Farey sequence

Farey diagram to F9 represented with circular arcs. In the SVG image, hover over a curve to highlight it and its terms.
Farey diagram to F9.
Symmetrical pattern made by the denominators of the Farey sequence, F9.
Symmetrical pattern made by the denominators of the Farey sequence, F25.

In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction,[a] which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size.

With the restricted definition, each Farey sequence starts with the value 0, denoted by the fraction ⁠0/1⁠, and ends with the value 1, denoted 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.[2]

  1. ^ Niven, Ivan M.; Zuckerman, Herbert S. (1972). An Introduction to the Theory of Numbers (Third ed.). John Wiley and Sons. Definition 6.1.
  2. ^ Guthery, Scott B. (2011). "1. The Mediant". A Motif of Mathematics: History and Application of the Mediant and the Farey Sequence. Boston: Docent Press. p. 7. ISBN 978-1-4538-1057-6. OCLC 1031694495. Retrieved 28 September 2020.


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