Arithmetics in number systems with a negative base

Abstract

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 − 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 = 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

Confluent Parry numbers, their spectra, and integers in positive- and negative-base number systems

“…1. For d −β (ℓ) = (m 0) k a · · · , a < m we have (β + 1)z = −β 2k+1 −β 2k+2 + mβ 2k+1 + · · · + mβ 2 + aβ + a >−1, see (20) +mβ + m − aβ − a > > −β 2k+1 + (m − a)β + m − (a + 1) > −β 2k+1 .…”

“…An analogous finiteness property for negative base system has not yet been sufficiently explored. So far, the only known class of numbers β > 1 such that Fin(−β) = Z[β, β −1 ] were the zeros of x 2 − mx + n, m − 2 ≥ n ≥ 1, as shown in [20].…”

Rhapsody in fractional

“…The position k is the logarithm of the corresponding weight, which is given by k = log b b k . Positive integer base PNS have been commonly used, but other bases are possible for representing numbers, namely negative integer [27,40], improper fractional [34], irrational [9,3,61], transcendental [27] and complex bases. Negative integer bases have the advantage that no minus sign is needed to represent negative numbers.…”

SUBSTITUTIONS OVER INFINITE ALPHABET GENERATING (−β)-INTEGERS

“…Note that the general class of quadratic Pisot numbers for Ito-Sadahiro case l = − β β+1 was also studied in [9].…”