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Arithmetics in numeration systems with negative quadratic base

Zuzana MasákováTomáš Vávra — 2011

Kybernetika

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...

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