Happy Pi Approximation Day

I hope you’re ready for your big Pi Approximation Day party today.

You might have observed Pi Day on March 14. It gets its name from 3.14, the first three digits of the ratio of a circle’s circumference to its diameter. Always on the lookout for excuses to eat pie, some geeky math types also celebrate the number on July 22. The fraction 22⁄7 has a value of 3.142857, so it has the same first three digits as pi.

Both 3.14 and 22⁄7 are approximations of pi, so the two days deserve the same title. In fact, 22⁄7 is closer to pi than 3.14 is. So if you’re an aspiring pedant, you can choose to celebrate July 22 as Pi Day and March 14 as Not Quite as Close to Pi Day. (Either way, you’ll enjoy more pie.) But what does it mean to be an approximation of pi—and why does it matter?

Pi is irrational. That is, the decimal expansion never ends and never repeats, so any number of decimal places we write out is an approximation. (Of course, we can write the number exactly using just one symbol: π.)

Each decimal digit we know makes any computation involving pi more precise. But how many of them do we actually need for sufficient accuracy? Of course it depends on the application. When we round pi to the integer 3, we are about 4.51 percent off from the correct value. If we use it to estimate the circumference of an object with a diameter of 100 feet, we will be off by 4 ½ feet. When we add the tenths place, and use the approximation 3.1 for pi, our error is only about 1.3 percent. The approximation 3.14 is about ½ percent off from the true value, and the fairly well known 3.14159 is within 0.000084 percent.

Believe it or not, in 2006 Akira Haraguchi of Japan recited 100,000 digits of pi from memory in 16 ½ hours, stopping for five minutes every hour to replenish his strength with onigiri rice balls. And the world record for number of digits of pi computed is 10 trillion, at least as of October 2011. Pi computation can be used to test computer precision, but I think this is a symptom of pi-mania rather than a legitimate need for pi. Other numbers could be used just as meaningfully, but we choose to use pi.

It seems that we know, and strive to discover, many, many more digits of pi than we need for any practical application on Earth, or even in the part of space we can hope to get to right now. I guess the endlessness of the decimal representation just fascinates people. Haraguchi, the pi reciter, told The Japan Times that his memorization of pi is part of his quest for eternal truth. For some, it is probably a challenge: How far can I go? We want to push our limits, and memorizing pages of numbers seems pointless until we give it the halo of pi.

Coming back down to Earth, Pi Day and Pi Approximation Day are great reasons to ponder the beauty of pi, simply because, even though it doesn’t follow any logical pattern, is the foundation for one of the most perfect shapes — the circle.