Kerry Mitchell <lkmitch@gmail.com> wrote:
Is there a standard, or at least, reasonable, definition of the modulo function for complex numbers?
I don't know if it's any kind of standard, but it has occurred to me that, modulo n^2+1, -1 has a real square root. For instance -1 is congruent to 9 mod 10, so i is 3 mod 10, and -i is 7 mod 10. (Or vice versa if you prefer.) Hence i+1 is 4 mod 10, etc. I don't know if this is useful for anything, but it doesn't seem to lead to any inconsistencies. Everyone's first thought when they learn about i is to ask what it *is*. To which I can answer that at least I know its last digit, which is 3. :-) Since no other power of 10 is 1 plus a square, I unfortunately can't find any other digits of i. The same is true in all other bases, thanks to Mihailescu's theorem (better known as Catalan's conjecture).