In my textbook on elementary number theory from a class last year, as well as elsewhere through my academic experience and even posts here, I often see the greatest common divisor notated as $(a,b)$ (where this represents the greatest common divisor of $a,b$).
My question is, why is such an unusual notation used? I say it's unusual (if not a bit absurd), for a couple of reasons:
It is unintuitive. It does not grant the reader an immediate knowledge of what it means. In particular, I can easily see it being misinterpreted as an ordered pair of points, and not as a function (which it basically is and I touch on in a second) or representing a single value. Good notation would be immediately understood, no? One should not struggle with parsing through the context or meaning.
A clearer notation is often used, and just three letters away. You could certainly think of $(a,b)$ as an ordered pair, if we consider it a function. I often see the alternate $gcd(a,b)$: a function, which takes in two inputs ($a$ and $b$), and outputs their greatest common divisor (or factor, if you prefer). Merely adding the three letters makes the notation clearer by clarifying it is a function, and using a relatively well-accepted acronym. (Or, at least, I often hear it abbreviated "g.c.d." in school.)
I've mulled this over for a while and I can't think of much without borrowing inspiration from other "unusual" notations. For example:
Common understanding in context: consider the $\log(x)$ function. Depending on context, without a base stated as seen here, it can be presumed to mean whatever is most "appropriate." It could be base $e$ (common in mathematics), base $10$ (common in engineering), or base $2$ (common in computer science). The ambiguity is essentially resolved by context in other words - personally, not a fan of that (I like the explicit stating of bases for full clarity), but if such a motivation holds elsewhere, it might stand to reason that $(a,b)$ could just be taken as a shorthand and "understood" in the context of number theory to simply be the greatest common divisor. That it is in fact used despite $gcd(a,b)$ being clearer and somewhat common also does seem to suggest this playing a role.
Perhaps it just started that way and it kept being used. For example, 3Blue1Brown has made a nice video (based on a MSE post) regarding how logarithms, roots, and exponentiation can be unintuitive from a notation perspective, and proposes an alternative, easier to understand notation. I haven't been in middle school in nearly a decade, but guessing from discussions online, that obviously hasn't really picked up too much. In that sense, perhaps it's like that for $(a,b)$: like we still notate exponentiation, etc., the same way today despite how counterintuitive it is (at least for students), perhaps it just "stuck" and proliferated?
I wasn't able to easily find any reason why online, and it bugs me a bit, so I ask:
Why do we so unintuitively notate the greatest common divisor of two numbers $a,b$ as $(a,b)$?
I of course recognize that this is somewhat moot in light of how easily it's made clear - as established above, all I have to do is append the front of the parenthetical by $gcd$ or $gcf$ to make it way better. And of course, people do that. But I'm more curious in where this ambiguous former notation arose and, in light of a better notation, why it is still used today.