Let $K$ be a field. The equivalence of valuations $|\cdot|_1$ and $|\cdot|_2$ of $K$ may be defined so that

1. $|\cdot|_1$ is not the trivial valuation;
2. if $|a|_1 < 1$ then $|a|_2 < 1 \qquad \forall a \in K.$

It it easy to see that these conditions imply symmetry for both valuations (use $\frac{1}{a}$ ). Also, we have always $$|a|_1 \leqq 1 \, \Leftrightarrow \, |a|_2 \leqq 1;$$ so both valuations have a common valuation ring in the case they are non-archimedean. (The equivalence of the more general Krull valuations is defined to mean that they have common valuation rings.) Further, both valuations determine a common metric on $K$ .