mediant

The mediant of two fractions a/b and c/d (with positive denominators b > 0 and d > 0) is the fraction (a+c)/(b+d). It is a simple linear combination of the numerators and denominators, distinct from the sum of the fractions themselves.

A key property is about ordering. If a/b < c/d, then a/b < (a+c)/(b+d) < c/d, provided the denominators are positive. If a/b > c/d, the inequalities reverse. In particular, the mediant lies strictly between the two original fractions whenever they are distinct and positive.

Example: the mediant of 1/3 and 2/5 is (1+2)/(3+5) = 3/8, which lies between 1/3 (0.333…) and 2/5 (0.4).

Applications and connections: the mediant arises in the construction of Farey sequences and the Stern-Brocot tree, where new fractions between neighboring terms are formed by taking mediants. It is also used to approximate real numbers by rationals and to explore the distribution of rationals within an interval. The mediant operation helps generate all reduced fractions between two bounds by iteratively inserting mediants with increasing denominators.

Limitations and generalizations: the mediant is not guaranteed to provide the closest rational approximation to a target in any fixed metric, and it behaves differently when signs of the fractions vary. Generalizations include weighted mediants, defined as (ma+nc)/(mb+nd) for integers m, n, which extend the concept to broader linear combinations.