How did a question about compound interest reveal the number e?

By asking what happens when interest is added more and more often. In 1683, Jacob Bernoulli studied a simple puzzle: an account starts with $1 and pays 100 percent interest a year. Credited once, at the end of the year, it is worth $2. But compound interest pays interest on interest, so crediting it in smaller, more frequent slices does better: compounded weekly, the dollar grows to about $2.69, and compounded daily, to about $2.71. Bernoulli noticed that the result approaches a limit as the intervals shrink. That limit, reached with continuous compounding, is 2.718281828…, the number now called e.

Compound interest is interest paid on interest: once a bank or investment adds interest to your balance, the next round of interest is calculated on that larger amount, not just on what you started with. Put $1,000 in an account paying 5% a year and after one year you have $1,050; after two years you have $1,102.50, not $1,100, because the second year's interest was earned on the new, bigger balance.

The idea itself is ancient. A Babylonian clay tablet from roughly 2000 to 1700 BC may contain the first recorded compound interest problem, though it took until the medieval period for mathematicians to study it seriously. For centuries it had a terrible reputation: compound interest charged by lenders was once considered the worst kind of usury, and Roman law, along with the common laws of many other countries, condemned it severely. Still, merchants needed it: around 1340, the Florentine merchant Pegolotti published a table of compound interest rates, and in 1494 Luca Pacioli's math textbook gave the famous 'Rule of 72,' a shortcut for estimating how many years it takes an investment to double just by dividing 72 by the interest rate. In 1613, Richard Witt wrote the first book devoted entirely to the subject, packed with 124 worked examples.

Today, the math hasn't changed, but the paperwork has: many countries now legally require banks to disclose a single, comparable annual compound interest rate, so ordinary savers can tell which account is actually offering the better deal.