Jason Rosenhouse: Yummy, Delicious Pi!

RosenhouseHere is a classic bar bet for you: take a wine glass, the kind with a really long stem. Ask whoever is near you to guess whether the height of the glass or the circumference at the top is greater. Most people will choose the height. In fact, they will regard it as obvious that the height is greater. But they will be wrong! Unless it is a very oddly-shaped glass, the circumference will be significantly greater. (Of course, you will need a piece of string to convince your mark of that.) It is a remarkably effective optical illusion.

As we all learned in grade school, the circumference of a circle is pi times the diameter, and pi is just a little greater than three. So the circumference at the top will be three times longer than the diameter. Any glass taller than that would be unpleasant to drink from.

Apparently knowing something about pi can make you money. Who said math isn’t practical?

I remember being fascinated by pi as a kid. When my father—a chemical engineer—first told me about it, I asked him if there was also a number called cake. The number pi is typically defined as a sort of geometrical object: it is the ratio of the circumference of a circle to its diameter. We could also say that pi is the area of a circle whose diameter is one. Yet somehow it keeps appearing in the most unexpected of places.

For example, suppose you pick two whole numbers at random, by which I mean the usual numbers like 1, 2, 3, 4, and so on. Sometimes the two numbers will share a common factor, like 4 and 6, which share a common factor of 2. Other times the two numbers will share no common factor (other than 1), like 3 and 7. Pairs like the second are said to be relatively prime. It turns out the probability that a pair of randomly chosen numbers is relatively prime is 6 divided by pi squared. Not a circle in sight, yet there is pi!

Or imagine that you have a very large sheet of notebook paper whose lines are one inch apart. Suppose you take a one-inch needle and drop it from a height onto the paper. The probability that the needle hits a line is 2 divided by pi. Only lines this time. Still no circles. This is called the Buffon needle problem, if you were curious.

One of the first things you learn about pi is that it is an irrational number, which means it is an infinite, non-repeating decimal. My sixth grade math teacher told me it was just crazy that a number should behave like that, and that is why it is called irrational. You can imagine my disappointment when I later learned that it is irrational only in the sense that it cannot be expressed as a ratio of whole numbers. I like my teacher’s explanation better. You can find fractions that are good approximations, like 22/7 or 355/113, but approximations are not the real thing.

The fact that pi is an infinite, non-repeating decimal, and that it cannot be written simply in terms of whole numbers, makes it difficult to write down at all. That is why we just give it a name, pi, and call it a day. We could as easily have called it Harry the number if we wanted to, but perhaps that lacks gravitas.

Pi is one of the special numbers of mathematics. Another is e, which is typically defined in ways that require calculus, and which have nothing to do with circles. This is another of those strange, irrational numbers that seems to keep popping up in unexpected places. Still another is i, which is defined to be the square root of minus 1, a number so bizarre it is commonly said to be imaginary. And we certainly should not forget the two most special numbers of them all, by which I mean 1 and 0.

Perhaps having experienced social ostracism at the hands of more normal numbers, the five special numbers have gotten together to create one of the most remarkable equations in mathematics. It is called Euler’s identity, and says:

e+1=0

It is remarkable that these five special numbers, defined in contexts entirely separate from one another, should play together so well. At the risk of seeming melodramatic, religions have started over less.

So take a moment this March 14 to give some thought to the most delicious number we have: pi. We will not have another perfect square day until May 5, 2025 (a date that will be written 5/5/25). And since e is 2.72 when rounded to two decimal places, we will never have an e day until February is granted 72 days. Or perhaps someday we will dramatically increase the size of the calendar, and then we will have e day on the second day of the twenty-seventh month.

But pi day comes every year. Enjoy it!

Jason Rosenhouse is a professor of mathematics at James Madison University in Harrisonburg, Virginia. He is the author or editor of six books, including The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brainteaser, and Among the Creationists: Dispatches From the Anti-Evolutionist Frontline. His book Taking Sudoku Seriously, coauthored with Laura Taalman, received the 2012 Prose award, from the American Association of Publishers, for popular science and mathematics. With Jennifer Beineke, he is the editor of the Mathematics of Various Entertaining Subjects series, published by Princeton University Press and the Museum of Mathematics in New York. He is currently working on a book about logic puzzles, to be published by Princeton.