## You are here

Homebijection between unit interval and unit square

## Primary tabs

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 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

$P\;:=\;(0.x_{1}x_{2}x_{3}\ldots,\,0.y_{1}y_{2}y_{3}\ldots)$ |

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

$p\;:=\;0.x_{1}y_{1}x_{2}y_{2}x_{3}y_{3}\ldots$ |

Then it’s apparent that

$\displaystyle P\mapsto p$ | (1) |

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

$|I\!\times\!I|\;\leq\;|I|.$ |

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

$|I\!\times\!I|\;\geq\;|I|.$ |

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. this MAA article.

Remark 2. Since the mapping $g\!:\,I\to\mathbb{R}$ defined by

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

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 $2^{{\aleph_{0}}}$.

## Mathematics Subject Classification

03E10*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections