Generating Intuitions for Exponential Growth
For some rates, this works really well. At 2% annual growth, the rule gives 35 years, and the actual value is 35.003 years. Other times it fails horribly. At 70% growth, the rule predicts doubling in one time step, but it actually takes 1.3.
How does the heuristic perform in general? Not that well. It’s accurate at 2% growth, but then quickly converges to being off by 0.3 timesteps.
So why was the rule ever popular? An early version is attributed to 15th century Italian accountant Luca Pacioli. Coincidentally the same guy who failed to teach Leonardo DaVinci math. In Summa de arithmetica, he writes:
In wanting to know of any capital, at a given yearly percentage, in how many years it will double adding the interest to the capital, keep as a rule [the number] 72 in mind, which you will always divide by the interest, and what results, in that many years it will be doubled. Example: When the interest is 6 percent per year, I say that one divides 72 by 6; 12 results, and in 12 years the capital will be doubled.
For 6 percent, the error is only 0.1, which is not yet too bad. In general, it’s helpful to think of the Rule of 72 as a heuristic that works decently for a certain range of values.
