A real number \(\theta > 1\) is said to be a beta-number if the orbit of \(x = 1\) under the \(\theta\)-transformation, \(T_ \theta : x \to \theta x \text{mod} 1\) is a finite set. It is a simple beta-number if the orbit eventually contains \(x = 0\). Beta numbers are algebraic integers and the location of their Galois conjugates (other than \(\theta\) itself) is of interest. Let \(\Phi\) be the closure of the set of conjugates of all beta- numbers and \(\Phi_ 0\) be the closure of the set of all conjugates of simple beta-numbers. Let \({\mathcal G}\) be the set of zeros \(\lambda\) with \(| \lambda | 1\) whose other conjugates lie in \(\Phi\).