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 2.718 281 828 459 045 235 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

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

In calculus, ex 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

e to the power of pi times i, plus 1, equals zero

an equation

an equation

an equation

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.

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