Today is March 14, or 3/14 if you live in the United States. You may have heard today referred to as “Pi Day.”

The Greek letter π, spelled out phonetically in English as “pi,” is used in math to stand for a particular number. The number π is just a little bigger than the number 3.14, hence the celebration of 3/14.

Some people celebrate with actual pies (even weird fish ones):

Most people have heard about π at some point, but I thought I’d talk a little bit about how I think mathematicians think about π.

There are a lot of different ways you can approach the concept of a number. You can think about counting discrete objects, say, hats in a hat shop, and get the integers. You can think about ratios between integers, and get the set of all rational numbers (in other words, fractions).

Then there are the real numbers, a set of infinitely many numbers that includes integers like 1 and 3, as well as less obvious choices such as -1/4, √7, and π. And there are even broader number systems beyond that.

From the mindset of whole numbers and fractions, π can seem kind of mystical: it can’t be expressed on paper using any sort of fraction or decimal. In other words, π is not 3.14, and it’s not 3.14159, and it’s not 3.14159265358979. It’s close to those numbers I wrote down. But π is a different number, and no many how many digits you use, you’ll never be able to write it down precisely.

This is emphatically not why mathematicians care about π.

 

First, a bit of context for π. Suppose you had a perfect circle and measured its circumference (blue), i.e. the length around the outside of the circle. Then suppose you measured its diameter (yellow), the length the line you get by connecting two points on the circle that are opposite each other. If you divide those two numbers – the first length divided by the second length, or the circumference divided by the diameter – you always get the same number. The number you get is called π.

That is one possible way to define π. In other words, this definition lets you identify what number you’re talking about, unambiguously, without writing it down in terms of decimals. Saying “π is what you get when you take circumference divided by diameter” is a lot like saying “4 is what you get when you take 1+1+1+1.”

Then there are much more fun contexts for π, like Euler’s formula, that take some more mathematical background to appreciate.

I find it interesting that, in the larger context of math, π isn’t mysterious for being impossible to write down as a fraction or decimal. It’s not even special for that! Surprisingly, the set of irrational numbers (those, like π, that cannot be written down as a fraction or a decimal) is huge. It’s actually larger than the set of integers or fractions.

The size of the set of real numbers, which holds the irrational numbers, is actually a different level of infinity than the size of the set of integers or fractions.

That shocked me when I first heard it. It still kind of does.

So, happy Pi Day! Here’s to circles, and to the real numbers.

If only there was a dessert pun for two-to-the-aleph-null…