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Complex Numbers with Three Radix Expansions

Published online by Cambridge University Press:  20 November 2018

William J. Gilbert
William J. Gilbert*
Affiliation:
University of Waterloo, Waterloo, Ontario
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1. Introduction. This paper deals with the lack of uniqueness of the representations of the complex numbers in positional notation using Gaussian integers as bases.

Kátai and Szabó [3] proved that all the complex numbers can be written in radix form using the base –n + i with the natural numbers 0, 1, 2, …, n2 as digits. They remarked that they did not assert the uniqueness of these representations but gave no further indications of any multiple expansions. The geometry of these complex bases [2] indicates that some numbers have two expansions in a given base, while a few numbers even have three different expansions.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 6 , 01 December 1982 , pp. 1335 - 1348
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Gilbert, W. J., The fractal dimension of sets derived from complex bases, submitted.Google Scholar
2. Gilbert, W. J., Fractal geometry derived from complex bases, The Mathematical Intelligencer 4 (1982), 78–86.Google Scholar
3. I, Kâtai and J, Szabô, Canonical number systems for complex integers, Acta Sci. Math. (Szeged) 37 (1975), 255–260.Google Scholar