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 additional thoughts
Suppose you make the 8's for integers to be located along the number line crossing the line at their respective integer values, and they're small enough so they don't cross each other. Then place smaller 8's within these 8's to represent fractional parts, in the manner described by Larry, to expand the decimal fraction.
Minor difficulties can be handled, such as the need for an outside 8 for each of the ranges (0,1) and (-1,0) as both ranges have a zero integer part.
But numbers such as pi have an infinite expansion. Where would it's 8 be? Actually, it would have a location, in the same sense that the circle example has circles of definite radius, but no size, so it would be a point, rather than a figure-8.
Posted by Charlie
on 2005-02-15 19:04:45