The financial application of 72 dates back to the Renaissance. Franciscan friar and mathematician Luca Pacioli documented the technique in his 1494 masterwork, Summa de arithmetica, geometria, proportioni et proportionalita. Pacioli noted that an investor seeking to estimate how quickly capital doubles at compound interest could divide the number 72 by the annual percentage yield.
The underlying science hinges on exponential growth and natural logarithms. To solve for doubling time ($t$) at an interest rate ($r$):
$$(1 + r)^t = 2$$
Taking the natural logarithm of both sides isolates the time component:
$$t \cdot \ln(1 + r) = \ln(2)$$
$$t = \frac{\ln(2)}{\ln(1 + r)}$$
The natural logarithm of 2 equals roughly 0.693147. For continuous compounding at decimal rates, the theoretical numerator should be 69.3. Yet for annual discrete compounding in everyday commerce, 69.3 proves inconvenient for mental math. Raising the numerator to 72 compensates for discrete compounding intervals while offering divisibility by almost every common single-digit interest rate.
