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)?

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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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