p-adic number

Remarks

Every $x \in \Q_p$ with $x \neq 0$ has a unique expansion $x = \sum_{n \ge m} a_n p^n$ with $m \in \Z$, digits $a_n \in \set{0, 1, \ldots, p - 1}$ and $a_m \neq 0$. Then $\abs{x}_p = p^{-m}$. Example: $-1 = \sum_{n \ge 0} (p - 1) p^n$ in $\Q_p$. Check: the partial sum up to $p^{N-1}$ is $p^N - 1$, and $\abs{(p^N - 1) - (-1)}_p = p^{-N}$ tends to $0$. Non-example: the formal sum $\sum_{n \ge 0} p^{-n}$ is not a $p$-adic number, because its terms have $\abs{p^{-n}}_p = p^n$ and do not tend to $0$. The elements with $\abs{x}_p \le 1$ are the p-adic integers $\Z_p$. By Ostrowski's theorem, the completions of $\Q$ are $\R$ and the fields $\Q_p$.