# Modulo complex number

I was wondering what would happen if we tried to do a modulo operand with complex numbers? For instance, what would be the answer (if any) to the next statement?

$x \mod (a + bi)$

can it be done? if so, could anyone explain?

• complex numbers form a field.So, the remainder is always 0. What exactly do you want your modulo operand to do? Though it is possible to define "modulo operations" on certain subrings of $\mathbb C$. – Mohan Jan 9 '13 at 20:26

If you wanted to investigate some of this yourself, you might start by looking at the possible residues you can get $\pmod{1+i}$, in the ring of Gaussian integers. You have to be a bit careful, since, for example $(1+i)$ is a factor of $2$ in the Gaussian integers, but you should be able to discover things like: every Gaussian integer is equivalent to either 0 or 1 $\pmod{1+i}$. (But it would also be possible to say that every Gaussian integer is equivalent to either 0 or $i$.)