\begin{definition}[Definition:P-adic Norm/Rational Numbers/Definition 2]
Let $p \in \N$ be a prime.
The '''$p$-adic norm''' on $\Q$ is the mapping $\norm {\,\cdot\,}_p: \Q \to \R_{\ge 0}$ defined as:
:$\forall r \in \Q: \norm r_p = \begin {cases} 0 & : r = 0 \\ \dfrac 1 {p^k} & : r = p^k \dfrac m n: k, m, n \in \Z, p \nmid m, n \end {cases}$
\end{definition}