##  [Farey Sequence](/farey-sequence-0) 

 Definition

The ascending sequence of reduced rational numbers between 0 and 1 whose denominators do not exceed a given positive integer N, arranged in increasing order and including 0/1 and 1/1.

 

 

 

 

 

 





## Principle

Principle

Adjacent terms a/b and c/d in the Farey sequence of order N satisfy bc − ad = 1; mediants and the Farey addition rule generate intermediate fractions under denominator bounds.

 

 

 

 

 





## Demonstration

Demonstration

For N = 5 the Farey sequence F5 is 0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1; adjacent pairs satisfy the determinant condition and mediants with denominator ≤5 appear between neighbors.

 

 

 

 

## Misapplication

Misapplication

Assuming that the mediant of any two fractions always belongs to the same Farey sequence without checking the denominator bound or adjacency can produce a fraction not in the sequence.

 

 

 

 

 





## Consequence

Consequence

Farey sequences organize rationals by denominator complexity, yield identities used in modular and analytic number theory, and connect to geometric constructions (e.g., Ford circles) and continued fractions.

 

 

 

 

## Reversal

Reversal

The Stern–Brocot tree produces all positive rationals without a fixed denominator bound by iterated mediants, contrasting with Farey sequences that fix a maximum denominator and list only reduced fractions in an interval.

 

 

 

 

 





## Boundary

Boundary

Definition is specific to the interval [0,1] and to denominators ≤ N; one may extend definitions to other intervals or to reduced rationals without an upper denominator bound, but then properties like adjacency determinants change.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Distinguish Farey sequences from Stern–Brocot enumerations and from sorted lists of rationals by denominator alone; Farey emphasizes adjacency and the bc−ad=1 condition under a denominator cap.

 

 

 

 

 





## Synthesis

Synthesis

A Farey sequence of order N is the ordered list of reduced rationals in [0,1] with denominators at most N, characterized by the unit determinant condition for neighbors and by mediant generation constrained by denominator limits.