Each Farey sequence starts with the value 0, denominated by the fraction 0/1, and ends with the value 1, denominated by the fraction 1/1 (although some authors omit these terms).

A Farey sequence is sometimes called a Farey series, which is not strictly correct, because the terms are not summed.

Table of contents |

2 History 3 Properties 4 References |

*F*= {0/1, 1/1}_{1}*F*= {0/1, 1/2, 1/1}_{2}*F*= {0/1, 1/3, 1/2, 2/3, 1/1}_{3}*F*= {0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1}_{4}*F*= {0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1}_{5}*F*= {0/1, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 1/1}_{6}*F*= {0/1, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 1/1}_{7}*F*= {0/1, 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8, 1/1}_{8}

*The history of 'Farey series' is very curious*-- Hardy & Wright (1979) Chapter III*... once again the man whose name was given to a mathematical relation was not the original discover so far as the records go.*-- Beiler (1964) Chapter XVI

From this, we can relate the lengths of *F _{n}* and

If *a/b* and *c/d* are neighbours in a Farey sequence, with *a/b* < *c/d*, then their difference *c/d*-*a/b* is equal to 1/*bd*. Since *c/d*-*a/b*=(*bc*-*ad*)/*bd*, this is equivalent to saying that *bc*-*ad*=1.

Thus 1/3 and 2/5 are neighbours in *F _{5}*, and their difference is 1/15.

The converse is also true. If *bc*-*ad*=1 for positive integers *a*,*b*,*c* and *d* with *a*<*b* and *c*<*d* then *a/b* and *c/d* will be neighbours in the Farey sequence of order min(*b,d*).

If *p/q* has neighbours *a/b* and *c/d* in some Farey sequence, with *a/b*<*p/q*<*c/d* then *p/q* is the mediant of *a/b* and *c/d* - in other words, *p/q*=(*a*+*b*)/(*c*+*d*). And if *a/b* and *c/d* are neighbours in a Farey sequence then the first term that appears between them as the order of the Farey sequence is increased is (*a*+*b*)/(*c*+*d*), which first appears in the Farey sequence of order *b*+*d*.

Thus the first term to appear between 1/3 and 2/5 is 3/8, which appears in *F _{8}*.

Fractions that appear as neighbours in a Farey sequence have closely related continued fraction expansions. Every fraction has two continued fraction expansions - in one the final term is 1; in the other the final term is greater than 1. If *p/q*, which first appears in Farey sequence *F _{q}*, has continued fraction expansions

For every fraction *p/q* (in its lowest terms) there is a Ford circle C[*p/q*], which is the circle with radius 1/2*q*^{2} and centre at (*p/q*,1/2*q*^{2}). Two Ford circles for different fractions are either disjoint or they are tangent to one another - two Ford circles never intersect. If 0<*p/q*<1 then the Ford circles that are tangent to C[*p/q*] are precisely the Ford circles for fractions that are neighbours of *p/q* in some Farey sequence.

Thus C[2/5] is tangent to C[1/2], C[1/3], C[3/7], C[3/8] etc.

- Beiler, Albert H. (1964)
*Recreations in the Theory of Numbers*(Second Edition). Dover. ISBN 0486210960 - Hardy, G.H. & Wright, E.M. (1979)
*An Introduction to the Theory of Numbers*(Fifth Edition). Oxford University Press. ISBN 0198531710