\begin{definition}[Definition:P-adic Number]
Let $p$ be any prime number.
Let $\norm {\,\cdot\,}_p$ be the p-adic norm on the rationals $\Q$.
Let $\CC$ be the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$.
Let $\NN$ be the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.
Let $\Q_p$ denote the quotient ring $\CC \, \big / \NN$.
Let $\norm {\, \cdot \,}_p:\Q_p \to \R_{\ge 0}$ be the norm on the quotient ring $\Q_p$ defined by:
:$\ds \forall \sequence {x_n} + \NN: \norm {\sequence {x_n} + \NN }_p = \lim_{n \mathop \to \infty} \norm{x_n}_p$
Each left coset $\sequence {x_n} + \NN \in \CC \, \big / \NN$ is called a '''$p$-adic number'''.
The '''p-adic numbers''' is the valued field $\struct {\Q_p, \norm {\,\cdot\,}_p}$.
\end{definition}