Density of rational and irrational numbers Since $\mathbb{Q}$ is countable and $\mathbb{R} \backslash \mathbb {Q}$ is not, what does this tell us about the density of rational and irrational numbers along the real number line? Saying that there exists more irrational numbers than rational numbers seems rather vague becuase we're comparing infinites. How do we even define density here? 

One can have any combination of (countably infinite, uncountable) and (dense in the line, not dense in the line). (Unless, as was suggested in the comments, you have some notion of density other than the one defined by Captain Falcon.)
Countably infinite and dense in the reals: Rationals
Countably infinite and not dense in the reals: Integers
Uncountable and dense in the reals: Irrationals
Uncountable and not dense in the reals: Unit interval
