# Farey sequence

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

The first 5 Farey sequences are

1 $\frac{0}{1}<\frac{1}{1}$ $\frac{0}{1}<\frac{1}{2}<\frac{1}{1}$ $\frac{0}{1}<\frac{1}{3}<\frac{1}{2}<\frac{2}{3}<\frac{1}{1}$ $\frac{0}{1}<\frac{1}{4}<\frac{1}{3}<\frac{1}{2}<\frac{2}{3}<\frac{3}{4}<\frac{% 1}{1}$ $\frac{0}{1}<\frac{1}{5}<\frac{1}{4}<\frac{1}{3}<\frac{2}{5}<\frac{1}{2}<\frac{% 3}{5}<\frac{2}{3}<\frac{3}{4}<\frac{4}{5}<\frac{1}{1}$

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 $b,d\leq n$), 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 FareySequence 2013-03-22 12:47:16 2013-03-22 12:47:16 ariels (338) ariels (338) 7 ariels (338) Definition msc 11B57 msc 11A55 ContinuedFraction