Arithmetic in Complex Bases
William J. Gilbert · Mathematics Magazine · 1984
In this note, we introduce a novel way of doing complex arithmetic that does not involve separating the complex numbers into their real and imaginary parts. This method uses the representation of complex numbers in positional notation using a complex base - n + i, for a positive integer n, with natural numbers as digits. Addition, subtraction and multiplication can be performed directly in this positional notation and is similar to real decimal arithmetic; the main difference is in the carry digits. However, division is more complicated and the construction of a good algorithm for long division is a challenging unsolved problem. We say that an integer z (real or complex) is represented in the base b with digits from the set 6D