E (mathematical constant)
 The title of this article given above is incorrect, due to technical limitations. The correct title is e (mathematical constant).
 e ≈ 2.71828 18284 59045 23536 02874 ...
Table of contents 
2 Properties 3 History 4 Humorous use of e 5 External links 6 Reference 
Definitions
The three most common definitions of e are the following.
 1. Define e by the following limit.

 2. Define e as the sum of the following infinite series.

 Here n! stands for the factorial of n.
 3. Define e to be the unique number x > 0 such that

In 1975, the Swiss Felix A. Keller discovered the following formula that converges in e ("Keller's Expression" Steven Finch, mathsoft):   :   This formula was published for the first time 1998 on Steven Finch's  website.
Properties
The exponential function is important because it is, up to multiplication by a scalar, the unique function which is its own derivative and is hence commonly used to model growth or decay processes.
The number e is known to be irrational and even transcendental. It was the first number to be proved transcendental without having been specifically constructed for this purpose (c.f. Liouville number); the proof was given by Charles Hermite in 1873. It is conjectured to be normal. It features in Euler's Formula, one of the most important identities in mathematics:
The infinite continued fraction expansion of e contains an interesting pattern that can be written as follows:
Proofs
See the following articles for proofs of properties of e: Proof of the equivalence of the definitions of e
 Proof that e is irrational
 Proof that e is transcendental
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.
The exact reasons for the use of e are unknown, but it may be because the letter e is the first letter of the word exponential. Another view is that the letters a, b, c, and d were already frequently used for other purposes, and e was the first available letter. It is unlikely that Euler choose the letter because it is his first initial, since he was a very modest man, always trying to give proper credit to the work of others.
Humorous use of e
In the IPO filing for Google, Inc, in 2004, rather than a typical roundnumber amount of money, the company announced its intention to raise $2,718,281,828, which is, of course, e billion dollars to the nearest integer.External links
Reference
 Eli Maor, e: The Story of a Number, ISBN 0691058547