the number e

Imagine a bank that pays 100% interest a year, but you get to compound it more and more often — yearly, monthly, daily, every instant. As you compound infinitely often, one dollar does not grow without bound; it settles on a specific number, about 2.71828. That special number is called e, the natural base for growth.

More precisely, e is the value that (1 + 1/n)^n approaches as n grows without limit. It is an irrational number (its decimal never ends or repeats) and in fact a transcendental number (it is not a root of any polynomial with integer coefficients). Like π, e is one of the truly fundamental constants of mathematics, and it appears far beyond finance.

The reason e earns the title natural is a calculus fact: the function e^x is the one exponential whose rate of change at every point equals its own height. That self-matching property makes e the most convenient base for describing any process of continuous growth or decay, which is why it shows up throughout science.