# E mathematical constant

The mathematical constant e is the base of the natural logarithm function. Its value to the 29th decimal digit is:

e = 2.71828 18284 59045 23536 02874 7135...

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

e is 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.

## Definitions

The three most common definitions of e are listed below.

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

These different definitions have been proven to be equivalent.

## Properties

The exponential function ex is important because it is the unique function (up to multiplication by a constant) which is its own derivative, and therefore, its own primitive:

$\displaystyle \frac{d}{dx}e^x=e^x$ and
$\displaystyle \int e^x\,dx=e^x + C$ , where C is a constant.

e is known to be both irrational (proof) and transcendental (proof). 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:

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

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

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

described by Richard Feynman as Euler's jewel.

The infinite continued fraction expansion of e contains an interesting pattern (sequence A005131 in OEIS) that can be written as follows:

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

## 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.

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

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, 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 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.1

## 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 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 responsible for a mysterious billboard [1] 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. (The first 10-digit prime in e is 7427466391, which surprisingly starts as late as at the 101st digit.) [2]

## 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: "The number e"; University of St Andrews Scotland (2001)

## Notes

Template:Ent O'Connor, "The number e"