# 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.

## 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.

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If one were to ask what area the Ford circles cover, one would need to think about whether every rational number is in a Farey sequence and how many fractions there are that have each possible denominator.
The relation mr - np = 1 calls to mind a property of Farey sequences, named after but not first discovered by a geologist, John Farey.
In algebra, a Farey sequence is the sequence of all fractions between 0 and 1 in which both the numerator and denominator are nonnegative and have no common divisors other than 1.
Each lesson centers on a key mathematical concept or application, with subjects including algebraic expressions and sequences, codes based on a simple fold, Benford's law, roots for divisibility tests and properties of numbers, the Farey sequences of order for fractions, interpretation of graphs, possibility tests and factors, estimation scales and units, symmetry, coordinates, averages and range, constructions, measurements, proofs, transformations, and sophisticated geometric forms.

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