centralmediant

The central mediant is a concept from number theory and the theory of Farey sequences that provides a way to insert fractions between two given fractions. If two fractions a/b and c/d are given in lowest terms with b, d positive, the central mediant is defined as (a + c)/(b + d). This fraction always lies strictly between a/b and c/d provided that a/b < c/d. Because the numerator and denominator of the mediant are sums of the numerators and denominators of the endpoints, the mediant is called “central” to emphasize its symmetry and its position midway in terms of fraction size, even though it is not the arithmetic mean of the values.

The central mediant is a fundamental tool in the construction of Farey sequences. Starting from the first two fractions 0/1 and 1/0 (or the first two terms 0/1 and 1/1 in practical use), repeated insertion of centrally mediant fractions generates all reduced fractions between 0 and 1 with denominators up to a given bound. Each step creates a fraction that is the mediant of adjacent terms, guaranteeing that the list remains in ascending order and that no further reduction is possible. The algorithm underpinning the construction is closely related to Stern–Brocot trees, where each node’s left and right children are the mediants of the node with its left and right parents, respectively.

Properties of the central mediant include monotonicity: if a/b ≤ c/d, then a/b ≤ (a + c)/(b + d) ≤ c/d. Uniqueness holds in the sense that the mediant of distinct fractions with denominators bounded by n is the unique fraction with the smallest denominator lying between them. The mediant also satisfies a mediant inequality: for any real numbers α and β with α < θ < β, there exists a fraction m/n obtained as a mediant of two reduced fractions that lies between α and β, though this result is often proved using the mediant concept.

Applications of the central mediant spread across analytic number theory, where it facilitates proofs concerning density and distribution of rational numbers. In algebraic dynamics, the mediant maps play a role in describing kneading sequences of unimodal maps. Additionally, the central mediant appears in algorithms for rational approximation and in constructing visibility graphs on integer lattices where lattice points are connected according to Farey adjacency.

Although the mediant was first systematically studied in the 19th century by mathematicians such as J. J. Sylvester, its simple definition and powerful structural role in generating ordered sets of fractions keep it a staple in modern combinatorial and number‑theoretic teaching.