For geeks, there There are many great holidays on the calendar. There is of course Mole Day (10/23) to celebrate Avogadro’s number, which is huge (on the order of 10).23) and extremely important in physics. E is day (2/7) for Euler’s ubiquitous number (e = 2.718…). But the best one is Pi Day, celebrated on March 14 because the infinitely long decimal approximation of pi starts at 3.14. There’s a lot to say about pi—I’ve been writing Pi Day posts for 14 years. (Here is a partial list).
What is pi (or as the Greeks would say, Ï€)? By definition, it is the ratio of the circumference and diameter of the circle. It’s not clear why it should be special, but Pi appears in several cool places that have nothing to do with Circles. But one of the strangest things about pi is that it is an irrational number. This means that it is a value that cannot be expressed as a fraction of two integers. Oh sure. The number 22/7 (22 ÷ 7) is a reasonable approximation, but it is not pi.
But wait a second. When we say that pi is irrational, what we are really saying is that it is irrational in the system of numbers we use, which is the base-10, or decimal, system. But there is nothing inevitable about that system. As you probably know, computers use the base-2, or binary, number system. Base-10 was probably chosen in the analog era because we have 10 fingers to count. (Fun Fact: Latin Origin score Is digitusmeaning “finger.”)
So can there be any number system in which pi is rational? The answer is yes.
Wait, what’s the number system?
Let’s review how the number system works. Imagine you are a bean counter from the Neanderthal era. For each successive bean, you write a different symbol on the wall of your cave. For 200 beans, you need 200 symbols. It’s mind-numbing, and that’s why you call them “numbers.”
One day you meet a clever Homo sapiens who says, “You’re trying too hard!” They have a new system with only 10 symbols, written as 0 to 9, which can represent any quantity of beans. Once you reach 9, you move one space 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. What is its actual meaning, we can write it as follows:
Illustration: Rat Allen
