The Background: I was thinking about ways to conceptualize the way rational and irrational numbers interact within the real line while helping somebody in an elementary real analysis class. It came to my mind that one way of thinking about this was to represent an irrational number x as:
$r \in \mathbb{Q}$ $i \in \mathbb{R} - \mathbb{Q}$ $r + i = x$
As a notational simplification from this point let $\mathbb{R} - \mathbb{Q} := \mathbb{I}$ and also: $\mathbb{C} - \mathbb{R} = \hat{\mathbb{C}}$ I then noticed that there were some similarities (and differences between this set under real-number addition and multiplication and C when the real line was deleted) mainly:
$\forall a,b \in \mathbb{I}$ $\forall z,w \in \hat{\mathbb{C}}$
$a + b = r_a + r_b +i_a + i_b = r_c + i_c = c \in \mathbb{R}$ similar to: $z + w = \mathcal{Re}(z) +\mathcal{Re}(w) + i(\mathcal{Im}(z) +\mathcal{Im}(w)) = u \in \mathbb{C}$ and $ab = (r_a + i_a)(r_b + i_b) = r_ar_b + r_ai_b+ i_ar_b + i_ai_b = r_c + i_c \in \mathbb{R}$ similar to: $zw = \mathcal{Re}(z)\mathcal{Re}(w) - \mathcal{Im}(z)\mathcal{Im}(w) + i(\mathcal{Re}(z)\mathcal{Im}(w) + \mathcal{Re}(w)\mathcal{Im}(z)) = u \in {\mathbb{C}}$
The primary difference I noticed between this and complex number arithmetic was that the value $i_ai_b$ could be rational or irrational, and the sign change.
It also seemed to me that there would be a way to compensate for these differences in some homomorphism/isomorphism between $\hat{\mathbb{C}}$ and $\mathbb{I}$ or alternatively maybe just one quadrant of $\hat{\mathbb{C}}$
I am aware that multiplication and addition lack closure (that is the product/sum of two rationals can be irrational, and the product/sum of two complex numbers can be real), and that there is also no additive or multiplicative identity present in either set (which is why I think such a homomorphism/isomorphism may exist in the first place). I figure if such an homomorphism/isomorphism does exist then it will make explaning the lack of these properties in $\mathbb{I}$ easier when talking to student with some experience with complex algebra, but little analysis background. (as well as provide an interesting little tidbit to toss around)
The Question: Has anyone seen a construction that of a homomorphism/isomorphism between $\mathbb{C - R}$ and $\mathbb{R - Q}$, or can anybody point me in the right direction for developing one? Alternatively is there a counter-argument that disproves this possibility that I'm missing somewhere?
I'm also looking for more of a conceptual notion of homomorphism/isomorphism than a particular one (since these sets don't have nice algebraic properties, or the identity elements that make such a construction a homomorphism in traditional definitions).