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I came across an old mathematical paper on the web, published in the 1980s (I can't seem to find it again). The paper was about complex number arithmetic, and it talked about expressing complex numbers in complex bases. I would like to express a number in a complex base in the following manner:

$x + iy = \sum_n a_n(p + iq)^n$

$x + iy$ is a complex number expressed in base $p + iq$

$a_n, n=\{0,\ldots,\infty\}$ are all integers.

Given integers, $x,y,p,q$, I would like to find $a_n, n=\{0,\ldots,\infty\}$ Is there a systematic way to do this? Formulating this as a system of equations gives us 2 equations and infinite unknowns.

Complex division doesn't give integral remainders, so I dropped the idea of trying to find the coefficients by recursive division as in the case of real numbers.

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Have you looked at complex bases? –  Rick Decker Jul 3 '13 at 0:26

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