bijection between unit interval and unit square
The real numbers in the open unit interval can be uniquely represented by their decimal expansions, when these must not end in an infinite string of 9’s. Correspondingly, the elements of the open unit square are represented by the pairs of such decimal expansions.
be such a pair representing an arbitrary point in and let
Then it’s apparent that
But since contains more than one horizontal open segment equally long as (and accordingly there is a natural injection from to ), we must have also
The conclusion is that
Remark 1. Georg Cantor utilised continued fractions for constructing such a bijection between the unit interval and the unit square; cf. e.g. http://www.maa.org/pubs/AMM-March11_Cantor.pdfthis MAA article.
Remark 2. Since the mapping defined by
is bijective, we can conclude that the sets and , i.e. the set of the points of a line and the set of the points of a plane, have the same cardinalities. This common cardinality is .
|Title||bijection between unit interval and unit square|
|Date of creation||2015-02-03 21:45:39|
|Last modified on||2015-02-03 21:45:39|
|Last modified by||pahio (2872)|