# the number *e*

The positive, irrational (proved by Euler, 1737), transcendental
(proved by Hermite, 1873) number which is the
limit as n tends to infinity of

,

approximately 2**.**718 281 828 459 045 235 36…

According to Cajori, the first mathematician to call this number *e* was Leonard Euler, in
manuscripts dated 1728 or 1729, a letter of 1731, and then in a publication of 1736.ยน

*e* is the base of natural logarithms, so ln e = 1.

In calculus, *e*^{x} is its own derivative.

## Newton, 1665

(! is the factorial sign, indicating the product of all the integers greater
than zero up to the one marked, so for example 5! = 1×2×3×4×5 =
120.)

## Euler, 1737

1. Leonard Euler.

*Mechanica sive motus scientia analytice exposita.*

St. Petersburg, 1736.

Pages 251, 256.

## Further Reading

Eli Maor.

*e. The Story of a Number.*

Princeton, NJ: Princeton University Press, 1994.

http://turnbull.dcs.st-and.ac.uk/history/HistTopics/e.html

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Last revised: 22 October 2002.