e is the unique number where the derivative of eˣ equals eˣ itself: the function that is its own derivative. This makes e the natural base for exponential growth and decay, appearing wherever quantities change at a rate proportional to their current value.
e is the limit of (1 + 1/n)ⁿ as n approaches infinity, discovered by Jacob Bernoulli while studying compound interest in 1683. Euler introduced the notation e in 1731. It is irrational (proved by Euler) and transcendental (Hermite, 1873).
e appears in probability (the normal distribution), statistics, finance (continuous compounding), physics (radioactive decay, the harmonic oscillator), and almost every area of analysis.