Arithmetics in numeration systems with negative quadratic base

Zuzana Masáková; Tomáš Vávra

Kybernetika (2011)

  • Volume: 47, Issue: 1, page 74-92
  • ISSN: 0023-5954

Abstract

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We consider positional numeration system with negative base - β , as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when β is a quadratic Pisot number. We study a class of roots β > 1 of polynomials x 2 - m x - n , m ≥ n ≥ 1 , and show that in this case the set Fin ( - β ) of finite ( - β ) -expansions is closed under addition, although it is not closed under subtraction. A particular example is β = τ = 1 2 ( 1 + 5 ) , the golden ratio. For such β , we determine the exact bound on the number of fractional digits appearing in arithmetical operations. We also show that the set of ( - τ ) -integers coincides on the positive half-line with the set of ( τ 2 ) -integers.

How to cite

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Masáková, Zuzana, and Vávra, Tomáš. "Arithmetics in numeration systems with negative quadratic base." Kybernetika 47.1 (2011): 74-92. <http://eudml.org/doc/197050>.

@article{Masáková2011,
abstract = {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 $x^2-mx-n$, $m\ge n\ge 1$, 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 $\beta =\tau =\frac\{1\}\{2\}(1+\sqrt\{5\})$, 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.},
author = {Masáková, Zuzana, Vávra, Tomáš},
journal = {Kybernetika},
keywords = {numeration systems; negative base; Pisot number; numeration systems; negative base; Pisot number},
language = {eng},
number = {1},
pages = {74-92},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Arithmetics in numeration systems with negative quadratic base},
url = {http://eudml.org/doc/197050},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Masáková, Zuzana
AU - Vávra, Tomáš
TI - Arithmetics in numeration systems with negative quadratic base
JO - Kybernetika
PY - 2011
PB - Institute of Information Theory and Automation AS CR
VL - 47
IS - 1
SP - 74
EP - 92
AB - 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 $x^2-mx-n$, $m\ge n\ge 1$, 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 $\beta =\tau =\frac{1}{2}(1+\sqrt{5})$, 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.
LA - eng
KW - numeration systems; negative base; Pisot number; numeration systems; negative base; Pisot number
UR - http://eudml.org/doc/197050
ER -

References

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