\begin{definition}[Definition:Canonical P-adic Expansion]
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for prime number $p$.
Let $a$ be a $p$-adic number, that is left coset, in $\Q_p$.
Let $\ds \sum_{n \mathop = m}^\infty d_n p^n$ be the unique $p$-adic expansion that is a representative of $a$ where:
:$m \in \Z_{\le 0}$
:$\forall n \in \Z_{\ge m}: d_n$ is a $p$-adic digit
:$m < 0 \implies d_m \ne 0$
For $m < 0$, the '''canonical $p$-adic expansion of $a$''' is the expression:
:$\ldots d_n \ldots d_3 d_2 d_1 d_0 . d_{-1} d_{-2} \ldots d_m$
with infinitely many $p$-adic digits before the point and finitely many $p$-adic digits after the point.
In the case that $m = 0$, the '''canonical $p$-adic expansion of $a$''' is the expression:
:$\ldots d_n \ldots d_3 d_2 d_1 d_0$
with infinitely many $p$-adic digits to the left and no point or $p$-adic digits after the point.
\end{definition}