Farey sequence¶

The increasing sequence of reduced rational numbers between zero and one whose denominators do not exceed a chosen order.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4513
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Elementary Number Theory → Mathematics

Core Idea¶

Adjacent terms a/b and c/d in the order-n Farey sequence satisfy bc-ad=1 under the standard convention, and mediants generate new terms as the order increases. All eligible reduced fractions are enumerated, duplicates are removed through coprimality and exact rational comparison sorts them; determinant and mediant relations expose the sequence's local structure. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

The Pizza Fraction Line

Pretend you cut pizzas into 1, 2, or 3 equal slices and write down every amount you can make, from no pizza to a whole pizza. You leave out repeats, since two halves is the same as one whole. Then you line them up from smallest to biggest. That neat line of fractions is a Farey sequence.

Ordered Simple Fractions List

A Farey sequence of order n lists every fraction from 0 to 1 whose bottom number is n or less, in order from smallest to largest. Each fraction is written in simplest form, so 2/4 becomes 1/2 and appears only once. For order 3, the list is 0/1, 1/3, 1/2, 2/3, 1/1. To find new fractions when n grows, add the tops and add the bottoms of two neighbors: between 1/3 and 1/2 you get 2/5. Neighbors also obey a neat cross-multiplying rule.

Bounded-Denominator Fraction Sequence

The Farey sequence of order n is the increasing list of all reduced fractions between 0 and 1, endpoints included, whose denominators are at most n. 'Reduced' means the top and bottom share no common factor, which removes duplicates like 2/4. Neighboring terms a/b and c/d have a special property: bc − ad = 1. For example, for 1/3 and 1/2, 1·3 − 1·2 = 1. When you raise the order, new terms appear as mediants of neighbors, (a + c)/(b + d). That fraction always lies between the two and appears once the order allows its denominator. These rules give the sequence a very tidy local structure.

For a positive integer n, the Farey sequence of order n consists of all reduced fractions in a stated interval, usually [0,1] with endpoints included, whose denominators are at most n, listed in increasing order. Construction amounts to enumerating eligible fractions, using coprimality of numerator and denominator to eliminate duplicates, and sorting by exact rational comparison. Under the standard convention, adjacent terms a/b < c/d satisfy the determinant identity bc − ad = 1. As the order increases, new terms arise as mediants (a+c)/(b+d) of adjacent terms, inserted where their denominator first falls within the bound. These determinant and mediant relations describe the sequence's local structure. The identity depends on explicit conventions for the order, interval, endpoints, reduction, denominator bound, and ordering, and changing any of them gives a different object.

Scope of Application¶

Farey sequence belongs to elementary number theory and is useful where the analyst can specify the typed elementary number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the positive order n, interval convention, reduced fractions, denominator bound, endpoint inclusion and increasing rational order are explicit. The scope is broad within that domain but bounded by the need for the positive order n, interval convention, reduced fractions, denominator bound, endpoint inclusion and increasing rational order are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the positive order n, interval convention, reduced fractions, denominator bound, endpoint inclusion and increasing rational order are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Farey sequence can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Farey sequence. Farey sequence compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed elementary number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the positive order n, interval convention, reduced fractions, denominator bound, endpoint inclusion and increasing rational order are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of elementary number theory because they reuse the typed elementary number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, All eligible reduced fractions are enumerated, duplicates are removed through coprimality and exact rational comparison sorts them; determinant and mediant relations expose the sequence's local structure., and type the carrier, state every parameter and convention in the definition, test that the positive order n, interval convention, reduced fractions, denominator bound, endpoint inclusion and increasing rational order are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Farey sequence Domain-specific

Parents (1) — outgoing typed relations

Farey sequence is a kind of Local Sequence Legality Prime

It remains its own entry because its identity is fixed by the positive order n, interval convention, reduced fractions, denominator bound, endpoint inclusion and increasing rational order.

Hierarchy path (1) — routes to 1 parentless root

Farey sequence → is a kind of Local Sequence Legality → is a kind of Local-to-Global Aggregation

Neighborhood in Abstraction Space¶

Farey sequence sits in a crowded region of the domain-specific corpus (13^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Numeral Systems & Integer Sequences (22 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25