Definition:P-adic Norm

The p-adic metric is a metric on the set of real numbers which yields a different topology than the regular Euclidean metric.

Let $$p \in \left\{{2,3,5,7,11,13,17,...}\right\}$$ 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 $$x\in \mathbb{Q}$$ (the set of rationals) with numerator $$a$$ and denominator $$b$$, define $$ord_p(x)=ord_p(a)-ord_p(b)$$.

Define a map $$|*|_p:\mathbb{Q} \to \mathbb{R}_+$$ as

$$|x|_p = \begin{cases} \tfrac{1}{p^{ord_p(x)}} & \mbox{if }x \neq 0, \\ 0 &\mbox{if }x = 0\end{cases}$$

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

$$|*|_p$$ forms a norm on the real numbers, which induces a metric by

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

The real numbers under this metric is called $$\mathbb{Q}_p$$.