The Rule of 70 and Why Nothing Grows Exponentially for Long

The companion calculator converts a growth-rate constant into a doubling time. That conversion is one of the most portable ideas in all of quantitative science, the same math describes multiplying bacteria, compounding interest, and spreading epidemics, and it comes with a famous mental shortcut and an equally famous warning.

Doubling Time: One Number, Many Domains

Any quantity that grows by a constant percentage per period has a doubling time, a fixed interval in which it doubles, no matter how large it has already become. That last part is the surprising bit: a bacterial culture doubles every 20 minutes whether it has ten cells or ten million, and a savings account at a fixed rate doubles in the same number of years whether it holds a hundred or a hundred thousand. Doubling time captures the tempo of exponential growth in a single, intuitive figure, which is exactly why it is preferred over the abstract growth constant.

The Rule of 70: Doing It in Your Head

You rarely need a calculator for a rough answer, because of a handy approximation: divide 70 by the percentage growth rate per period and you get the approximate doubling time.

The 70 comes from the natural logarithm of 2 (about 0.693) expressed as a percentage. Some people use 72 instead because it divides evenly by more numbers, handy for mental finance math. Either way, it turns doubling time into arithmetic you can do while listening to someone quote a growth rate.

Why Our Intuition Fails Us

Humans are notoriously bad at feeling exponential growth, because it stays deceptively small and then erupts. The classic parable is the pond where lily pads double daily and cover the whole pond on day 30, the pond is only half covered on day 29, giving one day's warning after a month of apparent calm. The old chessboard story, one grain on the first square doubling to the last, ends in more grain than exists on Earth. Both dramatize the same trap: constant doubling looks gentle for a long time, then overwhelms almost instantly.

The Warning: Exponentials Always End

The deepest lesson is that sustained exponential growth is physically impossible in the real world. The E. coli that double every 20 minutes would, in a couple of days of unchecked growth, theoretically outweigh the planet, which obviously never happens. Something always intervenes, food runs out, space fills, resources cap the climb, and the curve bends into the S-shaped logistic form. Exponential growth is only ever the early phase of a process that will be limited: the log phase of a culture, the opening days of an outbreak, the first stretch of a new market. The doubling time is real, but the doubling cannot continue forever.

Using Doubling Time Well

Take this calculator's doubling time as an exact translation of a growth constant into an intuitive interval, and lean on the rule of 70 for quick mental estimates from any percentage rate. Just carry the caveat that a constant doubling time describes only the unrestricted phase of growth, sooner or later, resource limits bend every exponential into a curve that levels off.