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If you are confused about why you are here go back to the calling page

The real line has dimension 1 and the plane has dimension 2,
so obviously they are quite different. However, it is
perhaps surprising that as *sets* they are
equivalent
. To show this we must associate each point in the plane
uniquely with a point on the line and vice versa.

In this note I'll show one way of associating points in the
open interval from 0 to 1 with a point in the interior of
the unit square where *0 < x < 1* and *0<
y <1*. Extending the result to the whole plane and
line is a good **exercise.**

be the coordinates of a point the unit square. As usual, to ensure uniqueness, we assume that if any of these numbers could end with an infinite string of zeros or nines, we pick the zeros. The corresponding numberx=0.x1x2x3x4... and y=0.y1y2y3y4...

z=0.x1y1x2y2x3y3x4y4...

Conversely, given a number *z* we use the first,
third, fifth ,... digit to define *x* and the
second, fourth, sixth digit to define *y.*

Clearly this procedure establishes the desired
correspondence. Moreover, the procedure can be extended in
an obvious manner to the cube or a higher dimensional
equivalent. Indeed, the real line considered as just a set
is equivalent to *n* -dimensional space for all
finite values of *n*.

Fine print, your comments, more links, Peter Alfeld, PA1UM

[16-Aug-1996]