Math intuitions
The rule of 72
Divide 72 by the annual growth rate in percent to get the number of years a quantity takes to double.
Stated without proof in Luca Pacioli's Summa de arithmetica (Venice, 1494); the logarithmic justification came after Napier.
The rule says that a quantity growing at r percent a year doubles in about 72 divided by r years. At 6 percent that is 12 years; at 9 percent, 8 years; at 12 percent, 6. It runs backwards too: a city that doubled in 24 years was growing at about 3 percent a year. The oldest known statement is in Luca Pacioli's Summa de arithmetica of 1494, where he tells the reader who wants to know in how many years a sum at a given percentage will have doubled, capital and interest together, to hold the number 72 in mind and divide it by the rate. He gives no derivation. Giovanni Vacca, who drew attention to the passage in 1915, noted that the reason behind it stayed a mystery to the arithmeticians who followed, Tartaglia among them.
The derivation is one line. If a quantity multiplies by (1 + r) each year, it doubles when (1 + r) to the power t equals 2, so t equals ln 2 divided by ln(1 + r). For small r, ln(1 + r) is close to r, and ln 2 is 0.6931, so t is about 69.3 divided by r in percent; that is exact for continuous compounding. With annual compounding the true answer is slightly longer, because ln(1 + r) is a little less than r, and 72 matches it almost exactly near 8 percent: exact doubling at 8 percent takes 9.01 years, and 72 over 8 is 9. At 2 percent the exact figure is 35.0 years against the rule's 36; at 6 percent, 11.9 against 12; at 12 percent, 6.12 against 6; at 18 percent, 4.19 against 4. And 72 divides neatly by 2, 3, 4, 6, 8, 9 and 12, which 69 does not.
The rule assumes the rate holds, and that is the assumption that fails most often: a startup growing 20 percent a month doubles every 3.6 months, so three years of it would be ten doublings, a factor of 1,024, which almost never happens. It works on anything that changes by a fixed fraction, including shrinkage. Inflation of 3 percent halves money's purchasing power in about 24 years; a 2 percent annual fee halves a fund's eventual value, relative to a fee-free one, in about 36 years. At high rates use the exact formula: at 72 percent the rule says one year and the true answer is 1.28 years.
Example
In the Summa de arithmetica, geometria, proportioni et proportionalità, printed in Venice in 1494, Luca Pacioli set the problem of how many years it takes for a sum lent at some percentage a year to come back doubled, capital and interest together. His answer was to keep the number 72 in mind and divide it by the rate, with no explanation of why. In 1915 Giovanni Vacca, writing in the Napier Memorial Volume of the Royal Society of Edinburgh, pointed out that the correct constant is 100 times the natural logarithm of 2, which is 69.31, and asked whether this merchant's rule, in print 120 years before Napier published logarithms, might have been one of the suggestions that led him there.
Where it applies
Use it to sanity-check any growth claim in seconds: a database growing 10 percent a month doubles every seven months, so a disk that is half full today is full in about seven months. For a cloud bill or a model's training compute, ask for the growth rate and convert it to a doubling time before deciding whether the trend is a problem.
Think about it
Take one number in your life that is growing at a known rate: how many years until it doubles, and does that date worry you?
Applied it to something real? Log it.
Test yourselfA fund returns 9 percent a year. Roughly how long until the money doubles, and where does the 72 come from?
About 8 years (72 divided by 9). Doubling takes ln 2 over ln(1 + r) years, roughly 69.3 divided by the rate in percent; 72 is used because it is close and divides neatly.
Related models
- CompoundingReturns earned on past returns: a steady rate applied to a growing base turns small repeated gains into large ones, mostly at the end.
- Exponential versus linear growthLinear growth adds the same amount each step; exponential growth multiplies by the same factor, and the two look alike only at the start.
- Orders of magnitudeEstimate to the nearest power of ten from a few facts you know; being within a factor of three is often enough to decide.