Definition:Mediant

Definition
Let $r, s \in \Q$ be rational numbers.

Let $r$ and $s$ be expressed as:
 * $r = \dfrac a b, s = \dfrac c d$

where $a, b, c, d$ are integers such that $b > 0, d > 0$.

Then the mediant of $r$ and $s$ is $\dfrac {a + c} {b + d}$.

Also see

 * Divided by Positive Element of Field of Quotients for justification that such expressions for $r$ and $s$ are possible


 * Mediant is Dependent upon Representation


 * Applesellers' Problem

Warning
From Mediant is Dependent upon Representation, the mediant of two rational numbers depends on the specific integers used to form the fractional expressions used.

Hence for the mediant to be defined on the rational numbers, it is commonplace to stipulate that the rational numbers are expressed in canonical form.

Care should be taken to express whether this is the case.