Mediant (mathematics)

In mathematics, the mediant of two fractions

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. It is sometimes called the freshman sum, as it is a common mistake in the usual addition of fractions.

In general, this is an operation on fractions rather than on rational numbers. That is to say, for two rational numbers q₁, q₂, the value of the mediant depends on how the rational numbers are expressed using integer pairs. For example, the mediant of 1/1 and 1/2 is 2/3, but the mediant of 2/2 and 1/2 is 3/4.

A way around this, where required, is to specify that both rationals are to be represented as fractions in their lowest terms (with c > 0, d > 0). With such a restriction, mediant becomes a well-defined binary operation on rationals.

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

The notion of mediant can be generalized to n fractions, and a generalized mediant inequality holds, a fact that seems to have been first noticed by Cauchy. More precisely, the weighted mediant ${\displaystyle m_{w}}$ of n fractions ${\displaystyle a_{1}/b_{2},\ldots ,a_{n}/b_{n}}$ is defined by ${\displaystyle {\frac {\sum _{i}w_{i}a_{i}}{\sum _{i}w_{i}b_{i}}}}$ (with ${\displaystyle w_{i}>0}$). It can be shown that ${\displaystyle m_{w}}$ lies somewhere between the smallest and the largest fraction among the ${\displaystyle a_{i}/b_{i}}$.

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