On-line Multiplication and Division in Real and Complex Bases

Abstract

A positional numeration system is given by a base and by a set of digits. The base is a real or complex number aamp;#x03B2; such that |aamp;#x03B2;| > 1, and the digit set A is a finite set of real or complex digits (including 0). In this paper, we first formulate a generalized version of the on-line algorithms for multiplication and division of Trivedi and Ercegovac for the cases that aamp;#x03B2; is any real or complex number, and digits are real or complex. We show that if (aamp;#x03B2;, A) satisfies the so-called (OL) Property, then on-line multiplication and division are feasible by the Trivedi-Ercegovac algorithms. For a real base aamp;#x03B2; and alphabet A of contiguous integers, the system (aamp;#x03B2;, A) has the (OL) Property if #A > |aamp;#x03B2;|. Provided that addition and subtraction are realizable in parallel in the system (aamp;#x03B2;, A), our on-line algorithms for multiplication and division have linear time complexity. Three examples are presented in detail: base aamp;#x03B2; = 3+aamp;#x221A;5/2 with alphabet A = {-1, 0, 1}, base aamp;#x03B2; = 2i with alphabet A = {-2, -1, 0, 1, 2} (redundant Knuth numeration system), and base aamp;#x03B2; = -3/2 + aamp;#x0269; aamp;#x221A;3/2 = -1 + aamp;#x03C9;, where aamp;#x03C9; = exp 2iaamp;#x03C0;/3, with alphabet A = {0, ±1, ±aamp;#x03C9;, ±aamp;#x03C9;2} (redundant Eisenstein numeration system).