TITLE: Do We Need the Digits of $\pi$?
QUESTION [41 upvotes]: 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?

REPLY [3 votes]: 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.