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.
 A: Easiest solution is to just say
$$
x\in\mathbb{R}^+
$$
Expresses both conditions in one hit.
A: There are ordered fields which strictly extend the real numbers, there $x>0$ is meaningful, but need not imply $x\in\Bbb R$. 
A: Of course, everything depends on context. I usually prefer to say things like

For any $x\in\mathbb{R}$ with $x>0$, ...

instead of

For any $x>0$, ...

to remove ambiguity, but I'm not insistent on it; I might be willing to sacrifice the "$x\in\mathbb{R}$" if it's making an orphan at the end of a paragraph, for example.
In contrast, everyone knows what

For any $\epsilon>0$, ...

almost always means, and it doesn't really add anything to say $\epsilon\in\mathbb{R}$. Of course, if you want to write $\epsilon>0$ and $\epsilon$ is not an element of $\mathbb{R}$, then it is all the more incumbent upon you to warn the reader of this non-standard use.
A: My favorite would be "$x$ is a positive real number". Simple, clear, unambiguous, and no strange symbols to decipher.
It's a bit long, maybe, but who cares -- the point of writing is clarity, not brevity.
A: It really depends on context. But be safe; just say $x > 0, x\in \mathbb R$. 
Omitting the clarification can lead to misunderstanding it. Including the clarification takes up less than a centimeter of space. Benefits of clarifying the domain greatly outweigh the consequences of omitting the clarification. 
Besides one might want to know about rationals greater than $0$, or integers greater than $0$, and we would like to use $x \gt 0$ in those contexts, as well. 
ADDED NOTE: That doesn't mean that after having clarified the context, and/or defined the domain, you should still use the qualification "$x\in \mathbb R$" every time you subsequently write $x \gt 0$, in a proof, for example. But if there's any question in your mind about whether or not to include it, error on the side of inclusion.
A: 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.
A: One might be able to decode that notation, but why force this puzzle on the reader?  
