For consecutive Farey fractions we have $$\frac{a}{b}, \frac{a}{b} + frac{c}{d}$ the mediant is obtained via a "simple's man addition": $$ \frac{c}{d} frac{p}{q} = \frac{a+c}{b+d} $$ which since $\frac{a}{b},\frac{c}{d}$ are consecutive if and only if $det\begin{pmatrix}a & c\\ b & d\end{pmatrix} = 1$ also turns out to be the rule of invariance of the determinant when you add a column to another column.
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For consecutive Farey fractions we have $$ \frac{a}{b} + \frac{c}{d} = \frac{a+c}{b+d} $$ which since $\frac{a}{b},\frac{c}{d}$ are consecutive if and only if $det\begin{pmatrix}a & c\\ b & d\end{pmatrix} = 1$ also turns out to be the rule of invariance of the determinant when you add a column to another column. |
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