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 x2−mx−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=12(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 (τ2)-integers.
We consider positional numeration systems with negative real base −β, where β>1, and study the extremal representations in these systems, called here the greedy and lazy representations. We give algorithms for determination of minimal and maximal (−β)-representation with respect to the alternate order. We also show that both extremal representations can be obtained as representations in the positive base β2 with a non-integer alphabet. This enables us to characterize digit sequences admissible as greedy and lazy (−β)-representation. Such a characterization allows us to study the set of uniquely representable numbers. In the case that β is the golden ratio and the Tribonacci constant, we give the characterization of digit sequences admissible as greedy and lazy (−β)-representation using a set of forbidden strings.