a shortcut on trial

The Wrong Constant That Keeps Being Right

Everyone who digs one layer down learns the smart-sounding correction: doubling time is really ln 2 ≈ 69.3 divided by the rate, and 72 was just picked because it divides nicely. But the correction is the misconception: 69.3 is exact only if interest compounds continuously, and in the annual world where real accounts live, doubling time runs ≈ 69.3/R + 0.35, a line that 72/R rides almost perfectly. Drag a rate slider and watch exact, 72/R, and 69.3/R recompute with signed errors: at 8%, 72 is off by 0.07% while the 'true' 69.3 misses by 3.8%. Flip the compounding toggle to continuous and the winner flips with it.

The exact annual-compounding curve is t = ln(2) / ln(1 + R/100). Expand its denominator near zero and the first two terms are 69.3147/R + 0.3466. That small added shelf is why a slightly larger numerator can win. The neat integer is not the derivation, but in the practical 4% to 12% band it shadows the corrected curve remarkably well.

The regime is the trick

Flip the bench to continuous compounding. Now the exact curve is t = ln(2)/(R/100), so 69.3/R is nearly exact at every rate and 72/R is about 3.87% high. The familiar shortcut is not a universal law. It is a well-placed approximation for periodic compounding.

At 18% annual compounding, a balance with no payments or new charges doubles in 4.1878 years; 72/18 says 4.0000, understating the wait by 4.49%. This is arithmetic, not a forecast or financial advice. At a constant 3% annual loss of purchasing power, the exact halving time is 22.7566 years, not 23.4. Decline follows ln(0.5)/ln(1 − r), so it must not borrow the growth error table.

There are not two regimes. There is one curve.

The bench has two buttons because that is how the shortcut is taught: annual, or continuous, pick a side. But those are not two regimes with two constants. They are two points on a single curve, and the curve turns out to have only one argument, which is not the number people think.

Write p for the interest rate per compounding period: the quoted annual rate divided by the number of times a year it actually compounds. A balance grows by a factor of (1+p) each period, so after t years and m periods a year it has grown by (1+p)^mt. Set that equal to 2 and solve. Then ask what numerator D would make the shortcut t ≈ D/R exactly right, where R = 100 m p is the rate as a bank quotes it:

The m cancels. The ideal numerator does not depend on the quoted rate and the compounding frequency separately. It depends only on their quotient, the rate per period. Continuous compounding is not a rival convention with a rival constant; it is simply the limit p → 0, where D → 100 ln 2 = 69.3147180560.

Which means these six accounts, which no one would call the same account, all have exactly the same best numerator:

A card charging 24% a year compounded monthly and a savings account paying 2% a year compounded once want the same rule of thumb. Not a similar one. The same one, to the last digit either engine can print. That is the fact the two-button framing hides.

There is a second cancellation, and it is what makes the rest of this page finite. If you use numerator N instead of the ideal D, your answer is N/R instead of D/R, so your relative error is

The rate cancels out of the error too. So there is no per-rate tuning to do and no trade-off between big rates and small ones: the best whole-number numerator is simply the whole number nearest D(p), and the boundary between N and N+1 sits exactly where D(p) = N + ½. Everything below is that one equation, solved.

The numerators nobody proposed

Now solve D(p) = N + ½ for every N in turn and you get the complete map of which whole number is best, and where. For a growing quantity it looks like this.

Eleven numerators, each owning about three percentage points, and the ones the world kept are not the ones that win. There has never been a rule of 71. There is no rule of 73, no rule of 74, no rule of 77. Three of those four are prime.

71 is the most accurate whole number for every per-period rate between 3.439385% and 6.370968%, and nobody has ever proposed it, because you cannot divide by 71 in your head.

That band is not an exotic corner of the rate axis. It is the middle of it. So the numbers folklore preserved were not selected for accuracy. Something else did the selecting, and it is visible the moment you make the constraint explicit.

The criterion was never accuracy

The problem a person actually has at a dinner table is not "minimise relative error". It is "give me a whole number of years without a pencil". That second problem carries a hard constraint the first one does not: the numerator has to be divisible by the rate. Which turns the whole question into a small, completely finite contest. For each whole-percentage rate, list the numerators that divide it evenly, and see which of those lands closest to the ideal.

Read the "best" column down and the folklore falls out of it. 72 wins at 4%, 6%, 8%, 9%, 12%, 18% and 24%. Seven of the first twenty-five whole rates, more than any other number takes. 70 wins at 1%, 2%, 5%, 7%, 10% and 14%. Six. Every other winner in the table takes one rate or two, and then never again.

There are two rules of thumb in the world rather than one because this contest has two winners. There are not ten rules because nothing else wins more than twice.

The two of them between them cover thirteen of the first twenty-five whole rates. And they are not interchangeable: turn the divisibility constraint on and each one owns a contiguous stretch of the rate axis, with a single boundary between them.

So the honest statement of the rule of 72 is not "72 is close to 69.3". It is this: 72 is the most accurate numerator you can actually divide by, for any quantity compounding between about 4.9% and about 12.3% per period. And the honest statement of the rule of 70 is the same sentence with the band below that boundary. They were never rivals. They are two adjacent tiles of one map, and each field kept the tile its own rates land on.

The other branch, where 72 is never right

Everything so far was a quantity going up. Run it the other way, for a quantity losing p of itself each period, and one sign flips in the derivation:

And because −ln(1−p) is larger than p where ln(1+p) is smaller, the numerator now falls below 69.3147 instead of climbing above it. The two branches leave the same point in opposite directions. Growth walks up towards 72 and past it. Decline walks down, and never comes back.

72 never appears, and it cannot: the decline numerator is below 69.3147 at every positive rate, so the nearest whole number to it is at most 69.

That is the unconditional half, and it holds whatever you allow. The list above is the conditional half, and it is worth being exact about what it depends on. Those five numerators are the best ones admissible under the threshold this page has been using, which is that a numerator must divide at least five of the whole rates from 1% to 30%. Push the threshold higher in the instrument above, up to ten, and only 60, 72, 84 and 90 survive it. 72 then reappears on the shrinking row, not because it has become right but because everything nearer has been ruled out. That is a real consequence of a free choice, and the instrument is set up to let you see it rather than to hide it.

At a steady 10% a year the ideal numerator for halving is 65.7881347896. The rule of 72 overstates how long your money keeps half its value by 9.44%; even the rule of 70 is 6.40% long. This is the quiet failure mode of a rule learned on a savings account and carried across to inflation: it errs in the same direction every time, always telling you that you have longer than you have, and getting worse exactly as the situation gets worse.

It is also, at last, why the two professions disagree. An economist reaches for 70 and an investor reaches for 72, and it is easy to read that as a difference of taste or of rigour. It is neither. Growth at eight per cent and decline at eight per cent want numerators on opposite sides of the constant, and the rates each field spends its life looking at land in different tiles of the map above.

Why 72 rather than 71, in a number

"It divides nicely" is the standard explanation and it is correct, but it is never stated as a quantity, which makes it sound like a shrug. Here is the quantity. Count how many of the whole rates from 1% to 30% each candidate divides evenly:

Which leaves the two criteria on the table at once, pulling in different directions, and the interesting question is whether anything is good at both. Put every whole number from 55 to 100 on two axes, worst-case error over a per-period band against divisibility, and ask which are not beaten on both at once.

Over 4% to 12% per period, nothing in the entire range 55 to 100 is both more accurate than 72 and more divisible than 72. Its nearest accuracy rivals, 71 and 73, are prime; its nearest divisibility rival, 60, is ten times less accurate. Over the tighter band 6% to 10%, 72's worst case is under one per cent. So 72 is not a compromise between two goods. In the band it is used in, it is the corner where both are already maximised, and there is nothing to trade.

Where real rates actually fall

Bands are abstract until you drop something into them. Every rate below is a published figure from the body that publishes it, on the date given, and the only thing this page does to it is divide it by its own compounding frequency and read off the map. The sources are listed under the table, and one of them is worth reading the fine print of before you look at the numbers.

Two of those rows land the opposite way round from the folklore, and they are the two people most often reach for a rule about.

The rule of 70 is the credit card rule, and it is very nearly exact. The rule of 72 is the long-run equity rule. It is the other way round from how they are usually taught.

The card is the cleanest case in the table. The Federal Reserve publishes 22.15%, and its own footnote says the figure is an annual percentage rate as specified by Regulation Z, which means it is a nominal annual rate: the number you get by taking the rate that actually compounds and multiplying it up. A card that compounds monthly is therefore charging 1.8458% per period, the ideal numerator there is 69.9525, and 70 divided by 22.15 gives 3.1603 years against a true 3.1581. Seven hundredths of one per cent. Whereas the rule people actually reach for in that situation, 72, gives 3.2506 years, and if you instead read 22.15% as though it were an effective annual rate the ideal numerator jumps to 76.7355 and 70 becomes 8.78% too small.

So the same published percentage supports three different right answers depending on a fact the publication does not print. That is not a flaw in the rule. It is the thing the rule is a function of, showing through.

Sources for the table, each read on 2026-08-23. FDIC, National Rates and Rate Caps, "Monthly Rate Cap Information as of August 17, 2026", savings 0.38%; the release says "rates paid" and states neither APR nor APY, so the yearly reading here is ours. Federal Reserve, Statement on Longer-Run Goals and Monetary Policy Strategy, "adopted effective January 24, 2012; as reaffirmed effective January 27, 2026", inflation at 2 percent measured by the annual change in the PCE price index. US Bureau of Labor Statistics, Consumer Price Index, July 2026, released 12 August 2026, "over the last 12 months, the all items index increased 3.4 percent before seasonal adjustment". Freddie Mac, Primary Mortgage Market Survey, "the 30-year fixed-rate mortgage averaged 6.65% as of August 20, 2026"; the survey does not call this an APR and states no compounding convention, so the monthly reading is our inference from ordinary US amortisation and is marked as such. Federal Reserve, G.19 Consumer Credit, released 7 August 2026, credit card plans, accounts assessed interest, 2026 Q2 preliminary, 22.15%; footnote 5 states "interest rates are annual percentage rates (APR) as specified by the Federal Reserve's Regulation Z", and the monthly compounding is our reading, not the release's. MSCI, MSCI World Index factsheet, gross returns in USD as of 31 July 2026, 9.02% annualised since 31 December 1987 and 13.29% annualised over ten years; annualised geometric returns are effective annual rates with one period a year by construction. Past returns are given here as arithmetic to place on a curve, and are not a forecast of anything.

1494, and the clause the retellings drop

The rule is usually traced to Luca Pacioli's Summa de arithmetica, geometria, proportioni et proportionalità, printed in Venice in 1494, and usually traced there at second hand. The first cut of this page said so, cited an encyclopaedia for it, and stated plainly that no primary source had been reached. So the second cut went and got it.

It is on folio 181 recto, article 44, the last article on the page, in Distinctio nona, Tractatus quintus. The running head on that leaf is mis-set, reading "Distincto nona. Tracatus quintus" with a letter dropped from each of two words, and the same typo appears in every copy checked, which is how you know they are one setting of the type. Here is the passage as it is actually printed, with the compositor's abbreviations expanded in square brackets:

A uoler sape[re] ogni q[uantit]a a tãto p[er] c[ento] l[a]no i[n] q[ua]ti ãni sira to[r]nata dopia fra p[r]o e capitale, tien p[er] regola 72 a mête, q[ua]le sêpre p[ar]tirai p[er] lo i[n]teresse, e q[ue]llo ne uê i[n] tãti ãni serãno redopiati el capitale, a far capo al[l]ãno. Ex[emplu]m: q[ua]do lo i[n]teresse e 6 p[er] c[ento] l[a]no, dico ch[e] p[ar]ta 72 i[n] 6, ne uê 12, e i[n] 12 anni dirai che la ditta q[uantit]a prestata, cioe capitale sia q[ua]to si voglia, sira dopiato a 6 p[er] c[ento] a capo d[a]nno; e a 8 p[er] c[ento] p[ar]ti 72 in 8, ne uê 9, e in tanti si redopiara ditta quantita 7c[etc], in aliis.

Three things in that are not in the version that circulates.

First, there are two worked examples, not one. Six per cent, where 72 over 6 gives 12 years, and eight per cent, where 72 over 8 gives 9. Almost every retelling carries only the first. The second is the better one: at 8% the exact answer is 9.0065 years, so Pacioli's 9 is short by two and a third days, while at 6% the exact answer is 11.8957 years and his 12 is long by thirty-eight.

Second, and this is the one that matters, the passage says a far capo all'anno. Reckoning up at the end of the year. It is a stipulation of the compounding period, sitting in the middle of the sentence that states the rule, and the modernised Italian quotation that circulates online silently drops it. That is not a small editorial loss. Everything above this section is the demonstration that the compounding period is the only argument the rule has. The clause five centuries of retelling threw away is precisely the clause without which the rule cannot be applied correctly, and the modern habit of reaching for 72 whatever the account does is, exactly, the consequence of having thrown it away.

Third, his own two examples sit on either side of a boundary he had no way of seeing. At 8% the ideal numerator is 72.0517 and 72 is the nearest whole number, close to the middle of its band. At 6% the ideal numerator is 71.3740, and the nearest whole number is 71, not 72. Pacioli's first example is therefore an instance of the whole-quotient contest above: at 6% the most accurate numerator is unusable because 71 will not divide by 6, so the divisible one wins, and it costs 0.877%. He could not have known that, and he chose correctly anyway, which is what selection by use looks like from the inside.

The gap

John Napier published logarithms in 1614, one hundred and twenty years after 1494. Until then there was no way to compute ln 2 / ln(1+p) at all, so whoever first wrote 72 down could not have derived it, could not have checked it against 100 ln 2, and had no way of knowing that the constant it approximates is irrational. The rule is older than the mathematics that explains it by more than a century, and the reason it is any good is not that someone reasoned it out. It is that a numerator you can divide by gets used, a numerator you cannot does not, and being wrong by a few per cent over a ledger's worth of years is a thing a working arithmetic notices.

Where to see it, and what is ours. Two digitised copies of the 1494 first edition were read for this, and both are free to open. Smithsonian Libraries (Dibner), archive.org summadearithmeti00paci, leaf 381. Herzog August Bibliothek Wolfenbüttel, shelfmark 83-1-Quod-2°, image 00379. Note that the two scans use different leaf offsets for the same printed folio, 381 and 379, because their front matter differs; the offsets above were derived here by reading the printed folio numbers off the running heads, and are not documented at either source. A third 1494 copy, Universidad de Sevilla BUS Inc. 146, is at archive.org A335068, leaf 380, at a resolution too low to read the article with confidence. The catalogue record is ISTC il00315000, imprint "Venice: Paganinus de Paganinis, 10-20 November 1494".

One correction to the record, offered carefully. The folio number "181" is repeated all over the internet for the 1494 edition, generally without a citation. The best scholarly source that actually reproduces the page, C. G. Lewin, "The emergence of compound interest", British Actuarial Journal 24 (2019), e34, doi:10.1017/S1357321719000254, gives its folio for the 1523 Toscolano edition, not 1494: its reference entry reads "the rule of 72 appears on f 181 of the 1523 edition, from which Figure 3 is taken." That the two editions carry it on the same folio number is a coincidence that the literature does not establish, and it was checked here directly against the 1494 leaves linked above rather than inherited. Lewin also gives the range over which the rule is serviceable as "between 3% and 12%", which is a judgement about tolerable accuracy rather than about which numerator is best; on the stricter criterion used above, 72 is the best divisible numerator from 4.901790% to 12.314469% per period, and the upper ends agree closely.

What is still not known. Whether Pacioli devised the rule. He states it without derivation, which many sources read as evidence that it predates him, and that reading is an inference rather than a citation: no earlier text containing it has been identified here or, as far as this page can tell, anywhere. Lewin's survey runs from Babylon through Fibonacci, Chuquet and Widmann to 1620 and names no predecessor, which is the strongest negative available and is not the same thing as a negative result. A Babylonian tablet poses a doubling-time problem at 20%, but as a single solved instance, not as a general rule.

What changed in the second cut

This layer was laid on 2026-07-23 and re-cut on 2026-08-23. The first cut stands unaltered above: its bench, its numbers and its argument are as they were. What the second cut adds, and one thing it corrects:

• Added: the collapse. The ideal numerator is a function of the rate per compounding period alone. The first cut framed annual and continuous as two regimes; they are two points on one curve, and so is every other compounding convention.
• Added: the complete map. Which whole number is most accurate, over which exact stretch of rates, for growth and for decline, with and without a divisibility constraint.
• Added: the criterion. The whole-quotient contest, in which 70 and 72 are the top two winners and nothing else wins more than twice, which is the first account this page can give of why exactly two rules survived.
• Added: the decline branch. 72 is never the best admissible numerator for a shrinking quantity, at any rate.
• Added: seven published rates, each from the body that publishes it and each placed on the curve, which is where the rule of 70 turns out to be the credit-card rule and the rule of 72 the long-run equity one.
• Added, and this is the one that took the longest: the primary source. The first cut said in as many words that it could not verify one. Folio 181 recto of the 1494 Summa has now been read directly in two digitised copies, both linked, with the leaf offsets worked out here because neither library documents them. It carries two worked examples rather than the one usually quoted, and it carries a clause about the compounding period that the modern quotation drops.
• Added: a second engine, in another language at 60 digits, and a comparison that fails if the two disagree.
• Corrected in emphasis: the first cut said 72 "was just picked because it divides nicely" is a misconception. That is right about the accuracy claim and wrong about the selection. Divisibility really is what picked the survivors; the correction is that this does not make 72 inaccurate, because in the band where it is used, divisibility and accuracy pick the same number.