system
Given the following post, respond as if you're commenting on Reddit:
human
Is it possible to form any real number angle in a 2d grid system?

Say you have a grid system (2d) with an x an y axis. Is it possible to form any angle with a vector, measuring the angle between the x-axis and the vector? I was wondering about it the other day when discussing complex numbers.
gpt
If you mean that the vector must have integer components, then no, you can't form every real number angle. There will be a countable number of such vectors, and as such there will be at most a countable number of angles, while the total number of angles in a circle is uncountable.

The same argument holds if you let the vector components be rationals.

The consequence is kind of amazing. There is a straight line through the origin that does not hit a rational point anywhere (except the origin, obviously).