Definition For \(b\) an integer greater than \(1\), \(w\) a positive integer, and \(n\) a nonnegative integer \(\underline{\phantom{\hspace{1in}}}\),   the base \(b\) fixed-width \(w\) expansion of \(n\) is \[(a_{w-1} \cdots a_1 a_0)_{b,w}\] where \(a_0, a_1, \ldots, a_{w-1}\) are nonnegative integers less than \(b\) and \[n = \sum_{i=0}^{w-1} a_{i} b^{i}\]