If we make a rectangular grid with integer coordinates, it's possible to assign a unique angle to any rational number, using the definition $\tan \phi=y/x$ for $\phi \in (-\pi/2, \pi/2)$.
For positive rationals it would look something like this:
It's obvious that the lines corresponding to the rational numbers can't fill all the space here (the gaps are especially noticeable around the numbers with small denominators/numerators).
Is it correct to say that all the angles corresponding to the irrational numbers have infinite $y,x$ coordinates? Does it fit with the usual constructions of the real numbers as limits of Cauchy sequences or Dedekind cuts?