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ehilge
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Homework Statement
The goal of the problem is to eliminate all of the points on a plane. This is done by picking a point, and then all of the points of a radius that are not an irrational number (cannot be expressed by a/b, like pi) are eliminated. So for a single point, a set of concentric rings all at radii of irrational numbers would remain. I need to figure out how many points would be selected to eliminate all other points on the entire plane and where those points would be located at.
Homework Equations
none that I know of
The Attempt at a Solution
You would need at least three points (which I will call a, b and c). If there were only two points (a and b), then all of the points on the plane would be eliminated except for the points where the radius away from both a and b is an irrational number. Now, with the introduction of point c, all of the points would be eliminated except for the points where the radius to each point a, b and c, is an irrational number. I'm thinking that this would take care of most of the points but I can't think of a way to ensure that all points on the plane are a rational distance from at least one of the points a, b, or c. Adding more points (d, e, ...) would increase the odds of eliminating every point, but I don't think would ensure that all points were eliminated. So, That means, there either has to be an infinite number of points selected or that the points were infinitely far away from each other, such that the distance between them would be twice the distance of the largest irrational number. But neither of these solutions seem to make much sense to me in the context of the problem.
So, Does anyone have an idea of how the orientation of these points would have to be to eliminate the entire plane? Any thoughts would be greatly appreciated.
Thanks!
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