Non-Standard Numeration Systems

P. Ambrož


We study some properties of non-standard numeration systems with an irrational base ß >1, based on the so-called beta-expansions of real numbers [1]. We discuss two important properties of these systems, namely the Finiteness property, stating whether the set of finite expansions in a given system forms a ring, and then the problem of fractional digits arising under arithmetic operations with integers in a given system. Then we introduce another way of irrational representation of numbers, slightly different from classical beta-expansions. Here we restrict ourselves to one irrational base – the golden mean ? – and we study the Finiteness property again. 


numeration system; beta expansion; tau-adic expansion

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ISSN 1210-2709 (Print)
ISSN 1805-2363 (Online)
Published by the Czech Technical University in Prague