Definition:P-adic Number

Definition
Let $\left({\Q, \left\vert{\,\cdot\,}\right\vert_p}\right)$ be defined as in P-adic Norm not Complete on Rational Numbers.

The p-adic numbers, denoted $\Q_p$, are the unique complete normed vector space that extends $\left({\Q, \left\vert{\,\cdot\,}\right\vert_p}\right)$

Also see

 * $p$-adic Norm not Complete on Rational Numbers for a proof that $\left({\Q, \left\vert{\,\cdot\,}\right\vert_p}\right)$ is not a complete normed vector space.


 * Completion Theorem for a proof that the completion of $\left({\Q, \left\vert{\,\cdot\,}\right\vert_p}\right)$ exists and is unique.