}.
This is very close to the following limit for 1/e:
\frac{1}{e} = \lim_{n\to\infty} \left(1-\frac{1}{n}\right)^n.
Derangements;
p_n = 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+(-1)^n\frac{1}{n!}.
Asymptotics:
The number e occurs naturally in connection with many problems involving asymptotics. A prominent example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers e and π enter:
n! \sim \sqrt{2\pi n}\, \left(\frac{n}{e}\right)^n.
A particular consequence of this is
e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}.
e in calculus:
The principal motivation for introducing the number e, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms.[8] A general exponential function y=ax has derivative given as the limit:
\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).
The limit on the right-hand side is independent of the variable x: it depends only on the base a. When the base is e, this limit is equal to one, and so e is symbolically defined by the equation:
    \frac{d}{dx}e^x = e^x.
Consequently, the exponential function with base e is particularly suited to doing calculus. Choosing e, as opposed to some other number, as the base of the exponential function makes calculations involving the derivative much simpler.
Another motivation comes from considering the base-a logarithm.[9] Considering the definition of the derivative of logax as the limit:
    \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),
where the substitution 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 a, and if that base is e, the limit is one. So symbolically,
    \frac{d}{dx}\log_e x=\frac{1}{x}.
 Stochastic representations;
In addition to the deterministic analytical expressions for representation of e, as described above, there are some stochastic protocols for estimation of e. In one such protocol, random samples X1, X2, ..., Xn of size n from the uniform distribution on (0, 1) are used to approximate e. If

    U= \min { \left \{ n \mid X_1+X_2+\cdots+X_n > 1 \right \} },

then the expectation of U is e: E(U) = e.[12][13] Thus sample averages of U variables will approximate e.
Known digits:
The number of known digits of e has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements.
Number of known decimal digits of e Date     Decimal digits     Computation performed by
1748     18     Leonhard Euler
1853     137     William Shanks
1871     205     William Shanks
1884     346     J. Marcus Boorman
1946     808     Unknown
1949     2,010     John von Neumann (on the ENIAC)
1961     100,265     Daniel Shanks and John Wrench
1978     116,000     Stephen Gary Wozniak (on the Apple II
1994 April 1     10,000,000     Robert Nemiroff & Jerry Bonnell
1997 May     18,199,978     Patrick Demichel
1997 August     20,000,000     Birger Seifert
1997 September     50,000,817     Patrick Demichel
1999 February     200,000,579     Sebastian Wedeniwski
1999 October     869,894,101     Sebastian Wedeniwski
1999 November 21     1,250,000,000     Xavier Gourdon
2000 July 10     2,147,483,648     Shigeru Kondo & Xavier Gourdon
2000 July 16     3,221,225,472     Colin Martin & Xavier Gourdon
2000 August 2     6,442,450,944     Shigeru Kondo & Xavier Gourdon
2000 August 16     12,884,901,000     Shigeru Kondo & Xavier Gourdon
2003 August 21     25,100,000,000     Shigeru Kondo & Xavier Gourdon
2003 September 18     50,100,000,000     Shigeru Kondo & Xavier Gourdon
2007 April 27     100,000,000,000     Shigeru Kondo & Steve Pagliarulo
2009 May 6     200,000,000,000     Shigeru Kondo & Steve Pagliarulo
2010 February 21     500,000,000,000     Alexander J. Yee [26]
2010 July 5     1,000,000,000,000     Shigeru Kondo & Alexander J. Yee
In computer culture:
In contemporary internet culture, individuals and organizations frequently pay homage to the number e.
For example, 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 e billion dollars to the nearest dollar. Google was also responsible for a billboard[28] 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 e}.com (now defunct). Solving this problem and visiting the advertised web site 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. The first 10-digit prime in e is 7427466391, which starts as late as at the