Mediant (mathematics)

In mathematics, the mediant of two fractions, generally made up of four positive integers

${\displaystyle {\frac {a}{c}}\quad }$ and ${\displaystyle \quad {\frac {b}{d}}\quad }$ is defined as ${\displaystyle \quad {\frac {a+b}{c+d}}.}$

That is to say, the numerator and denominator of the mediant are the sums of the numerators and denominators of the given fractions, respectively. It is sometimes called the freshman sum, as it is a common mistake in the early stages of learning about addition of fractions.

Technically, this is a binary operation on valid fractions (nonzero denominator), considered as ordered pairs of appropriate integers, a priori disregarding the perspective on rational numbers as equivalence classes of fractions. For example, the mediant of the fractions 1/1 and 1/2 is 2/3. However, if the fraction 1/1 is replaced by the fraction 2/2, which is an equivalent fraction denoting the same rational number 1, the mediant of the fractions 2/2 and 1/2 is 3/4. For a stronger connection to rational numbers the fractions may be required to be reduced to lowest terms, thereby selecting unique representatives from the respective equivalence classes.

In fact, 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 ${\displaystyle bc-ad=1}$ 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.

The Stern–Brocot tree provides an enumeration of all positive rational numbers via mediants in lowest terms, obtained purely by iterative computation of the mediant according to a simple algorithm.