Is there a standard name for the "average" of the fractions $a/b$ and $c/d$ as $$ \frac{a+c}{b+d} \, \large ? $$ I understand that such an average is not unique in the sense that although $(2a)/(2b) = a/b$, this type of "average" of $2a/(2b)$ and $c/d$ is not the same as the above (in general).
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The place this happens naturally is in simple continued fractions. Convergents $p/q$ always have $\gcd(p,q) = 1.$ Given two consecutive convergents, your $a/b$ followed by $c/d,$ the result of the next "digit" being equal to $1$ is precisely that the next convergent is the mediant. If the next "digit" is some $k,$ the new convergent is $$ \frac{a + kc}{b + kd} $$ which is also between the first two.
The next set of relationships is that $ad-bc = \pm 1$ for an actual continued fraction. If that holds, then $c(b+kd)-d(a+kc) = \mp 1$ The matrix version is $$ \left( \begin{array}{rr} a & c \\ b & d \end{array} \right) \left( \begin{array}{rr} 0 & 1 \\ 1 & k \end{array} \right) = \left( \begin{array}{rr} c & a+kc \\ d & b+kd \end{array} \right) $$
Whatever the determinant of the first matrix, it is negated in the product because the determinant of the second matrix is $-1$
Will Jagy
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