Before there was a PI key on our Bowmar Brains, many of us would approximate the value of PI using the formula 355/113. Some of us also memorized the first nine digits of Pi – 3.14159265 and input that into our calculators. Either way, the result would be off by less than 1 part per million, good enough for most applications short of space launches.
Scientific American has this offering about How Much Pi Do You Need?:
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?
This interesting article goes on to discuss how many decimal places of PI accuracy are needed for space operations, navigation (GPS) and quantum mechanics.