Arithmetics in numeration systems with negative quadratic base

Details Arithmetics in numeration systems with negative quadratic base Arithmetics in numeration systems with negative quadratic base

2010-11-05

arxiv.org/abs/1011.1403v1

Mathematics

Number Theory

Scientific paper

We consider positional numeration system with negative base $-\beta$, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when $\beta$ is a quadratic Pisot number. We study a class of roots $\beta>1$ of polynomials $x^2-mx-n$, $m\geq n\geq 1$, and show that in this case the set ${\rm Fin}(-\beta)$ of finite $(-\beta)$-expansions is closed under addition, although it is not closed under subtraction. A particular example is $\beta=\tau=\frac12(1+\sqrt5)$, the golden ratio. For such $\beta$, we determine the exact bound on the number of fractional digits appearing in arithmetical operations. We also show that the set of $(-\tau)$-integers coincides on the positive half-line with the set of $(\tau^2)$-integers.

Affiliated with

Also associated with

No associations

Rating

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

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-454814