Irrational P-adics $\mathbb{Q}_p$ is completion of $\mathbb{Q}$ by defining a new metric. So, with respect to this new metric they are complete.
I just want to be sure, are there p-adic rationals? If there are P-adic irrationals, what sense they make. I mean, how are they constructed?
Edit My question was if P-adic irrationals exist. I made some type rendering a little confusing question. Sorry. I did not intend to ask if p-adic rationals are existant. 
Thank You.

The rationals with a given metric embed in the completion with respect to that metric (this is true of any metric space of course). So, given a $p$-adic metric on $\mathbb Q$, we have that $Q$ embeds in the completion. The completion is the set of $p$-adic numbers. So, we may identify the rationals in the $p$-adic numbers and thus consider elements in the completion that are not in the rationals to be irrational $p$-adic numbers. The existence of such irrationals is immediate: The set of rationals is countable, but the completion is uncountable.
