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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?

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    $\begingroup$ 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 $\mathbb C$. $\endgroup$ – Mohan Jan 9 '13 at 20:26
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(This used to be a comment, but is getting a bit too long now.)

The quick answer is: Yes, it is quite possible and indeed very productive to define the idea of congruence modulo a complex number rather than an ordinary positive integer.

A bit more detail:

A good example to start with would be to look at for example with the well-known Gaussian Integers, which are covered in many elementary books on number theory.

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 $i$.)

You can read about some of this in the first 3 or 4 chapters of Ireland and Rosen's book "A classical Introduction to Modern Number Theory" - a real classic, which can be tough going, although the early chapters are more approachable.

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