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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.

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    $\begingroup$ Maybe it comes from ring theory where in PID domain the ideal generated by two elements, say $a$ and $b$ is the ideal generated by their gcd. In symbols $(a,b)=(gcd(a,b))$. $\endgroup$
    – jijijojo
    Dec 29, 2018 at 23:44
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    $\begingroup$ I agree with you that it's a stupid and ambiguous notation. As for why it is sometimes written like that, maybe our sister site specifically dedicated to history of mathematics would be a better place to ask. $\endgroup$
    – Arthur
    Dec 29, 2018 at 23:48
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    $\begingroup$ But as a notation $(a,b)$ is more universal in the sense it does not depend of the language. For instance in french they use $pgcd(a,b)$ (plus petit commun diviseur) or sometimes $a\wedge b$. $\endgroup$
    – jijijojo
    Dec 29, 2018 at 23:49
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    $\begingroup$ Probably a shorthand that made its way into the literature; writing gcd constantly could become cumbersome. That, and it coincides with the gcd for ideals. Context gets rid of the ambiguity, I see no issue with it. Edit: $(((a,b),(c,d)), ((e,f),(g,h)))$; an ordered pair of open intervals whose end points are the gcds of some integers? $\endgroup$ Dec 29, 2018 at 23:54
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    $\begingroup$ All notation is unusual if you not used to it. $\endgroup$
    – Somos
    Dec 30, 2018 at 0:44

1 Answer 1

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The pair / tuple notation used both for gcds and ideals serves to highlight their similarity. Just as in the domain $\,\Bbb Z,\,$ in any PID we have the ideal equality $\,(a,b) = (c)\iff \gcd(a,b) \cong c,\,$ where the congruence means "associate", i.e. they divide each other (differ by only a unit factor). Thus in a PID we can equivalently view $\,(a,b)\,$ as denoting either a gcd or an ideal, and the freedom to move back-and-forth between these viewpoints often proves useful.

Gcds and ideals share many properties, e.g. associative, commutative, distributive laws, and

$$ b\equiv b'\!\!\!\pmod{\!a}\,\Rightarrow\, (a,b) = (a,b')$$

Using the shared properties and notation we can give unified proofs of theorems that hold true for both gcds and ideals, e.g. in the proofs below we can read the tuples either as gcds or ideals

$$(a,b)\,(a^2,b^2)\, =\, (a,b)^3\ \ \ {\rm so}\ \ \ (a,b)=1\,\Rightarrow\, (a^2,b^2) = 1$$

$\quad \color{#c00}{ab = cd}\ \Rightarrow\ (a,c)\,(a,d)\, =\ (aa,\color{#c00}{cd},ac,ad)\, =\, \color{#c00}a\,(a,\color{#c00}b,c,d)\,\ [= (a)\ \ {\rm if}\ \ (a,c,d) = 1] $

Such abstraction aids understanding generalizations and analogies in more general ring-theoretic contexts - which will become clearer when one studies divisor theory, e.g. see the following

Friedemann Lucius. Rings with a theory of greatest common divisors.
manuscripta math. 95, 117-36 (1998).

Olaf Neumann. Was sollen und was sind Divisoren?
(What are divisors and what are they good for?) Math. Semesterber, 48, 2, 139-192 (2001).

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