Compound Interest: The Formula, Rule of 72, and Historical Origins
Compound interest A = P(1+r/n)^(nt). Rule of 72 estimates doubling time (from 1494). Time is exponentially more powerful than rate. The constant e was discovered by Bernoulli in 1683 through studying this formula.
Compound interest formula: A = P(1 + r/n)^(nt) Where A = final amount, P = principal, r = annual rate, n = compounding frequency, t = years. Rule of 72: Divide 72 by the annual interest rate to estimate doubling time. At 8%: 72 ÷ 8 = ~9 years to double. This approximation appeared in Luca Pacioli's Summa de arithmetica (1494). Why time dominates: The exponent (nt) means time has an exponential effect on growth. Doubling the time period has a dramatically larger impact than doubling the interest rate. This is the mathematical basis for the investment advice to "start early." Historical origins: - Earliest known compound interest calculation: Babylonian clay tablet, circa 2000-1700 BC - Roman law condemned compound interest as "the worst kind of usury" (anatocism) - Jacob Bernoulli discovered the mathematical constant e (≈2.71828) in 1683 specifically by studying compound interest — asking what happens as compounding becomes continuous - Richard Witt's Arithmeticall Questions (1613) was the first book entirely devoted to compound interest Continuous compounding formula: A = Pe^(rt) — this uses Bernoulli's constant e and represents the theoretical maximum compound growth.