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

 If you were building a fence around a giant circular swimming pool with a radius of 100 meters and used that approximation to estimate the amount of fencing you would need, you would be half a millimeter short. Half a millimeter is tiny compared with the total fence length, 628.3185 meters. Being within half a millimeter is surely sufficient, and the tools you are using to make the fence probably introduce more uncertainty into your structure than your approximation of pi.