RAIRO - Theoretical Informatics and Applications

Research Article

Rational base number systems for p-adic numbers

Christiane Frougnya1 and Karel Kloudaa2

a1 LIAFA, CNRS UMR 7089, Case 7014, 75205 Paris Cedex 13, and Université Paris 8, France. [email protected]

a2 Faculty of Information Technology, Kolejní 550/2, 160 00 Prague, Czech Republic; [email protected]


This paper deals with rational base number systems for p-adic numbers. We mainly focus on the system proposed by Akiyama et al. in 2008, but we also show that this system is in some sense isomorphic to some other rational base number systems by means of finite transducers. We identify the numbers with finite and eventually periodic representations and we also determine the number of representations of a given p-adic number.

(Received November 2 2010)

(Accepted July 4 2011)

(Online publication August 22 2011)

Key Words:

  • Rational base number systems;
  • p-adic numbers.

Mathematics Subject Classification:

  • 11A67;
  • 11E95