Hostname: page-component-7bb8b95d7b-l4ctd Total loading time: 0 Render date: 2024-09-17T08:30:42.909Z Has data issue: false hasContentIssue false
  • English
  • Français

The Division Algorithm in Complex Bases

Published online by Cambridge University Press:  20 November 2018

William J. Gilbert
William J. Gilbert*
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, e-mail:wgilbert@uwaterloo.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

11A6358F08
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 1 , 01 March 1996 , pp. 47 - 54
Copyright
Copyright © Canadian Mathematical Society 1995

References

1. Bamsley, M. F., Fractals everywhere, 2nd éd., Academic Press, New York, 1993.Google Scholar
2. Gilbert, W. J., Radix representations of quadratic fields, J. Math. Anal. Appl. 83(1981), 264–274.Google Scholar
3. Gilbert, W. J., Complex numbers with three radix expansions, Canad. J. Math. 34(1982), 1335–1348.Google Scholar
4. Gilbert, W. J., Arithmetic in complex bases, Math. Mag. 57(1984), 77—81.Google Scholar
5. Hutchinson, J. E., Fractals and self similarity, Indiana Univ. Math. J. 30(1981), 713—747. 6.1. Kâiti and J. Szabô, Canonical number systems for complex integers, Acta Sci. Math. (Szeged) 37(1975), 255–260.Google Scholar
7. Knuth, D. E., The art of computer programming, Vol 2, Seminumerical algorithms, 2nd éd., Addison- Wesley, Reading, Massachusetts, 1981.Google Scholar
8. Manila, D. W., Basic digit sets for radix representation, J. Assoc. Comput. Mach. 29(1982), 1131–1143.Google Scholar
9. Prusinkiewicz, P. and Sandness, G., Koch curves as attractors and repellers, IEEE Comput. Graphics Appl. (6) 8(1988), 26–40.Google Scholar