HaloO,

Jonathan Lang wrote:
That said, I'm still trying to wrap my head around how the Euclidiean
definition would work for complex numbers.  What would be the quotient
and remainder for, e.g., 8i / 3; 8 / 3i; (3 + 4i) / 3; 8 / (4 + 3i);
or (12 + 5i) / (3 + 4i)?


I assume you are intending the Gaussian Integers Int[i], i.e. complex
numbers with Int coefficients. There you have to solve the equations

    a = q * b + r

with q and r from Int[i] and N(r) < N(b) where N(x + yi) = x**2 + y**2.
This yields for your numbers e.g.

  a = 8i, b = 3  =>  q = 2i, r = 2i

But what comes as a surprise to me is that these q and r are not unique!
q = 3i and r = -i works as well. So there is an additional constraint on
r that enforces a unique pair. E.g. x >= 0 and y >= 0 for r = x + yi.
Here are my results for the rest of your examples:

 a = 8,       b = 3i      =>  q = -2i,    r = 2
                              q = -3i,    r = -1

 a = 3 + 4i,  b = 3       =>  q = 1 + i,  r = i

 a = 8,       b = 4 + 3i  =>  q = 1 - i,  r = 1 + i

 a = 12 + 5i, b = 3 + 4i  =>  q = 2 - i,  r = 2
                              q = 3 - i,  r = -4i
                              q = 2 - 2i, r = -2 + 3i

I cannot give an algorithm how to calculate the remainder.
Even less do I know how to generalize it to full Complex.

Regards, TSa.
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