  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.
