Definition:P-adic Norm/Rational Numbers/Definition 2

Definition
Let $p \in \N$ be a prime. Let $k, m, n \in \Z : p \nmid m, n$.

Let $\displaystyle r := p^k \frac m n$.

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 \\ p^{-k} & : r \ne 0 \end{cases}$

Also see

 * Equivalence of Definitions of P-adic Norms