You have an ink stamp that is so amazingly precise that, when inked and pressed down on the plane, it makes every circle whose radius is an irrational number (centered at the center of the stamp) black.
Is it possible to use the stamp three times and make every point in the plane black?
If it is possible, where would you center the three stamps?
(In reply to
re(2): alternation implies 1-1 by Cory Taylor)
I'm sorry that it wasn't helpful.
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I think some are misreading what DJ wrote. He said:
"...looking at a single radius from a single point, the distances must actually alternate rational and irrational points."
I don't want to put words in his mouth (or in anyone's who read this post), but it seems to imply that one point (along the radius) is rational, and the next point is irrational.... the next point is rational, and so on.
But that's not what that statement means.
What would the "next point" mean? If I center at the origin, what's the next point after (1,0)?
Points are not adjacent.
Cory, if you are clear on the issue of "no smallest positive number", then why don't you think that applies to this situation (namely, that there must be a rational number between any two irrationals and vice versa).
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BTW, I don't see it... where would you put this "missing" not of yours?
re(3): alternation implies 1-1
