I see this a lot in astrophysics and geophysics papers where someone wants to write that some parameter is of some order of magnitude. So say for example if some nondimensional number was in region of $10^3$ then they will write $R = O(10^3)$.
As far as I can tell by the definition of big oh notation this is wrong as $O(1) = O(10^3) = O(c)$ for any real number $c$.
Am I correct that this is not the correct use of this notation? What should be written instead? Also is this kind of fudging notation accepted or do mathematicians frown on this?