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I was reading today that someone found $\pi$ to the ten trillionth digit. Whenever I read that $\pi$ has been calculated to more digits, I ask myself whether this is useful. I know that there are conjectures out there about distributions of numbers in $\pi$ and such. So, I supposed knowing more digits helps us test conjectures. But, are there more reasons that we would want to know the digits? Anything really cool I'm ignoring or forgetting?

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    $\begingroup$ This should probably be community. Do I have the ability to do that? $\endgroup$ – Joe Johnson 126 Oct 17 '11 at 10:58
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    $\begingroup$ No you can't do it yourself. You need to flag for moderator attention and ask the moderators to convert the question into CW mode. However, I don't really see why it should be a community wiki. $\endgroup$ – t.b. Oct 17 '11 at 11:00
  • $\begingroup$ @t.b.: I figured CW because I'm not looking for one particular answer. $\endgroup$ – Joe Johnson 126 Oct 17 '11 at 11:07
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    $\begingroup$ This is a related thread, and I think the answers given there apply to this question, too. (P.S. I don't think this should be CW.) $\endgroup$ – J. M. is a poor mathematician Oct 17 '11 at 11:16
  • $\begingroup$ Thanks for all the great replies. I'll accept the highest answer, as I feel is customary with questions of this nature. $\endgroup$ – Joe Johnson 126 Oct 21 '11 at 20:24
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Not for any real-life calculations according to wikipedia

Practically, one needs only 39 digits of π to make a circle the size of the observable universe accurate to the size of a hydrogen atom.

It is however useful to test supercomputers for accuracy and as a memory intensive number-crunching benchmark.

Today the high precision calculation of $\pi$ finds practical use in testing 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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    $\begingroup$ Something went wrong with your first link. I tried to fix it by linking to the relevant part of the Wikipedia page on $\pi$. Please check that it's the link you intended. $\endgroup$ – t.b. Oct 17 '11 at 11:18
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    $\begingroup$ "any calculations" is a bit of a stretch; as mentioned in the answers to a related question, PSLQ and other integer-relation algorithms do require a whole pile of digits to assist in finding new relations between constants, as well as new formulae (and thus methods) to compute them... $\endgroup$ – J. M. is a poor mathematician Oct 17 '11 at 11:52
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    $\begingroup$ @J.M.: Although those algorithms would not need a trillion digits. I would guess at most a couple hundred. $\endgroup$ – Cheerful Parsnip Oct 17 '11 at 15:48
  • $\begingroup$ Can't you calculate prime numbers using pi? $\endgroup$ – qazwsx Mar 12 at 16:56
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A less mathematical reason for calculating more and more decimal places is because we know they are there. Man is inherently curious and always wants to see what's over the next hill, round the next bend etc.

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    $\begingroup$ More to the point, we are curious about something which we already know is very important. I could roll dice until I die to find out what will happen next, but that doesn't mean that I would find this very enlightening. The difference is that the digits of $\pi$ are more likely to have at least some interesting properties, compared to any typical set of dice-rolls. (Also, for what it's worth, I've met a number of Women who are also very curious.) $\endgroup$ – Niel de Beaudrap Oct 17 '11 at 11:50
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Knowing more and more digits of $\pi$ has absolutely no value to anybody. Digits of $\pi$, beyond the twentieth or so, are completely worthless. The only value in this enterprise, if any, lies in the process by which the digits are generated, not the digits themselves.

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  • $\begingroup$ "While such an exercise might seem frivolous, the fact is we learned a lot from the continual refinement of our algorithms to work efficiently at ultrahigh precision. The reward is a deeper understanding of the theory, and often a better algorithm for low-precision cases." $\endgroup$ – J. M. is a poor mathematician Oct 18 '11 at 0:54
  • $\begingroup$ @J.M.: That seems to back me up (if you ignore my "if any", that is). Who said it? $\endgroup$ – TonyK Oct 18 '11 at 7:36
  • $\begingroup$ It's a quote I cited in the other thread. $\endgroup$ – J. M. is a poor mathematician Oct 18 '11 at 9:32
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    $\begingroup$ Knowing more and more digits of $\pi$ is of value to me because whenever I lose an argument to my partner, I can always walk away with my head held high knowing that I probably remember more digits of $\pi$ than they do. $\endgroup$ – Trogdor Sep 27 '15 at 5:37
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Another reason is for its properties of randomness.

This can be used to test software for analyzing random sequences. It can also be used as a teaching aide. Example: if the digits of pi exhibit statistical randomness (it is believed they do), then at some point in pi's expansion there will be a sequence of one million consecutive 0's.
This surprises a lot of students.

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    $\begingroup$ It's not a good test for randomness tests because we lack a proof that π has the properties we want to test. $\endgroup$ – Dan Piponi Oct 17 '11 at 18:03
  • $\begingroup$ @user80: That's irrelevent, because we know those properties hold for all the digits we use to test (when used to test software, we won't use anywhere near the first trillion digits). Similarly, Goldbach's Conjecture is often used in proofs of algorithmic complexity: we don't know that it's true, but we know that it's true for much, much larger numbers than we'll ever use in practice, so it's still useful. $\endgroup$ – BlueRaja - Danny Pflughoeft Oct 17 '11 at 19:50
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Whenever we calculate the first $n$ digits of an irrational number, we simply estimate it and there is a small error in our approximation.

If we use that for practical applications, we should always be aware of the error and check if it is resonable or not for the practical application. If not we need more digits, so theoretically there is always a need for extra digits.

The above answers covered well why 1 billion digits are probably more than enough for $\pi$, anyhow in general it is hard to say how many digits we need to know to cover any possible application. A resonable number of digits shouldn't suffice, we always need an unresonable number.

Not for $\pi$, but here is a known example where 26 digits (well not exactly, they were 26 binary digits) were not enough for a practical application, and unfortunatelly some people died because the error in that application was too big:

http://ta.twi.tudelft.nl/users/vuik/wi211/disasters.html#patriot

I find this example interesting, because most people would think that 5-6 digits should suffice in all cases, and it is easy to understand why in this case the estimation wasn't good (of course people should had thought about it before it happend)...

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There are more applications of real numbers than simply measuring the lengths of things. So even if you have more digits of $\pi$ than you could ever measure in a practical experiment, there are still applications for those digits.

One example is in forming conjectures about equality between mathematical expressions, especially integrals.

For example consider the integral $I_n=\int_0^\infty\frac{\sqrt{x}\log^nx}{(1-x)^n}$. For small $n$ you can numerically integrate and with some guesswork find that the result is a polynomial in $\pi$ with rational coefficients. What about for large $n$? You can guess it's a polynomial in $n$ and then use an algorithm like LLL to conjecture rational coefficients based on numerical integration. The more digits of $\pi$ you have, the more confidence you have in the correctness of your result. If you find yourself unable to fit a polynomial in $\pi$ you might conjecture that you need a polynomial in more constants or of higher degree. But if you give yourself enough flexibility you can eventually find some expression that fits any real number simply by accident. But by using many digits of $\pi$ you make such accidents far less likely and you can be more confident in your conjecture.

Note, of course, that finding an identity holds to 100 decimal places still doesn't mean it's correct. But having found a plausible sequence of conjectures you may be better placed to find a proof.

For more on this subject see this set of notes on experimental mathematics.

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protected by Zev Chonoles Sep 27 '15 at 5:44

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