Let $\displaystyle p,q$ be two prime numbers. Define $\displaystyle d:\mathbb{Q} \times \mathbb{Q} \to [0,\infty)$ by $\displaystyle d(x,y) = \max\{|x-y|_p,|x-y|_q\}$. Is there an algebraic analogue to the completion of the rationals by this metric similar to the inverse limit of $\displaystyle \Bbb{Z}/p^n\Bbb{Z}$ for the completion of the rationals by a single p-adic metric? If not, is there a way to combine the p-adic and q-adic metrics that is more algebraically intuitive?