Arithmetics in number systems with negative base

Details Arithmetics in number systems with negative base Arithmetics in number systems with negative base

2010-02-04

arxiv.org/abs/1002.1009v2

Mathematics

Number Theory

13 pages

Scientific paper

We study the numeration system with negative basis, introduced by Ito and Sadahiro. We focus on arithmetic operations in the set ${\rm Fin}(-\beta)$ and $\Z_{-\beta}$ of numbers having finite resp. integer $(-\beta)$-expansions. We show that ${\rm Fin}(-\beta)$ is trivial if $\beta$ is smaller than the golden ratio $\frac12(1+\sqrt5)$. For $\beta\geq\frac12(1+\sqrt5)$ we prove that ${\rm Fin}(-\beta)$ is a ring, only if $\beta$ is a Pisot or Salem number with no negative conjugates. We prove the conjecture of Ito and Sadahiro that ${\rm Fin}(-\beta)$ is a ring if $\beta$ is a quadratic Pisot number with positive conjugate. For quadratic Pisot units we determine the number of fractional digits that may appear when adding or multiplying two $(-\beta)$-integers.

Affiliated with

Also associated with

No associations

Rating

Arithmetics in number systems with negative base does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Arithmetics in number systems with negative base, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Arithmetics in number systems with negative base will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-155128