Mediant is Dependent upon Representation
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Theorem
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 dependent upon the specific integers chosen for $a, b, c, d$.
Proof
Let $r = \dfrac 1 2$ and $s = 1$.
We have:
- $r = \dfrac 1 2 = \dfrac 2 4 = \dfrac 3 6$
Then the mediant of $r = \dfrac 2 4$ and $s = \dfrac 1 1$ gives:
- $\dfrac {2 + 1} {4 + 1} = \dfrac 3 5$
but the mediant of $r = \dfrac 1 2$ and $s = \dfrac 1 1$ gives:
- $\dfrac {1 + 1} {2 + 1} = \dfrac 2 3$
which is not the same as $\dfrac {2 + 1} {4 + 1}$.
$\blacksquare$