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Do We Need the Digits of $\pi$?

Given that at 39 digits, we have enough of $\pi$ to calculate the volume of the known universe with (literally!) atomic precision, what value is gained? Are there other formulas for which more digits of $\pi$ are useful? If so how many digits of $\pi$ do we need before there's no gain?

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You mean gained besides "knowing more than 40 digits of $\pi$"? –  Asaf Karagila Jan 13 '13 at 19:00
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Well, you need about 64,000 digits to cover every possible block of digits in a time ("HH:MM") or date ("MM/DD" or "DD/MM"). So, what I gain from knowing more than 39 digits is the foundation of my "piClock" app ---see piclock.com--- which actually uses the first million digits. (If there's objection to "advertising" this app here, I'll remove my comment.) –  Blue Jan 13 '13 at 19:08
    
In some cases, you may need to know more than $40$ digits of $\pi$ to get $40$ digits of your answer (involving $\pi$) precise, due to error terms that blow up. –  TMM Jan 13 '13 at 19:09
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@Blue I see no need for you to remove your (very relevant) link. –  Chris Taylor Jan 13 '13 at 19:30
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The most important reason for knowing digits of $\pi$ is explained in the book/film Contact by the late physicist Carl Sagan. However you have to know it in base 11! :) –  Maesumi Jan 13 '13 at 19:43
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marked as duplicate by TMM, Norbert, Davide Giraudo, whuber, Brandon Carter Jan 13 '13 at 20:50

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6 Answers

The practicality of knowing $\pi$ to so many digits has long since passed. I think the main reason people continue to calculate its digits is because there is a certain prestige that goes along with being able to calculate more digits than anyone else. It brings notoriety, especially when testing a new supercomputer.

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There is no practical gain in computing the circumference of a physical circle. As a matter of fact, most exercises in computing more and more digits of $\pi$ are rather some kind of computer benchmark tests (or may in fct detect computer malfunction to some extent).

In theory, it is at least feasible that a rather good approximation of $\pi$ might be needed for some intricate proof (say, of the Riemann hypothesis), but to repeat it: That would not be related to physical circle circumferences.

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As per wikipedia: Pi

For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, [as you point out,] thirty-nine digits are sufficient to perform most cosmological calculations, because that is the accuracy necessary to calculate the volume of the known universe with a precision of one atom.

Despite this, people have worked strenuously to compute $\pi$ to thousands and millions of digits. This effort may be partly ascribed to the human compulsion to break records, and such achievements with $\pi$ often make headlines around the world.

(This obsession with/compulsion to memorize/calculate more and more of the digits of $\pi$, may also, for at least a few, constitute a manifestation of OCD, and provide grounds for such a diagnosis!)

(To the credit of $\pi$ and its digits) They do have practical benefit:

... such as testing supercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π.

...[and can be applied to test the accuracy and]

...the "global integrity" of a supercomputer. A large scale calculation of pi is entirely unforgiving; it soaks into all parts of the machine and a single bit awry leaves detectable consequences.

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I could see it to be useful to gain insight on some of $\pi$'s properties. For example, we don't know whether $\pi$ is normal or not (normal number is 'morally' a number where each digit is equiprobable in every base), so a statistical analysis of known digits may hint at that (that would not prove it, obviously).

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The hunt for more digits of $\pi$ helps to spur research into analysis, especially in developing new methods for accelerating convergence of sums. See, for instance, Bailey-Borwein-Plouffe.

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$\pi$ is also used to randomly generate numbers. Maybe there are some applications there too.

π as a random number generator

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