Digital representation systems for canonical structures in the plane.

The two-dimensional systems given by complex numbers ℂ (i2 = −1), dual numbers 픸 (휀2 = 0) and hyperbolic numbers 픻 (j2 = 1) are, up to algebra isomorphism, the three possible associative algebra structures on ℝ2. 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, <italic>Acta Sci. Math.</italic> <bold>37</bold> (1975) 255–260] proved that all complex numbers can be written in radix expansion with the natural numbers 0, 1,…,n2 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 ℝ2 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. [ABSTRACT FROM AUTHOR]