Continuous compounding converges on one mysterious number
The constant e, roughly 2.71828, underpins natural logarithms and the exponential function that equals its own derivative. Jacob Bernoulli discovered it in 1683 while studying interest credited ever more frequently. It is irrational and transcendental, yet appears in growth, probability, and statistics.
The number e anchors the natural logarithm and the exponential function, approximately 2.71828. Like π, it is irrational and transcendental—no ratio of integers, no root of a nonzero polynomial with rational coefficients. It ranks with 0, 1, π, and i in Euler's identity e^(iπ) + 1 = 0.
Jacob Bernoulli introduced e in 1683 solving continuous compounding. An account at $1 with 100% annual interest yields $2 if credited once yearly, $2.25 if credited twice, $2.613 if monthly. As compounding intervals n grow, (1 + 1/n)ⁿ approaches e ≈ 2.718281828. More generally, e^(Rt) dollars accumulate after t years at rate R with continuous compounding.
Leonhard Euler used the letter e from 1727 or 1728; its first printed appearance was in Mechanica (1736). Euler proved e equals the infinite series 1 + 1/1! + 1/2! + 1/3! + .... The function y = eˣ is the unique function equal to its derivative with value 1 at x = 0.
Beyond finance, e surfaces in probability. A gambler playing n times with win probability 1/n loses all n bets with probability approaching 1/e ≈ 36.79%. Exponential growth x(t) = x₀e^(kt) models quantities whose rate of change is proportional to size. The standard normal distribution's density includes e^(−x²/2). John Napier's 1618 logarithm tables touched the constant before Bernoulli named its role in compounding. Gottfried Leibniz briefly used the letter b for the constant in letters to Huygens around 1690.
Source: E (mathematical constant)
