Science & Technology

How Many Digits of Pi Do We Actually Need?

You've probably seen those memory champs who can recite thousands of digits of pi in one go. Impressive? Yes. Excessive? Hell yes. Though pi famously has an infinite number of digits (it's irrational, after all), we certainly don't need all of them. Honestly, pi, you're a little much.

Does Size Really Matter?

If you need the refresher, pi (π) is the mathematical constant that represents the ratio of a circle's circumference to its diameter. In high school math class, you can get away with using the approximate 3.14 for your calculations, despite it having decimals that go on forever. When you're actually solving real-world, er, real-universe problems, 3.14 won't get you the precision you need. However, you don't need a memory champ on hand in order to use pi in these scenarios. Scientific American reports that NASA scientists keep the space station operational with only about 15 digits of pi. Anyone could memorize that.

The director and chief engineer for NASA's Dawn mission, Marc Rayman, puts the accuracy of using only 15 decimals of pi into perspective: "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," he continues. "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." Mind. Blown.

Related Video: When Pi Is Not 3.14

Go Colossally Huge, or Go Home

Just for fun, let's go big. Like, size-of-the-observable-universe big. According to Rayman, calculating the circumference of the visible universe doesn't even come close to requiring an unmemorizable number of digits. You want to calculate the size of the largest physical circle possible? Just about 40 digits of pi after the decimal point will get you there. And, boy, will it ever get you there.

"The radius of the universe is about 46 billion light years. Now let me ask a different question: How many digits of pi would we need to calculate the circumference of a circle with a radius of 46 billion light years to an accuracy equal to the diameter of a hydrogen atom (the simplest atom)? The answer is that you would need 39 or 40 decimal places," Rayman writes to NASA/JPL.

"If you think about how fantastically vast the universe is — truly far beyond what we can conceive, and certainly far, far, far beyond what you can see with your eyes even on the darkest, most beautiful, star-filled night — and think about how incredibly tiny a single atom is, you can see that we would not need to use many digits of pi to cover the entire range." No shade, memory champs.

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For a delightfully approachable take on high-level mathematics, check out "How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics" by Eugenia Cheng. We handpick reading recommendations we think you may like. If you choose to make a purchase, Curiosity will get a share of the sale.

Written by Joanie Faletto July 30, 2017

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