# Farey sequence

The $n$’th *Farey sequence* is the ascending sequence of all rationals
$\{0\le \frac{a}{b}\le 1:b\le n\}$.

The first 5 Farey sequences are

1 | $$ |
---|---|

2 | $$ |

3 | $$ |

4 | $$ |

5 | $$ |

Farey sequences are a singularly useful tool in understanding the convergents^{} that appear in continued fractions^{}. The convergents for any irrational $\alpha $ can be found: they are precisely the closest number to $\alpha $ on the sequences ${F}_{n}$.

It is also of value to look at the sequences ${F}_{n}$ as $n$ grows. If $\frac{a}{b}$ and $\frac{c}{d}$ are reduced representations of adjacent terms in some Farey sequence ${F}_{n}$ (where $b,d\le n$), then they are adjacent fractions; their difference is the least possible:

$$\left|\frac{a}{b}-\frac{c}{d}\right|=\frac{1}{bd}.$$ |

Furthermore, the *first* fraction to appear between the two in a Farey sequence is $\frac{a+c}{b+d}$, in sequence ${F}_{b+d}$, and (as written here) this fraction is already reduced.

An alternate view of the “dynamics” of how Farey sequences develop is given by Stern-Brocot trees^{}.

Title | Farey sequence |
---|---|

Canonical name | FareySequence |

Date of creation | 2013-03-22 12:47:16 |

Last modified on | 2013-03-22 12:47:16 |

Owner | ariels (338) |

Last modified by | ariels (338) |

Numerical id | 7 |

Author | ariels (338) |

Entry type | Definition |

Classification | msc 11B57 |

Classification | msc 11A55 |

Related topic | ContinuedFraction |