The Rule of 72 says that if you divide 72 by an annual rate of return, you get roughly the number of years it takes for money to double. It is the most durable piece of mental arithmetic in personal finance — and it is far older than most people using it realise. The first known written record of it is more than five hundred years old, and it was set down by a Franciscan friar who never explained where it came from.
Stranger still: the rule is older than the mathematics that justifies it. By the time anyone could prove why 72 works, people had been using it for 120 years.
The First Written Record: Venice, 1494
In 1494 the Italian mathematician Luca Pacioli published Summa de arithmetica, geometria, proportioni et proportionalita in Venice — a sweeping summary of the mathematics of his day, written as a textbook. Tucked inside it is the earliest known reference to the Rule of 72.
What makes that reference remarkable is what is missing from it. Pacioli states the rule and does not derive it or explain it. He simply uses it. Historians take that as evidence the rule was already in circulation — you do not stop to prove something your readers already treat as common knowledge. So 1494 is not when the Rule of 72 was invented. It is only the first time somebody bothered to write it down.
- Before 1494Somebody is already using the shortcut. We do not know who. Pacioli writes it down without explaining it, the way you would quote a fact everyone in the room already knows.
- 1494Luca Pacioli publishes Summa de arithmetica in Venice. It contains the first known written reference to the Rule of 72 — and no derivation.
- 1497Pacioli moves to Milan at Duke Ludovico Sforza’s invitation, where he teaches mathematics to Leonardo da Vinci, and the two live and work together.
- 1614John Napier publishes his work on logarithms — Mirifici Logarithmorum Canonis Descriptio. The mathematics that explains the rule finally exists, 120 years after the rule was written down.
- TodayThe rule is unchanged. Nothing about arithmetic, inflation or markets has made 72 ÷ rate any less useful than it was in Venice.
Who Was Luca Pacioli?
Pacioli (born around 1447, died 19 June 1517) became a Franciscan friar in the early 1470s and spent his life teaching mathematics. He is remembered today less for the Rule of 72 than for something else in that same 1494 book: he was the first person to publish a work describing double-entry book-keeping on the continent, which is why he is often called the father of accounting.
In 1497 he accepted an invitation from Duke Ludovico Sforza to work in Milan. There he met Leonardo da Vinci, taught him mathematics, collaborated with him and lived with him — an association that lasted until the two fled Milan around 1506. The man who wrote down the Rule of 72 was, quite literally, Leonardo’s maths tutor.
The Rule Is Older Than the Logarithm
Here is the part that should give you pause. The Rule of 72 is an approximation of a logarithm. Doubling time comes from the natural log of 2, which is about 0.693147; for interest compounding continuously, the exact answer is 69.3147 divided by the rate. That is where the number in the rule comes from.
But logarithms were not published until 1614, when John Napier (1550–1617) brought out Mirifici Logarithmorum Canonis Descriptio. Pacioli died in 1517, almost a century before that book existed. He could not have derived the Rule of 72 even if he had wanted to.
So the rule was not discovered by working out the mathematics and rounding the answer. It almost certainly went the other way round: merchants and lenders noticed a pattern in numbers they used every day, the pattern held, and it was passed along as a piece of practical craft. The proof arrived 120 years late and simply confirmed what everybody had already found by hand.
Why 72, and Not 69.3147?
If 69.3147 is the mathematically exact figure for continuous compounding, why has 72 survived instead? Because the rule exists to be done in your head, and 72 is built for that.
That divisibility is the whole trick. At 6% a year, 72 ÷ 6 = 12 years, and you can do it while somebody is still talking. The exact answer is 11.9 years — close enough that the difference would not change a single decision. At 8%, 9% and 12% the division is just as clean, and those happen to be the rates people actually argue about.
The Rule’s Cousins: 70, 69.3 and 78
Because 72 is a compromise, other versions exist for other situations, and each is genuinely better in its own range:
- The Rule of 70 is more accurate at low rates. At 2% a year, 70 ÷ 2 = 35 years, and the exact answer is 35. The Rule of 72 would have said 36.
- The Rule of 69.3 is the exact one for continuous compounding — mathematically right, and almost useless without a calculator.
- The Rule of 78 is closer at high rates. At 20%, 78 ÷ 20 = 3.9 years against an exact 3.8, while the Rule of 72 gives 3.6.
None of them displaced 72, and the reason is not mathematical. 72 is the one that sits in the middle of the rates ordinary savers and investors actually meet, while dividing cleanly enough to be done without writing anything down.
What Five Centuries Have Not Changed
Almost every piece of financial guidance from 1494 is now useless. Currencies have gone, instruments have been invented, whole markets have appeared. The Rule of 72 survived because it is not about money at all — it is about the arithmetic of anything that grows by a fixed percentage. That is why the same rule is used to estimate how fast prices double under inflation, or a population, or a number of subscribers.
It survived for a second reason too. The rule is designed for a person with no tools. Every technology since Pacioli — logarithm tables, slide rules, calculators, spreadsheets, phones — has made the exact answer easier to get, and none of them has made a shortcut you can do in a conversation any less useful.
Frequently Asked Questions
Who invented the Rule of 72?
Nobody knows. Luca Pacioli recorded it in 1494, which is the earliest known written reference, but he did not claim it and did not explain it — strong evidence that it was already in use before him. There is no named inventor.
Is the Rule of 72 still accurate?
It is as accurate as it ever was, because the arithmetic has not changed. It is closest between roughly 6% and 10% a year, runs slightly high below that band and slightly low above it. At 6% it says 12 years where the true answer is 11.9.
Did Pacioli invent double-entry book-keeping as well?
No — he was the first to publish a description of it on the continent, not its inventor. The method was already being used by Italian merchants. It is the same pattern as the Rule of 72: Pacioli was the person who wrote down what practitioners were already doing.
The Bottom Line
The Rule of 72 is a piece of practical knowledge that outlived the world it came from. It was in use before 1494, written down by a friar who taught Leonardo da Vinci, and only explained 120 years later by mathematics that had not been invented when he wrote it. It has needed no revision since. If you want to see it work, the Rule of 72 explained, with a calculator and a doubling table, sets the shortcut beside the exact answer at every rate.
Further Reading
- The Rule of 72: How to Estimate When Your Money Will Double
- What Is Compound Interest?
- Simple Interest Explained
- What Is Inflation?
- Investing Overview
This article is for educational and informational purposes only and is not investment advice. The Rule of 72 is an approximation; actual returns vary and are not guaranteed.