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:

enter image description here

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?

  • 3
    $\begingroup$ 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". $\endgroup$ – GEdgar Apr 1 '16 at 12:15
  • $\begingroup$ @GEdgar, In other words, would your answer to the title be 'irrational numbers do not fit anywhere on this picture'? $\endgroup$ – Yuriy S Apr 1 '16 at 12:31

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