# Farey sequence

A Farey sequence is a sequence of numbers named after the English geologist John Farey (1766–1826)
who wrote about such sequences in an article called "On a curious property
of vulgar fractions" in the *Philosophical Magazine* in 1816. Farey
says that he noted the "curious property" while examining the tables of *Complete decimal quotients* produced by Henry Goodwin. To obtain the
Farey sequence for a fixed number *n*, consider all rational
numbers between 0 and 1 which, when expressed in their lowest terms,
have denominator (the number on the bottom of a fraction) not exceeding *n*. Write the sequence in ascending order of magnitude beginning with
the smallest. The "curious property" is that each member of the sequence
is equal to the rational whose numerator (the number on top of a fraction)
is the sum of the numerators of the fractions on either side, and whose
denominator is the sum of the denominators of the fractions on either side.
For example, the Farey sequence for *n* = 5 is (0/1, 1/5, 1/4, 1/3,
2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1), from which it can be seen that 2/5 =
(1 + 1)/(3 + 2), 1/3 = (1 + 2)/(4 + 5), 1/2 = (2 + 3)/(5 + 5), 2/3 = (3 + 3)/(5 + 4), and
so forth. Farey wrote:

I am not acquainted whether this curious property of vulgar fractions has been before pointed out?; or whether it may admit of some easy or general demonstration?; which are points on which I should be glad to learn the sentiments of some of your mathematical readers ...

One "mathematical reader" was Augustin Cauchy,
who gave the necessary proof in his *Exercises de mathématique*, published
in the same year as Farey's article. Farey was not the first to notice the
property. C. Haros, in 1802, wrote a paper on the approximation of decimal
fractions by common fractions. He explains how to construct what is in fact
the Farey sequence for *n* = 99 and Farey's "curious property" is built
into his construction.