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 still thinking
"All rational numbers can be expressed as a fraction. The 8 would be small enough to not impinge on the 8's located at adjacent grid points."
Even though there are many more irrationals than rationals, you can always make the separation between rationals as small as you like--there is no "next rational". If your 8 at rational a/b were any finite size, c (as measured by the distance of its outer part from its center), presumably smaller than 1, you could take 1/(100*ceil(c)) and add it to a/b, and you'd get another rational number (or 1/ceil(c), etc.).
Likewise, irrational numbers can be as close as you like also, and in some intuitive sense they are closer than the rationals, though as we've seen above, the rationals are as close as you'd like also.
Posted by Charlie
on 2005-02-16 03:31:26