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

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
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