# bijection between unit interval and unit square

The real numbers in the open unit interval $I=(0,\mathrm{\hspace{0.17em}1})$ 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 $I\times I$ are represented by the pairs of such decimal expansions.

Let

$$P:=(0.{x}_{1}{x}_{2}{x}_{3}\mathrm{\dots},\mathrm{\hspace{0.17em}0}.{y}_{1}{y}_{2}{y}_{3}\mathrm{\dots})$$ |

be such a pair representing an arbitrary point in $I\times I$ and let

$$p:=\mathrm{\hspace{0.33em}0}.{x}_{1}{y}_{1}{x}_{2}{y}_{2}{x}_{3}{y}_{3}\mathrm{\dots}$$ |

Then it’s apparent that

$P\mapsto p$ | (1) |

is an injective^{} mapping from $I\times I$ to $I$. Thus

$$|I\times I|\le |I|.$$ |

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

$$|I\times I|\ge |I|.$$ |

The conclusion^{} is that

$$|I\times I|=|I|,$$ |

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

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 $g:I\to \mathbb{R}$ defined by

$$g(x)=\mathrm{tan}\left(\pi x-\frac{\pi}{2}\right)$$ |

is bijective^{}, we can conclude that the sets $\mathbb{R}$ and $\mathbb{R}\times \mathbb{R}$, 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 ${2}^{{\mathrm{\aleph}}_{0}}$.

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 |