# Rational numbers as angles - where do irrationals fit in?

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?

• Say it this way: Lines with irrational slope do not pass through any lattice point in the first quadrant. That is better than saying something I don't understand like "infinite $x,y$ coordinates". – GEdgar Apr 1 '16 at 12:15
• @GEdgar, In other words, would your answer to the title be 'irrational numbers do not fit anywhere on this picture'? – Yuriy S Apr 1 '16 at 12:31