Chiffres de e

The constant turned up before anyone was looking for it. An appendix to John Napier's 1618 work on logarithms contains a table of what are effectively natural logarithms, with no indication that their base was a number worth naming. Jacob Bernoulli came closer in 1683 while working on continuously compounded interest: he asked what (1 + 1/n)ⁿ approaches as n grows, showed the limit lies between 2 and 3, and thereby produced the first real approximation to e.

The constant got a letter before it got a decimal expansion. Leibniz, writing to Huygens around 1690, called it b. The e we use is Euler's, who was using it by the late 1720s; the usual guess is that he simply took the next vowel after a, which he had already spent. The frequently repeated claim that he named it after himself has no support — Euler was not given to that, and he never said so.

Euler is nonetheless responsible for most of what makes e interesting. In Introductio in analysin infinitorum (1748) he published the value to 18 decimal places, 2.718281828459045235, along with the series e = 1 + 1/1! + 1/2! + ⋯ that this page sums, the relation now written e^(ix) = cos x + i sin x, and the continued fraction [2; 1, 2, 1, 1, 4, 1, 1, 6, …], whose failure to terminate proves e irrational.

Transcendence took another century and a half. Charles Hermite settled it in 1873, making e the first number proved transcendental that had not been constructed for the purpose — Liouville's earlier examples were built to be transcendental. Hermite's method was adapted by Lindemann nine years later to dispose of π.

Propriétés principales

• e = 1 + 1/1! + 1/2! + 1/3! + ⋯, the series this page sums; the terms shrink fast enough that about 1,760 of them pin down 5,000 digits.
• e = lim(n→∞) (1 + 1/n)ⁿ, which is why it governs continuously compounded growth.
• The functions that are equal to their own derivative are exactly the constant multiples of eˣ.
• e is irrational. Euler proved it by showing its simple continued fraction, [2; 1, 2, 1, 1, 4, 1, 1, 6, …], does not terminate.
• e is transcendental, proved by Charles Hermite in 1873 — the first number shown transcendental that had not been constructed for that purpose.
• The digit block 1828 appears twice in a row at the start — 2.718281828 — and then does not recur immediately. It is a coincidence, not a pattern, and it makes the opening easy to memorise.
• The ten-digit prime 7427466391 begins at the 99th decimal place of e.

Où on les rencontre

• Continuously compounded interest, and every exponential growth or decay law written in the form A·e^(kt) — radioactive half-lives, capacitor discharge, population models.
• The normal distribution, whose density is proportional to e^(−x²/2), and the Poisson distribution, whose probabilities carry a factor of e^(−λ).
• The secretary problem: examining the first 1/e ≈ 36.8% of applicants and then taking the next one better than all of them is the optimal stopping rule.
• Stirling's approximation n! ≈ √(2πn)·(n/e)ⁿ, the standard tool for the factorials behind combinatorics and statistical mechanics.
• Google filed to raise exactly $2,718,281,828 in its 2004 IPO, and ran a recruiting billboard reading "{first 10-digit prime found in consecutive digits of e}.com" — the answer, 7427466391, starts at the 99th decimal place.