Farey sequence

The n’th Farey sequence is the ascending sequence of all rationals {0ab1:bn}.

The first 5 Farey sequences are

1 01<11
2 01<12<11
3 01<13<12<23<11
4 01<14<13<12<23<34<11
5 01<15<14<13<25<12<35<23<34<45<11

Farey sequences are a singularly useful tool in understanding the convergentsMathworldPlanetmath that appear in continued fractionsMathworldPlanetmath. The convergents for any irrational α can be found: they are precisely the closest number to α on the sequences Fn.

It is also of value to look at the sequences Fn as n grows. If ab and cd are reduced representations of adjacent terms in some Farey sequence Fn (where b,dn), then they are adjacent fractions; their difference is the least possible:


Furthermore, the first fraction to appear between the two in a Farey sequence is a+cb+d, in sequence Fb+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 treesMathworldPlanetmath.

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