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