Digital representation systems for canonical structures in the plane

Abstract

The two-dimensional systems given by complex numbers [Formula: see text] [Formula: see text], dual numbers [Formula: see text] [Formula: see text] and hyperbolic numbers [Formula: see text] [Formula: see text] are, up to algebra isomorphism, the three possible associative algebra structures on [Formula: see text]. The goal of this work is to investigate canonical numbers systems for the rings of integers in the two-dimensional systems yielding digital representation systems in the plane. Kátai and Szabó [Canonical number systems for complex integers, Acta Sci. Math. 37 (1975) 255–260] proved that all complex numbers can be written in radix expansion with the natural numbers [Formula: see text] as digits. In this paper, we will characterize all canonical number systems for the rings of integers in dual and hyperbolic numbers. Finally, using the associate matrices of the two-dimensional bases, we prove that all points in the plane [Formula: see text] can be written in digital representation systems with a large class of bases, including binary, octal, decimal and hexadecimal ones. In particular, we prove that a digital representation system in the plane is finite or periodic if and only if it represents a point with rational coordinates.