Definition:P-adic Number

Definition
Let $\norm {\,\cdot\,}_p$ be the p-adic norm on the rationals $\Q$.

By P-adic Norm is Non-Archimedean Norm then $\norm {\,\cdot\,}_p$ is a non-archimedean norm on $\Q$ and the pair $\struct{\Q, \norm{\,\cdot\,}_p}$ is a valued field.

The p-adic numbers, denoted $\struct{\Q_p, \norm{\,\cdot\,}_p}$, is the unique (up to isometric isomorphism) non-Archimedean valued field that completes $\struct{\Q, \norm{\,\cdot\,}_p}$.

Also see

 * $p$-adic Norm not Complete on Rational Numbers for a proof that $\struct{\Q, \norm{\,\cdot\,}_p}$ is not a complete valued field.


 * Completion Theorem for a proof that the completion of $\struct{\Q, \norm{\,\cdot\,}_p}$ exists and is unique up to isometric isomorphism.