First 100 trillion digits of pi8/27/2023 ![]() Whenever we calculate the first $n$ digits of an irrational number, we simply estimate it and there is a small error in our approximation. ![]() But having found a plausible sequence of conjectures you may be better placed to find a proof.įor more on this subject see this set of notes on experimental mathematics. Note, of course, that finding an identity holds to 100 decimal places still doesn't mean it's correct. But by using many digits of $\pi$ you make such accidents far less likely and you can be more confident in your conjecture. But if you give yourself enough flexibility you can eventually find some expression that fits any real number simply by accident. 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. The more digits of $\pi$ you have, the more confidence you have in the correctness of your result. 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. For small $n$ you can numerically integrate and with some guesswork find that the result is a polynomial in $\pi$ with rational coefficients. One example is in forming conjectures about equality between mathematical expressions, especially integrals.įor example consider the integral $I_n=\int_0^\infty\frac$. 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. There are more applications of real numbers than simply measuring the lengths of things.
0 Comments
Leave a Reply.AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |