Re: [math-fun] Complex modulo function?
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).
Can't its last digit also be 7? On Sat, May 16, 2020 at 9:59 AM Keith F. Lynch <kfl@keithlynch.net> wrote:
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).
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- http://golly.sf.net/ --
participants (2)
-
Keith F. Lynch -
Tomas Rokicki