Q and R (reals and rationals) are not isomorphic because some elements in R are not division rings correct? (such as pi)?
Unless I am much mistake, an element cannot be a division ring. It makes no sense!
Assume they are isomorphic, and let $\displaystyle \theta: \mathbb{R} \rightarrow \mathbb{Q}$. Where is $\displaystyle 2$ mapped to, and where is $\displaystyle \sqrt{2}$ mapped to?
(I am assuming you mean not isomorphic as rings, and not as groups?)