Does $\,x>0\,$ hint that $\,x\in\mathbb R\,$? 
Does $x>0$ suggest that $x\in\mathbb R$?

For numbers not in $\,\mathbb R\,$ (e.g. in $\mathbb C\setminus \mathbb R$), their sizes can't be compared. 
So can I omit "$\,x\in\mathbb R\,$" and just write $\,x>0\,$?
Thank you.

It's probably best to say something like "there is a real number $x > 0$ such that...," or "let $x > 0$ be a real number," etc. depending on what you want to say about $x$.  This avoids ambiguity, and also using words to describe the new symbol being introduced makes life easier for the reader.  The goal of writing (even mathematical writing) should not be to avoid redundancy.
