Math intuitions
Exponential versus linear growth
Linear growth adds the same amount each step; exponential growth multiplies by the same factor, and the two look alike only at the start.
The arithmetic is ancient. Gordon Moore's 1965 observation that the components on a chip doubled every year is the best documented modern case.
A quantity grows linearly when it gains the same amount each period and exponentially when it gains the same fraction. Start with 100 and add 10 a year: after 30 years you have 400. Start with 100 and add 10 percent a year: after 30 years you have 100 times 1.1 to the power 30, which is 1,745. For the first few years the two lines sit almost on top of each other, which is why the exponential one gets mistaken for a straight line. Gordon Moore saw this in 1965. The number of components on the cheapest integrated circuit had risen by roughly a factor of two per year, and the cheapest chip of his day held about 50. Ten more doublings multiply by 1,024, and he wrote that by 1975 the figure would be 65,000. Adding 50 a year would have given 550.
Exponential growth appears wherever the rate of increase depends on the amount already there: interest on a balance, cells dividing, an infection spreading, each chip generation paying for the tools that make the next. The useful handle is the doubling time. Something that doubles every two years is 2 to the power 10, about 1,000 times bigger, after 20 years and about a million times bigger after 40. Our World in Data's series of transistors per microprocessor runs from 2,308 in 1971 to 58.2 billion in 2021. That ratio is about 25 million, which is 2 to the power 24.6, so the count doubled every 2.0 years for fifty years. A linear extrapolation from 1971, adding 2,308 transistors a year, would have reached 118,000 by 2021 and missed by a factor of nearly half a million.
Nothing grows exponentially for ever. Growth that feeds on a finite resource, whether atoms on a wafer, susceptible people or customers, bends into an S-curve as the resource runs out, and the hard part is telling where you are on the curve. Fitting an exponential to the early part of an S-curve is how forecasts of epidemics and startups both go wrong. The physicist Al Bartlett spent decades telling audiences that the greatest shortcoming of the human race is its inability to understand the exponential function, and the failure runs both ways: people underestimate compounding growth, and they also extrapolate it past the point where the mechanism stops. The question to ask is what the growth feeds on and when that runs out.
Example
In the 19 April 1965 issue of Electronics, Gordon Moore of Fairchild Semiconductor noted that the number of components on the lowest-cost integrated circuit had increased at roughly a factor of two per year, and that the minimum cost in 1965 was reached at about 50 components per circuit. Seeing no reason the rate would not hold for at least ten years, he wrote that by 1975 the number of components per circuit for minimum cost would be 65,000. Our World in Data's series of transistors per microprocessor shows the pattern outlasting his horizon by decades: 2,308 transistors on a chip in 1971 and 58.2 billion in 2021, a doubling about every two years for half a century.
Where it applies
Any metric with a steady doubling time (requests per second, training compute, data volume, a backlog that breeds more backlog) will cross whatever fixed limit you have, and the crossing will look sudden. Plot it on a log scale: a straight line on a log axis is exponential growth, and a bend is the news.
Think about it
Which number in your work has been doubling at a steady rate, and what fixed limit does it hit if that continues for five more doublings?
Applied it to something real? Log it.
Test yourselfA quantity grows 10 percent a year from 100. Roughly where is it after 30 years, and why does a linear guess miss?
About 1,745 (100 × 1.1^30), against 400 if you add 10 a year. Each year's gain is a fraction of a larger base, so the gap widens every period.
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.
- The rule of 72Divide 72 by the annual growth rate in percent to get the number of years a quantity takes to double.
- NonlinearityCause and effect are not proportional: a push can do nothing or everything depending on where the system sits relative to a threshold.
- Feedback loopsA system's output feeds back into its input: balancing loops hold a level steady, reinforcing loops make change grow on itself.