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How many digits of pi do we really need? Eh, not that many, says NASA.

False color image of Pluto produced by the New Horizons composition team, using a pair of Ralph/LEISA instrument scans.
(NASA/JHUAPL/SwRI)

Happy pi day! The date 3/14 is, of course, a poor approximation for π, which has infinite digits (3.1415926...). But that raises a few questions: How many digits have we actually found? And how many do we need?

Back in 2010, Shigeru Kondo and Alexander Yee caused a stir when they announced they'd calculated pi to 5 trillion digits with homemade computers. (The whole project nearly collapsed when Kondo's daughter turned on a hair dryer and tripped a circuit breaker.) Since then, other researchers have used Yee's methods to calculate 22 trillion digits and counting.

It's a phenomenal bit of number crunching. It's also, for most everyday purposes, overkill.

Marc Rayman, the director and chief engineer for NASA's Dawn mission, recently made this clear in response to a question on Facebook. NASA, he explained, certainly doesn't need trillions of digits for its calculations. In fact, they get by with using just 15 — 3.141592653589793. It's not perfect, but it's close enough:

The most distant spacecraft from Earth is Voyager 1. It is about 12.5 billion miles away. Let's say we have a circle with a radius of exactly that size (or 25 billion miles in diameter) and we want to calculate the circumference, which is pi times the radius times 2. Using pi rounded to the 15th decimal, as I gave above, that comes out to a little more than 78 billion miles.

We don't need to be concerned here with exactly what the value is (you can multiply it out if you like) but rather what the error in the value is by not using more digits of pi. In other words, by cutting pi off at the 15th decimal point, we would calculate a circumference for that circle that is very slightly off. It turns out that our calculated circumference of the 25 billion mile diameter circle would be wrong by 1.5 inches.

Think about that. We have a circle more than 78 billion miles around, and our calculation of that distance would be off by perhaps less than the length of your little finger.

Going further, if you used 40 digits of pi, Rayman says, you could calculate the circumference of the entire visible universe — an area with the radius of about 46 billion light-years — "to an accuracy equal to the diameter of a hydrogen atom." That'll do!

Mathematicians have been able to calculate 40 digits of pi since the 1700s. Ever since then, they've been rocketing far beyond our wildest spaceship needs:

(Nageh/Wikimedia Commons)

Read more: Steven Strogatz's 2015 piece on why pi matters is pretty great.