Imagine you have a magic hole puncher. When you punch a point (x,y) on the Cartesian plane, every point that is a transcendental distance away from (x,y) is punched out. What is the minimum amount of points that you need to punch with your hole puncher to knock out the entire plane?
Prove that you can knock out every point on the plane with that number of punches and that this cannot be accomplished with any lower number of punches.

Note: If you make a punch at (x,y), (x,y) is not eliminated because it is a 0 distance from (x,y) which is an algebraic distance.