Editing Farey sequence (section)

{{Short description|Increasing sequence of reduced fractions}}
[[Image:Farey diagram horizontal arc 9.svg|thumb|300px|link={{filepath:Farey diagram horizontal arc 9.svg}}|Farey diagram to ''F''<sub>9</sub> represented with circular arcs. In [[Media:Farey diagram horizontal arc 9.svg|the SVG image]], hover over a curve to highlight it and its terms.]]
{{multiple image
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 | image1 = Farey diagram square 9.svg|caption1=Farey diagram to ''F''<sub>9</sub>.
 | image2 = Farey sequence denominators 9.svg|caption2=Symmetrical pattern made by the denominators of the Farey sequence, ''F''<sub>9</sub>.
 | image3 = Farey sequence denominators 25.svg|caption3=Symmetrical pattern made by the denominators of the Farey sequence, ''F''<sub>25</sub>.
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In [[mathematics]], the '''Farey sequence''' of order ''n'' is the [[sequence]] of completely reduced [[fraction]]s, either between 0 and 1, or without this restriction,{{efn|“''The sequence of all reduced fractions with denominators not exceeding n, listed in order of their size, is called the Farey sequence of order n.''” With the comment: “''This definition of the Farey sequences seems to be the most convenient. However, some authors prefer to restrict the fractions to the interval from 0 to 1.''” — Niven & Zuckerman (1972)<ref>{{cite book |author1-link=Ivan M. Niven |first1=Ivan M. |last1=Niven |first2=Herbert S. |last2=Zuckerman |title=An Introduction to the Theory of Numbers |edition=Third |publisher=John Wiley and Sons |year=1972 |at=Definition&nbsp;6.1}}</ref>}} which have [[denominator]]s 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 {{sfrac|0|1}}, and ends with the value 1, denoted by the fraction {{sfrac|1}} (although some authors omit these terms).

A ''Farey sequence'' is sometimes called a Farey [[series (mathematics)|''series'']], which is not strictly correct, because the terms are not summed.<ref>{{cite book|last1=Guthery |first1=Scott B. |year=2011 |title=A Motif of Mathematics: History and Application of the Mediant and the Farey Sequence |chapter=1. The Mediant |page=7 |publisher=Docent Press |location=Boston |language=en |isbn=978-1-4538-1057-6 |oclc=1031694495 |chapter-url=https://books.google.com/books?id=swb2c9enRJcC&pg=PA7 |access-date=28 September 2020}}</ref>