I think this might help: http://en.wikipedia.org/wiki/Continued_fraction
Basically what we do is to correspond each rational number canonically to a continued fraction.(see the wiki link). We can denote $Q_n$ the rational number that can be represented canonically by a continued fraction of length $n$. The restriction map will then give a projection from $Q$ to $Q_n$. Real number is then the inverse limit of this system. That is because an irrational number can be expressed uniquely as a continued fraction.
This is the best we can perhaps hope for since as I mention in the comment above the main actor here is not the arithmetic rule but the ordering. This is consistent with the way the set theorist view real number as $ \omega^ \omega$ as well.