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.)
(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