PlanetMath: Farey sequence

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Farey sequence

(Definition)

The '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	$\frac{0}{1} < \frac{1}{1}$
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 .

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

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

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.

"Farey sequence" is owned by ariels.

(view preamble)

See Also: continued fraction

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: Stern-Brocot trees, fraction, difference, adjacent fractions, terms, adjacent, representations, reduced, number, irrational, continued fractions, convergents, rationals, sequence
There are 3 references to this entry.

This is version 4 of Farey sequence, born on 2002-06-13, modified 2002-06-24.
Object id is 3102, canonical name is FareySequence.
Accessed 5597 times total.

Classification:

AMS MSC:	11B57 (Number theory :: Sequences and sets :: Farey sequences; the sequences $$)
	11A55 (Number theory :: Elementary number theory :: Continued fractions)

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

No messages.

Interact