## Definition

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
$m < 0 \implies d_m \ne 0$