For mediant in music, see mediant.

In mathematics, the mediant of two fractions

and

(where c > 0, d > 0) is

that is to say, the numerator and denominator of the mediant are the sums of the numerators and denominators of the given fractions, respectively.

Properties of the mediant:

• An important property (also explaining its naming) of the mediant is that it lies strictly between the two fractions of which it is the mediant: If a / c < b / d, then

This property follows from the two relations

and

• Assume that the pair of fractions a/c and b/d satisfies the determinant relation bc - ad = 1. Then the mediant has the property that it is the simplest fraction in the interval (a/c, b/d), in the sense of being the fraction with the smallest denominator. More precisely, if the fraction a' / c' with positive denominator c' lies (strictly) between a/c and b/d, then its numerator resp. denominator can be written as and with two positive real (in fact rational) numbers . To see why the λ_i must be positive note that

and

must be positive. The determinant relation

bc - ad = 1

then implies that both must be integers, solving the system of linear equations

for λ₁,λ₂. Therefore .

Mediants commonly occur in the study of continued fractions and in particular, Farey fractions . The nth Farey sequence F_n is defined as the (ordered with respect to magnitude) sequence of reduced fractions a/b (with coprime a, b) such that b ≤ n. If two fractions a/c < b/d are adjacent (neighbouring) fractions in a segment of F_n then the determinant relation bc - ad = 1 mentioned above is generally valid and therefore the mediant is the simplest fraction in the interval (a/c, b/d), in the sense of being the fraction with the smallest denominator. Thus the mediant will then (first) appear in the (c + d)th Farey sequence and is the "next" fraction which is inserted in any Farey sequence between a/c and b/d. This gives the rule how the Farey sequences F_n are successively built up with increasing n.

External links

• Mediant Fractions