Geodesic
The Division Algorithm in Complex Bases
Canadian mathematical bulletin, Tome 39 (1996) no. 1, pp. 47-54

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Complex numbers can be represented in positional notation using certain Gaussian integers as bases and digit sets. We describe a long division algorithm to divide one Gaussian integer by another, so that the quotient is a periodic expansion in such a complex base. To divide by the Gaussian integer w in the complex base b, using a digit set D, the remainder must be in the set wT(b,D) ∩ Z[i], where T(b,D) is the set of complex numbers with zero integer part in the base. The set T(b,D) tiles the plane, and can be described geometrically as the attractor of an iterated function system of linear maps. It usually has a fractal boundary. The remainder set can be determined algebraically from the cycles in a certain directed graph.
DOI : 10.4153/CMB-1996-006-1
Mots-clés : 11A63, 58F08 
Gilbert, William J. The Division Algorithm in Complex Bases. Canadian mathematical bulletin, Tome 39 (1996) no. 1, pp. 47-54. doi: 10.4153/CMB-1996-006-1
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     title = {The {Division} {Algorithm} in {Complex} {Bases}},
     journal = {Canadian mathematical bulletin},
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     year = {1996},
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     doi = {10.4153/CMB-1996-006-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1996-006-1/}
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