Number of feet in a mile

Here are a couple amusing things I’ve run across recently regarding the number of feet in a mile. Both are frivolous, but also have a more serious side.

Mnemonic

First, you can use “five tomatoes” as a mnemonic for remembering that there are 5280 feet in a mile.

As I’ve argued earlier, systems of units that seem awkward now were designed according to different criteria. If you pointed out to a medieval Englishman that the number of feet in a mile is hard to remember, he might ask “Why would you want to convert feet into miles?”

Almost integers

The number of feet in a mile is very nearly

e^{\pi \sqrt{67}/3}

The exact value of the expression above is 5280.0000088… The difference between a mile and exp(π √67 / 3) feet is less than the length of an E. coli bacterium.

When people see examples like this, an expression that can’t be an integer and yet is eerily close to being an integer, some ask whether there’s a reason. Others think the question is meaningless. “The number equals whatever it equals, and it happens to be near an integer. So what?”

But there is some deeper math going on here. I touch on the reason in this post. 67 is a Heegner number, and so exp(π √67) is nearly an integer. That integer is 5280³ + 744 and so exp(π √67 / 3) ≈ 5280.

Lurking in the background is something called the j-function. It has something to do with why some expressions are very nearly integers. Incidentally, the number 744 mentioned above is the constant term in the Laurent expansion for the j-function.