For geeks, there are a number of nice holidays on the calendar. There is in fact Mole Day (10/23) to commemorate Avogadro’s quantity, which is big (on the order of 1023) and massively essential in physics. There’s e Day (2/7) for Euler’s ubiquitous quantity (e = 2.718…). But the perfect is Pi Day, held on March 14 as a result of the infinitely lengthy decimal approximation of pi begins with 3.14. There’s a lot to say about pi—I’ve been writing Pi Day posts for 14 years. (Here’s a partial record).
What is pi (or because the Greeks would say, π)? By definition, it is the ratio of the circumference to the diameter of a circle. It’s not apparent why that ought to be particular, however pi reveals up in a bunch of cool locations that appear to have nothing to do with circles. But one of many weirdest issues about pi is that it is an irrational quantity. That means it is a worth that may’t be expressed as a fraction of two integers. Oh, certain. The quantity 22/7 (22 ÷ 7) is a good approximation, but it surely’s not pi.
But wait a second. When we are saying pi is irrational, all we’re actually saying is that it is irrational within the system of numbers we use, which is the base-10, or decimal, system. But there’s nothing inevitable about that system. As you most likely know, computer systems use a base-2, or binary, quantity system. Base-10 was most likely chosen within the analog period as a result of we now have 10 fingers to rely on. (Fun reality: The Latin root of digit is digitus, which suggests “finger.”)
So may there be a quantity system during which pi is rational? The reply is sure.
Wait, What’s a Number System?
Let’s overview how a quantity system works. Imagine you are a bean counter again in Neanderthal instances. For every successive bean, you write down a unique image on the wall of your cave. For 200 beans, you want 200 symbols. It’s mind-numbing, and so that you name them “numbers.”
One day you meet a intelligent Homo sapiens who says, “You’re working too hard!” They have a brand new system with simply 10 symbols, written as 0 to 9, which might signify any amount of beans. Once you attain 9, you simply transfer over one spot to the left and begin once more, the place every digit is now a a number of of 10. After that it is multiples of 100, and so forth in successively increased powers of 10.
Take the quantity 214: We have 2 a whole lot, 1 ten, and 4 ones. We can write what this actually means as the next:
Illustration: Rhett Allain