The Number e — Grow Money and Meet the Constant of Change

Alongside pi there is a second great constant, e = 2.71828..., first glimpsed by Jacob Bernoulli in 1683 studying compound interest. It is the number of continuous growth. You will grow it with counters.

Start with 1 counter -- one coin earning 100% interest in a year. Paid ONCE at the year's end it becomes 2 (double). Now pay it as 50% TWICE: after the first half-year you have 1.5, and 50% of that added gives 2.25 -- more! Lay out the counters and work it through.

Pay the interest monthly (12 small additions) and you reach about 2.61; daily gives 2.714; every second, almost 2.71828. The total does NOT run away to infinity -- it settles onto e = 2.71828. That settling point is what 'continuous growth' means. Record each result and watch it close in.

What your growing pile teaches. (1) Splitting growth into ever-smaller steps does not give ever-more money; it converges on e. (2) A far faster route to e is the endless sum 1 + 1/1! + 1/2! + 1/3! + ..., which nails it in a handful of terms. (3) e appears wherever things grow or fade smoothly: populations, radioactive decay, a cooling cup of coffee, charging batteries, continuously compounded money. (4) Its function e-to-the-x is the one curve that is its own rate of change, which makes it the natural language of calculus -- and in Euler's identity it binds e, pi, i, 1 and 0 in a single line often called the most beautiful in mathematics.