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E (mathematical constant)

The mathematical constant e (occasionally called Euler's number after the Swiss mathematician Leonhard Euler, or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms) is the base of the natural logarithm function. Its approximate value is:

e ≈ 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 95749 66967 6277

Alongside the number π and the imaginary unit i, e is one of the most important mathematical constants. It has a number of equivalent definitions; some of them are given below.

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Definitions

The three most common definitions of e are the following.

1. Define e by the following limit.
$e = \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n$
2. Define e as the sum of the following infinite series.
$e = \sum_{n=0}^\infty {1 \over n!} = {1 \over 0!} + {1 \over 1!} + {1 \over 2!} + {1 \over 3!} + {1 \over 4!} + \cdots$
where n! is the factorial of n.
3. Define e to be the unique number x > 0 such that
$\ln{x} = \int_{1}^{x} \frac{1}{t} \, dt = {1}$

These different definitions have been proven to be equal.

Properties

Many growth or decay processes can be modeled with an exponential function. The exponential function ex is important because it is the unique function which is its own derivative (up to a constant factor; the most general function that is its own derivative is kex, for any k).

e is known to be both irrational and transcendental. It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare 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:

$e^{ix} = \cos(x) + i\sin(x) \,\!$

The special case with x = π is known as Euler's identity:

$e^{i\pi} + 1 = 0 \,\!$

described by Richard Feynman as Euler's jewel.

The infinite continued fraction expansion of e contains an interesting pattern that can be written as follows:

$e = [1; 0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12,\ldots] \,$

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.

$\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n$

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, 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, the use of e was more common and eventually became the standard.

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.

Non-mathematical uses of e

In the IPO filing for Google Inc., 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 dollars to the nearest dollar.

Google was also the culprit of a mysterious billboard 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. Once solving this problem (the first 10-digit prime in e is 7427466391, which surprisingly starts as soon as the 101st digit), and visiting the web site advertised, there was an even more difficult problem to solve.

Donald Knuth, the famous computer scientist let the version-numbers of his book METAFONT approach e (i.e the versions are 2, 2.7, 2.71, 2.718, etc.)