# Definition:Mediant

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## 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

## 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.