closed r.e. sets with respect to finite state machines

Question

I would like to make a definition of regular computational numbers in $R$ (real numbers) as follows:

$x \in R$ is a regular computational number if there is a sequences of rational numbers $\{q_i\}$ that $q_i \rightarrow x$ and $q_i's$ are generated by a finite state machine (say Moore machine for example).

It is obvious that with this definition, regular computational numbers are closed under summation.

Is this set closed under multiplication too?