Definition:Farey Sequence

Definition

A Farey sequence is a chain of subsets of the reduced rational numbers lying in $\Q \cap \closedint 0 1$.

For $Q \in \Z_{>0}$, the Farey set $F_Q$ is the set of all reduced rational numbers with denominators not larger than $Q$:

$F_Q = \set {\dfrac p q: p = 0, \ldots, Q,\ q = 1, \ldots, Q,\ p \perp q}$

where $p \perp q$ denotes that $p$ and $q$ are coprime.

Let $F_Q$ denote the Farey sequence of all reduced rational numbers with denominators not larger than $Q$:

$F_Q = \set {\dfrac p q: p = 0, \ldots, Q,\ q = 1, \ldots, Q,\ p \perp q}$

The index $Q$ is called the order of $F_Q$.

$\dfrac 0 1, \dfrac 1 5, \dfrac 1 4, \dfrac 1 3, \dfrac 2 5, \dfrac 1 2, \dfrac 3 5, \dfrac 2 3, \dfrac 3 4, \dfrac 4 5, \dfrac 1 1$

This entry was named for John Farey.

The Farey sequence was first exploited by Charles Haros in $1801$ in creating tables of the decimal expansions for all vulgar fractions whose denominators are less than $100$.

In order to make sure he captured them all, he used a technique exploiting the properties of the mediant that originated with Nicolas Chuquet.

Some $15$ years later, John Farey rediscovered this property, and published a paper on the subject.

This was subsequently picked up on by Augustin Louis Cauchy, who reproved the results of Charles Haros while crediting John Farey with the technique.

1801: Charles Haros: Tables pour évaluer une fraction ordinaire avec autant de decimals qu'on voudra; et pour trouver la fraction ordinaire la plus simple, et qui approche sensiblement d'une fraction décimale (J. l'École Polytechnique Vol. 6, no. 11: pp. 364 – 368)

1816: John Farey: On a Curious Property of Vulgar Fractions (Phil. Mag. Vol. 47, no. 3: pp. 385 – 386)