# another proof of cardinality of the rationals

If we have a rational number^{} $p/q$ with $p$ and $q$ having no common factor,
and each expressed in base 10 then we can view $p/q$ as a base 11 integer,
where the digits are $0,1,2,\mathrm{\dots},9$ and $/$. That is, slash ($/$) is a symbol for a
digit. For example, the rational 3/2 corresponds to the integer $3\cdot {11}^{2}+10\cdot 11+2$.
The rational $-3/2$ corresponds to the integer $-(3\cdot {11}^{2}+10\cdot 11+2)$.

This gives a one-to-one map into the
integers so the cardinality of the rationals is at most the cardinality of
the integers. So the rationals are countable^{}.

Title | another proof of cardinality of the rationals |
---|---|

Canonical name | AnotherProofOfCardinalityOfTheRationals |

Date of creation | 2013-03-22 16:01:49 |

Last modified on | 2013-03-22 16:01:49 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 10 |

Author | Mathprof (13753) |

Entry type | Proof |

Classification | msc 03E10 |