# Farey Sequence

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## Farey sequence

[′far·ē ‚sē·kwəns] (mathematics)

The Farey sequence of order

*n*is the increasing sequence, from 0 to 1, of fractions whose denominator is equal to or less that*n*, with each fraction expressed in lowest terms.McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

The following article is from

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.## Farey Sequence

The Farey sequence of order *n* is the increasing sequence consisting of the fractions 0/1 and 1/1 and all the irreducible proper fractions whose numerator and denominator are greater than 0 and do not exceed *n*. For example, 0/1,1/3, 1/2, 2/3,1/1 is the Farey sequence of order 3.

If *a/b* and *a’/b’* are two consecutive terms in a Farey sequence, then *a’b – ab’* = 1. If *a/b, a’/b’*, and *a”/b”* are three consecutive terms in a Farey sequence, then *a’/b’ = (a + a”)/(b + b”*). The uses of Farey sequences include the approximation of irrational numbers by rational numbers and the reduction of binary quadratic forms.

The Farey sequence is named for the British scientist J. Farey, who reported some of its properties without proof in 1816.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.