When we have already studied the euclidean norm and the p-adic norms, we might wonder if it is possible to find other norms on the rational numbers.
What can turn out to be surprising is that these are the only norms on the rational numbers. In fact, any other norm is equivalent to one of these, in which equivalent refers that there is a real number $$s$$ so that the s-th power of this norm corresponds to the euclidean norm or a p-adic norm.
It is interesting to remember the idea of unicity forgetting this equivalence concept. The formalization of these concepts would take us to the theory of evaluations which does not make sense in our context. And so, we can finish our study assuming that the only norms on the rational numbers are the Euclidean norm and the p-adic norms.