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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.Let
x=0.x1x2x3x4... and y=0.y1y2y3y4...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 number 0 < z < 1 is
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