  It is a well known result that for $\beta\in(1,\frac{1+\sqrt{5}}{2})$ and
$x\in(0,\frac{1}{\beta-1})$ there exists uncountably many
$(\epsilon_{i})_{i=1}^{\infty}\in {0,1}^{\mathbb{N}}$ such that
$x=\sum_{i=1}^{\infty}\epsilon_{i}\beta^{-i}.$ When
$\beta\in(\frac{1+\sqrt{5}}{2},2]$ there exists $x\in (0,\frac{1}{\beta-1})$
for which there exists a unique $(\epsilon_{i})_{i=1}^{\infty}\in
{0,1}^{\mathbb{N}}$ such that $x=\sum_{i=1}^{\infty}\epsilon_{i}\beta^{-i}.$ In
this paper we consider the more general case when our sequences are elements of
${0,...,m}^{\mathbb{N}}.$ We show that an analogue of the golden ratio exists
and give an explicit formula for it.
