On Choosing The Best Units

Update (21 March 2011): A great version of the Gimli glider story is here.

In 1983, an Air Canada plane ran out of fuel on a scheduled flight. Running out of fuel is a very unusual situation for a modern airline; the problem here was an error in converting units between the Imperial system and the newly introduced metric system. Litres and kilos and gallons and pounds, oh my! Thanks to the skill of the pilots (and the experience of one of them with gliders; this led to the plane becoming known as the “Gimli glider,” since the plane landed at an abondoned air force base near Gimli, Manitoba), there were no serious injuries in the emergency landing.

Choosing the right units can help us to understand quantities that are otherwise obscure. For example, the speed of light in vacuum is such a large number in ordinary units (about $3 \times 10^8$ m/s) that it is hard to get a feel for. The problem is that the speed is so much greater than anything else in our experience. David Mermin, in his excellent book on special relativity, It’s About Time, mentions that the speed of light is about 1 foot per nanosecond.

Does that give you a better feel for how fast the speed of light is? It essentially exchanges a very large distance (300,000,000 m) in an ordinary amount of time (1 s) for an ordinary distance (1 foot) in a minute amount of time (1 ns). Maybe it helps, maybe it doesn’t, but it is certainly worth playing around with units to help us see our way around very large or very small quantities.

Of course, some people purposely use units to obscure understanding. I don’t know for sure that Verizon did this, but I wouldn’t put it past a large corporation to do just this. A fellow inquired about the cost of downloading data when he travelled from his native U.S. to Canada. He was told very clearly that the cost would be 0.002 cents per kilobyte of data. However, when his bill arrived, he saw that he was charged 100 times as much, $0.002/kB. His efforts at trying to explain to various customer service representatives that there is a difference between the two units is documented (including audio tapes of telephone conversations) here. It could be hilarious or depressing, depending on the listener.

Of course, all of this mess could have been avoided if Verizon had only quoted the cost in a reasonable unit; that is, one that would be far more likely to be understood by all. Rather than quote 0.002 cents per kB, why not use 2 cents per MB, which would be so much easier to understand? Or how about $20/GB? Most people can understand small whole numbers (say between 0 and a few hundred) best, so it would have been kind to stick to these.

Yes, we need to improve the basic numeracy of people in general, but we also have to encourage companies to quote numbers in sensible units.

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 22 January 2021.)