b-Parts and finite b-representation of real numbers

M. H. Hooshmand
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 19, 2013, Number 4, Pages 4–15
Full paper (PDF, 200Kb


Authors and affiliations

M. H. Hooshmand
Department of Mathematics, Shiraz Branch,
Islamic Azad University, Shiraz, Iran


The b-parts of real numbers and the generalized division algorithm were considered and discussed in [3]. Also some of their algebraic properties have been studied in [4]. In this paper we continue it and introduce a unique finite representation of real numbers to the base of an arbitrary real number b ≠ 0, ± 1 (namely finite b-representation), by using them. Finally we prove a necessary and sufficient conditions for the finite b-representation to be digital.


  • b-integer part
  • b-decimal part
  • Generalized division algorithm
  • Radix representation and expansion of real numbers
  • b-digital sequence

AMS Classification

  • 11A63
  • 11A67


  1. Apostol, T.M. Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
  2. Glendinning, P., N. Sidorov, Unique Representations of Real Numbers in Non-Integer Bases, Math. Res. Lett., Vol. 8, 2001, 535–543.
  3. Hooshmand, M.H. b-Digital Sequences, Proceedings of the 9th world Multiconference on Systemics, Cybernetics and Informatics (WMSCI 2005)- Orlando, USA, 142–146.
  4. Hooshmand, M.H., H. Kamarul Haili, Some Algebraic Properties of b-Parts of Real Numbers, Šiauliai Math. Semin., Vol. 3, 2008, No. 11, 115–121.
  5. Hooshmand, M.H., H. Kamarul Haili, Decomposer and Associative Functional Equations, Indag. Mathem., N.S., Vol. 18, 2007, No. 4, 539–554.

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Cite this paper

Hooshmand, M.H. (2013). b-Parts and finite b-representation of real numbers. Notes on Number Theory and Discrete Mathematics, 19(4), 4-15.

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