\begin{definition}[Definition:P-adic Expansion]
Let $p$ be a prime number.
A '''$p$-adic expansion''' is a series in the rational numbers $\Q$ of the form:
:$\ds \sum_{n \mathop = m}^\infty d_n p^n$
where:
:$m \in \Z_{\le 0}$
:$\forall n \in \Z_{\ge m}: d_n$ is a $p$-adic digit
:$m < 0 \implies d_m \ne 0$
\end{definition}