You Can Count on Pi

For the geeks, there are There are several fantastic holidays on the calendar. Of course there is mole day (10/23) to commemorate Avogadro’s numberwhich is huge (on the order of 10²³) and of enormous importance in physics. there is and day (2/7) for the ubiquitous Euler number (e = 2.718…). But the best one is Pi Day, which is celebrated on March 14 because the infinitely long decimal approximation of pi starts at 3.14. There is a lot to say about pi. I’ve been writing posts about Pi Day for 14 years. (Here is a partial list).

What is pi (or as the Greeks would say, π)? By definition, it is the relationship between the circumference and the diameter of a circle. It’s not obvious why that should be special, but pi appears in a lot of interesting places which seem to have nothing to do with circles. But one of the strangest things about pi is that it is an irrational number. That means it is a value that cannot be expressed as a fraction of two whole numbers. Oh, of course. The number 22/7 (22 ÷ 7) is a good approximation, but it is not pi.

But wait a second. When we say that pi is irrational, all we are really saying is that it is irrational in the number system we use, which is the base 10 or decimal system. But there is nothing inevitable about that system. As you probably know, computers use a base 2 or binary number system. Base 10 was probably chosen in the analog era because we have 10 fingers to count with. (Fun fact: the Latin root of digit is digitswhich means “finger”).

So could there be a number system in which pi is rational? The answer is yes.

Wait, what is a number system?

Let’s review how a number system works. Imagine you are a bean counter in the time of the Neanderthals. For each successive bean, you write a different symbol on the wall of your cave. For 200 beans, you need 200 symbols. It’s numbing, that’s why you call them “numbers.”

One day you meet an intelligent Homo sapiens who tells you: “You’re working too hard!” They have a new system with only 10 symbols, written 0 to 9, that can represent any number of beans. Once you reach 9, you simply move one place to the left and start again, where each digit is now a multiple of 10. After that, it is a multiple of 100, and so on in successively higher powers of 10.

Take the number 214: we have 2 hundreds, 1 ten and 4 ones. We can write what this actually means as follows: