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 Figure Eights (Posted on 2005-02-15)
Suppose that I drew an infinite number of disjoint closed curves in the plane (such as circles, squares, etc.). Suppose that I then tell you that there is one curve for each positive real number.

You would not have too much trouble believing my assertions at this point. For example, I could have drawn all circles with center at the origin. They are all disjoint, and for each positive real number x, there is a corresponding circle - namely, the circle of radius x.

But suppose that I also tell you that all the curves I drew were figure eights. Can you believe my assertions now?

(A figure eight is a curve in the plane obtained from the basic "8" shape by any combination of translation, rotation, expansion, or shrinking.)

 See The Solution Submitted by David Shin Rating: 4.2000 (5 votes)

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 re(4): still thinking / cardinality | Comment 18 of 34 |
(In reply to re(3): still thinking by David Shin)

If there were a way to map each irrational number to one and only one rational number, then we could make an 8 representing each irrational number and hide it inside the 8 of its unique mapped rational number.

 Posted by Larry on 2005-02-17 04:42:35

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