Is there an operator for adding the numerator and denominator of a fraction separately?

Question

Numbers in the Farey sequence are expressed as fractions e.g $F_5$:

$0\over1$ $1\over5$ $1\over4$ $1\over3$ $2\over5$ $1\over2$ $3\over5$ $2\over3$ $3\over4$ $4\over5$ $1\over1$

All of the $n\over5$ terms can be calculated by adding the numerators and denominators of the fractions immediately to either side of them, e.g. ${3\over5} = {1+2\over2+3}$ but obviously ${1\over2}+{2\over3}\neq{3\over5}$. Is there an operator to express this? ${a\over b} ? {c\over d} = {a+c\over b+d}$

I should point out that this is not for standard equalities, but for generating sequences, such as the above mentioned Farey sequence. It seems cumbersome to separate the numerators and denominators, perform calculations on them and then recombine them to get the next term.

Answer 1

Yes and no. On one hand, given the fact that we usually identify the fractions $\frac{a}b$, $\frac{2a}{2b}$, $\frac{3a}{3b}$, $\ldots$ - that is, since there are many ways to write a given rational number as the quotient of two integers, it is not clear that this is a particularly natural operation; for instance, keeping $?$ as the operation of adding numerators and denominators gives $$\frac{1}2?\frac{1}1=\frac{2}{3}$$ $$\frac{1}2?\frac{2}2=\frac{3}4$$ and yet, typically, we would consider the left-hand expressions to be equal, so it makes little sense for the results to be different. We could (and this suffices for the Farey sequence, since no fractions in it ever reduce) define $?$ to sum the numerator and denominator of the fraction in lowest terms, but that operator doesn't have a lot of nice properties - it is not even associative since $$\left(\frac{1}2?\frac{1}2\right)?\frac{1}3\neq\frac{1}2?\left(\frac{1}2?\frac{1}3\right)$$ $$\frac{2}4?\frac{1}3\neq\frac{1}2?\frac{2}5$$ $$\frac{1}2?\frac{1}3\neq\frac{1}2?\frac{2}5$$ $$\frac{2}5\neq\frac{3}7$$

However, we could consider that the notation $\frac{a}b$ actually refers to an ordered pair $(a,b)$ - so we make none of the usual identifications - and that, essentially, we perform this coordinate-wise addition to these pairs, and only afterwards do we consider it as an actual fraction. So, here, we define our operation in $\mathbb{Z}^2$ as the entirely sensible vector addition $(a,b)+(c,d)=(a+c,b+d)$ and consider the map $(a,b)\mapsto \frac{a}b$ only afterwards, recognizing that our operation is not well-defined in $\mathbb{Q}$.

Answer 2

The Wikipedia article on Farey sequences may be of interest to you. This question may be interesting as well.

The quantity $\frac{a + c}{b + d}$ is the mediant of the fractions $\frac ab$ and $\frac cd.$ I don't know of an operator symbol for mediant (like $+$ for sum or $\cdot$ for product) but there is a long tradition of making up symbols when it makes mathematical manipulations easier, so if you had reason to embed a "mediant operation" in some mathematical formula it would make sense to create a symbol for it.