The field of rational numbers has no other non-equivalent valuations than

• the trivial valuation,
• the absolute value, i.e. the complex modulus $|\cdot|_\infty$ and
• the $p$ -adic valuations $|\cdot|_p$ when $p$ goes through all positive primes.

Note. Any valuation $|\cdot|$ of the field $\mathbb{Q}$ defines a metric $d(x,\,y) = |x-y|$ in the field, but $\mathbb{Q}$ is complete only with respect to (the ``trivial metric'' defined by) the trivial valuation. The field has the proper completions with respect to its other valuations: the field of reals $\mathbb{R}$ and the fields $\mathbb{Q}_p$ of $p$ -adic numbers; cf. also $p$ -adic canonical form.