E (mathematical constant)
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 logarithms to the base e {\displaystyle e}. It is assumed that the table was written by William Oughtred. In 1661, Christiaan Huygens studied how to compute logarithms by geometrical methods and calculated a quantity that, in retrospect, is the base-10 logarithm of e, but he did not recognize e itself as a quantity of interest.

The constant itself was introduced by Jacob Bernoulli in 1683, for solving the problem of continuous compounding of interest. In his solution, the constant e occurs as the limit lim n → ∞ (1 + 1 n) n, {\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n},} where n represents the number of intervals in a year on which the compound interest is evaluated (for example, n = 12 {\displaystyle n=12} for monthly compounding).

The first symbol used for this constant was the letter b by Gottfried Leibniz in letters to Christiaan Huygens in 1690 and 1691.

Leonhard Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, and in a letter to Christian Goldbach on 25 November 1731. The first appearance of e in a printed publication was in Euler's Mechanica (1736). It is unknown why Euler chose the letter e. Although some researchers used the letter c in the subsequent years, the letter e was more common and eventually became standard.

Euler proved that e is the sum of the infinite series e = ∑ n = 0 ∞ 1 n ! = 1 0 ! + 1 1 ! + 1 2 ! + 1 3 ! + 1 4 ! + ⋯, {\displaystyle e=\sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots,} where n ! is the factorial of n. The equivalence of the two characterizations using the limit and the infinite series can be proved via the binomial theorem.
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