bijection between unit interval and unit square

The real numbers in the open unit interval  I=(0, 1)  can be uniquely represented by their decimal expansions, when these must not end in an infiniteMathworldPlanetmath string of 9’s.  Correspondingly, the elements of the open unit square I×I are represented by the pairs of such decimal expansions.


P:=(0.x1x2x3, 0.y1y2y3)

be such a pair representing an arbitrary point in I×I and let

p:= 0.x1y1x2y2x3y3

Then it’s apparent that

Pp (1)

is an injectivePlanetmathPlanetmath mapping from I×I to I.  Thus


But since I×I contains more than one horizontal open segment equally long as I (and accordingly there is a natural injection from I to I×I), we must have also


The conclusionMathworldPlanetmath is that


i.e. that the sets I×I and I have equal cardinalities, and the Schröder-Bernstein theorem even garantees a bijection between the sets.

Remark 1.  Georg Cantor utilised continued fractionsDlmfMathworldPlanetmath for constructing such a bijection between the unit interval and the unit square; cf. e.g. MAA article.

Remark 2.  Since the mapping  g:I  defined by


is bijectiveMathworldPlanetmath, 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 20.

Title bijection between unit interval and unit square
Canonical name BijectionBetweenUnitIntervalAndUnitSquare
Date of creation 2015-02-03 21:45:39
Last modified on 2015-02-03 21:45:39
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 18
Author pahio (2872)
Entry type Result
Classification msc 03E10
Related topic JuliusKonig
Related topic BijectionBetweenClosedAndOpenInterval