“Be kind, for everyone you meet is fighting a hard battle” - Often attributed to Plato but likely from Ian McLaren (pseudonym of Reverend John Watson)

Sunday, March 29, 2009

"Exponentially" really does mean something

In my never ending quest to understand what people believe about the world, and specifically about energy and fuel economy, I run across repeated misunderstanding of the word "exponentially." This or that "increases exponentially." Now some things do, in fact, increase exponentially. Money in a bank account at a fixed rate of interest comes to mind. The population of bacteria in a petri dish or humans on a finite planet (for a while) are other examples.

But it means more than "gets big quickly." People who should know better, or at least should learn better before using it this way frequently use it to mean this. An example is an article at the Planet Green web site entitled "Become a "Hypermiler," Save Even More Gas" by one Collin Dunn of Corvallis, OR. As usual, driving more slowly is first on his list. OK, I agree completely, but then he states that "The amount of drag your vehicle generates increases exponentially with each increase in speed; that is, driving a little faster generates a lot more drag, which requires more gas to overcome."

Well, it's just some guy writing an article for a Cable Channel web site, right? But he got this gem from the Toyota Open Road Blog where they go into detail to precisely state their error: "The amount of drag your vehicle generates is not linear – it does not increase at the same rate as your vehicle’s speed does. Instead, drag is more or less proportional to the square of speed. It increases exponentially." NO, NO, NO!

"More or less proportional to the square of speed" is correct. That is, if you know the drag at some speed, and want the drag at some other speed, take the ratio of the second speed to the first, square it, multiply the drag at the first speed by the squared ratio and there's your approximate drag at the second speed. For example, doubling the speed would approximately increase aerodynamic drag by a factor of four. This is NOT exponential, the mathematically astute refer to it as a power function. The variable (speed) is the base, the power (2 in this case) is the exponent. The exponent is fixed.

An exponential function has a fixed base, and the variable is the exponent. Now, depending on the ranges of the variables and the fixed numbers and the proportionality constants, the exponential function may be smaller than the power function at some values of the variable, but the exponential function is always ultimately larger for large values of the variable. So while something that increases exponentially really does get large very fast, at least after a while, it's not true that anything that increases quickly at any point increases exponentially. This most assuredly does include aerodynamic drag.

And don't even get me started on "mega."


Michael Tobis said...

What about "exponentially more", the comparative form? Like "with this modification, I get exponentially better mileage than previously." I've run into that one a couple of times.

To which I say, yes indeed. Much depends, however, on the exponent.

King of the Road said...

It's an honor to have such an esteemed blogger as yourself post a comment!

Yes, there is a whole universe of exponents.

Ed Davies said...

Pity, I was hoping you'd give a simple description of exponential functions. How about: "the rate of increase at a particular time is proportional to the size at that time". If there are more bacteria or humans then more descendants will be breed.

King of the Road said...


Yes, I should have done so. I'm working on an update to the post now that it's been linked. At the time is was more in the nature of a rant than educational!