\begin{definition}[Definition:P-adic Norm/Rational Numbers/Definition 1]
Let $p \in \N$ be a prime.
Let $\nu_p: \Q \to \Z \cup \set {+\infty}$ be the $p$-adic valuation on $\Q$.
The '''$p$-adic norm''' on $\Q$ is the mapping $\norm {\,\cdot\,}_p: \Q \to \R_{\ge 0}$ defined as:
:$\forall q \in \Q: \norm q_p := \begin{cases}
0 & : q = 0 \\
p^{-\map {\nu_p} q} & : q \ne 0
\end{cases}$
\end{definition}