Squeezing interest to a limit
Put $1 in an account paying 100% interest per year. Paid once at year's end you get $2. But suppose it is paid in two half-year installments of 50% each: you get 1·(1.5) = 1.5 at the half-year, then 1.5·(1.5) = 2.25 at the end — more, because the first half-year's interest *itself* earned interest. Pay it monthly, daily, every instant, and the year-end total keeps creeping up — but not without bound. It closes in on a single number.
Value after one year = (1 + 1/n)^n for n payments n = 1 (1 + 1/1)^1 = 2.00000 n = 2 (1 + 1/2)^2 = 2.25000 n = 4 (1 + 1/4)^4 = 2.44141 n = 12 (1 + 1/12)^12 = 2.61304 n = 365 (1 + 1/365)^365 = 2.71457 n = 1000000 (...)^1000000 = 2.71828... As n grows, the total settles on e = 2.718281828...
That limiting number is e = 2.718281828… It is an irrational number — its decimal never repeats — and in fact a transcendental number, meaning no polynomial equation with whole-number coefficients has it as a root. We carry it as the symbol e, just as we carry π, and reach for a calculator only at the last step.
Why this base is called natural
The natural exponential function is f(x) = e^x. It earns the name *natural* because it describes any quantity whose rate of growth is, at every instant, proportional to how much is already there — a savings account paid continuously, a colony of cells, a hot cup of coffee cooling toward room temperature. Anything that grows on what it has, growing without pause, is governed by e.
For continuous growth at rate r over time t, the model is A = A₀·e^(rt), where A₀ is the starting amount. A positive r grows; a negative r decays. Because e^x is itself an exponential function with base e > 1, its graph has the same shape as the curves of the last guide: always positive, hugging the x-axis on the left, rising on the right, and passing through (0, 1).