Exponential

I hear talk all the time about exponential increases. Consider three very recent examples from the New York Times, here:

“Since the 1960s there has been an exponential increase in artists working with maps,” she adds, using Jasper Johns’s “Map,” from 1963, by way of example.

here:

[...] historically black colleges and universities actually serve more students today than they did 50 years ago because there has been an exponential growth in students pursuing higher education [...]

and here:

But he said first responders generally did not have enough training to deal with diversions that could be “almost exponential” compared with those faced by most drivers.

I’ll note that two of those are quoting someone, and the third is in a blog, for which the Times has slightly more relaxed standards concerning informal writing. But the point isn’t to pick on the Times, but to note how often people refer to “exponential” growth or increase.

Here’s the thing: “exponential” isn’t just a fancy or intensive way to say “large”. It has a particular meaning.

In grade school, we were given a problem to work out: Suppose your parents gave you a penny on the first day of January, two pennies on the second day, four on the third, eight on the fourth, and so on, doubling the number of pennies each day. On January 31st, how many pennies would you have, all told?

When you’re in the sixth grade, that’s not an easy problem. Of course, any computer geek can solve that instantly now: on day “n” you get 2^n-1 pennies, and you have a total of 2ⁿ-1 pennies collected. So your final total at the end of January is 2³¹-1, which is 2,147,483,647 — more than two billion pennies.

We were astounded by that answer, once we got it. And that is exponential growth.

Strictly speaking, exponential growth requires an interval, some repetitions of that interval, and growth over each of those intervals by a common factor. It can be a doubling every day for a month, as in the problem with the pennies. It might be a five-fold increase each year for several years. The point is that the thing being measured is increasing in the exponent, a quantity of n in the first interval, n² in the next, n³ in the next, and so on.

In fact, exponential growth doesn’t even have to imply fast or large growth. Note that if a population increases by 1% every decade... that’s exponential growth, though no one would really think of it that way. In this case, the interval is ten years, and the common factor is 1.01.

In practice, of course, it’s often not that straightforward, and we allow for variations and approximations. But we must insist on the interval thing, and of a pattern of applicable growth over some reasonable number of intervals (it won’t do to just say that something doubled, and to call that “exponential growth”; one interval is not enough to establish a pattern).

The first New York Times example, the quote about artists working with maps, has a starting point, but no interval. There may have been a large increase in the number of art works involving maps, but without more information we can’t call it “exponential”. The same is true for the blog entry about black university students. If we had data to back it up, it would be valid to say that over the last 50 years there’s been an exponential increase each decade. But as it stands, he just means that it’s gone up by a lot.

And I can’t even decide what the third example means, to say that “diversions” are “almost exponential”. I suppose we might say something like that if the number of diversions that first responders have to deal with is the square or the cube of those bedeviling others — for every four diversions you and I have, a first responder has to cope with sixteen, or maybe 64. But that seems awkward, as well as unlikely.

The guy just means that they have a great deal more diversions (I would say “distractions”) than the average driver.

When that’s what we mean, that’s what we should say.