A 1683 study of interest compounded ever more frequently, and the limit it approaches (1683)
Jacob Bernoulli's compound-interest problem, the constant's first appearance
Science · Mathematics
e — Euler's number
1683 / 1748
Mathematics — constants and analysis
The base of natural logarithms, approximately 2.71828, and the one number whose exponential function is its own derivative. It first appeared in 1683 when Jacob Bernoulli studied continuously compounded interest, and it takes its usual symbol and its central place in analysis from Leonhard Euler, whose Introductio in analysin infinitorum of 1748 gave the series and the continued fraction. It is irrational, proved by Euler, and transcendental, proved by Charles Hermite in 1873, and it governs exponential growth and decay, radioactive half-lives, compound interest and the normal distribution.
How to spot them
- ▸The base of natural logarithms, roughly 2.71828
- ▸The unique base for which the exponential curve equals its own gradient at every point
- ▸Arose from a compound-interest question: the limit of (1 + 1/n) to the power n
- ▸Proved transcendental by Charles Hermite in 1873, nine years before the circle constant
- ▸The sum of the reciprocals of the factorials: 1 + 1 + 1/2 + 1/6 + 1/24 + …
Key facts
The base of the logarithms called 'natural', written ln
Natural logarithms; Napier's 1614 tables were closely related but not quite to this base
Hermite's 1873 proof that this constant is transcendental (1873)
Charles Hermite — the first number proved transcendental that was not built for the purpose
The secretary problem, in which you reject the first 37% of candidates on principle
The optimal-stopping rule: reject the first 1/e of applicants, then take the next best
The one curve that is its own derivative and its own integral
The exponential function, the defining property of this base
The derangement problem: the chance that no hat is returned to its owner
Tends to the reciprocal of the constant, about 0.3679
The sum of the reciprocals of the factorials, 1 + 1 + 1/2 + 1/6 + 1/24 + … (1748)
The series given in the Introductio in analysin infinitorum
Key ideas
- Compounding interest more and more often does not give unlimited money: £1 at 100% a year compounded continuously becomes about £2.72, and the limit of (1 + 1/n)^n is this constant.
- Because the exponential function is its own derivative, any quantity whose rate of change is proportional to its size — populations, radioactivity, cooling, capacitor discharge — is governed by it.
Quick-fire facts
- •Approximately 2.718281828459045; the repeated '1828' is a coincidence that makes it easy to memorise
- •First appeared in 1683 in Jacob Bernoulli's work on continuously compounded interest
- •The letter was used for it by Leonhard Euler in a 1731 letter to Goldbach and fixed by his Introductio in analysin infinitorum (1748)
- •Euler proved it irrational in 1737 by exhibiting its continued fraction, which has the pattern 2; 1, 2, 1, 1, 4, 1, 1, 6, 1 …
- •Charles Hermite proved it transcendental in 1873; Lindemann adapted the method for the circle constant in 1882
- •It is sometimes called Napier's constant after John Napier, who published logarithm tables in 1614, though Napier did not use this base
- •The 'e-Day' analogue of Pi Day is 7 February in British date order or 27 January in American
- •In the optimal-stopping or 'secretary' problem, the best strategy examines and rejects the first 1/e — about 37% — of the candidates
- •Google's 2004 IPO was for $2,718,281,828, and an earlier recruitment billboard asked for the first ten-digit prime in the constant's digits
- •Raised to the power of the circle constant times the imaginary unit, it gives −1 — the celebrated identity
Drill it — 7 cards
1 / 7new card
Sources
- MacTutor History of Mathematics Archive, University of St Andrews — The number e
- MacTutor History of Mathematics Archive, University of St Andrews — Leonhard Euler