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

e equals the limit as n goes to infinity of the quanity one plus one over n, the quantity to the nth power ,

approximately 2.71828182845904523536…

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 1736.¹

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

In calculus, e^x is its own derivative.

Newton, 1665

$e equals one plus the sum of an infinite series of fractions: one over one factorial, plus one over two factorial, plus one over three factorial, and so on.$

(! 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

an equation

1.
Leonard Euler.
Mechanica sive motus scientia analytice exposita.
St. Petersburg, 1736.

Pages 251, 256.

the number e

Newton, 1665

Euler, 1737

Further Reading