# Farey pair

Two nonnegative reduced fractions $a/b$ and $c/d$ make a *Farey pair* (with $$) whenever $bc-ad=1$, in other words, they are a Farey pair if their difference is $1/bd$. The interval $[a/b,c/d]$ is known as a *Farey interval.*

Given a Farey pair $a/b,c/d$, their *mediant* is $(a+c)/(b+d)$. The mediant has the following property:

If $\mathrm{[}a\mathrm{,}b\mathrm{,}c\mathrm{/}d\mathrm{]}$ is a Farey interval, then the two subintervals obtained when inserting the mediant are also Farey pairs. Besides, between all fractions that are strictly between $a\mathrm{/}b\mathrm{,}c\mathrm{/}d$, the mediant is the one having the smallest denominator.

Example.

Notice that $3/8$ and $5/11$ form a Farey pair, since
$8\cdot 5-3\cdot 13=40-391$. The mediant here is $8/21$.

Then $3/8$ and $8/21$ form a Farey pair: $8\cdot 8-3\cdot 21=64-63=1$. No fraction between $3/8$ and $5/11$ other than $8/21$ has a denominator smaller or equal than $21$.

Title | Farey pair |
---|---|

Canonical name | FareyPair |

Date of creation | 2013-03-22 14:54:42 |

Last modified on | 2013-03-22 14:54:42 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 6 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 11A55 |

Related topic | ContinuedFraction |

Defines | mediant |

Defines | Farey interval |