Definition:P-adic Norm

Definition
The p-adic norm is a norm on the set of real numbers which yields a different topology from the regular Euclidean Metric.

Let $p \in \left\{{2,3,5,7,11,13,17,\ldots}\right\}$ be any prime number.

For any nonzero integer $a$, define $ord_p(a)$ to be the highest power of $p$ which divides $a$.

For any $x \in \Q$ (the set of rationals) with numerator $a$ and denominator $b$, define $ord_p(x)=ord_p(a)-ord_p(b)$.

Define a map $|*|_p:\Q \to \R_+$ as


 * $|x|_p = \begin{cases} \dfrac 1 {p^{ord_p(x)}} & : x \neq 0 \\ 0 & : x = 0\end{cases}$

For any real number $x$ which is the limit of the Cauchy sequence $\left\{{x_1,x_2,\ldots}\right\}$, define $|x|_p = \lim_{n \to \infty} |x_n|_p$.

$|*|_p$ forms a norm on the real numbers, as is proved in P-adic Norm is a Norm which induces a metric by


 * $d(x,y) = |x-y|_p$

The real numbers under this metric is called $\Q_p$.