The first 5 Farey sequences are
Farey sequences are a singularly useful tool in understanding the convergents that appear in continued fractions. The convergents for any irrational can be found: they are precisely the closest number to on the sequences .
It is also of value to look at the sequences as grows. If and are reduced representations of adjacent terms in some Farey sequence (where ), 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 , in sequence , 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.
|Date of creation||2013-03-22 12:47:16|
|Last modified on||2013-03-22 12:47:16|
|Last modified by||ariels (338)|