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. --