Bill Gosper wrote:
If your developers are leery of GCD[1,π]→0, I think I can muster authoritative corroborators, if not corroborative literature. Motivation: The gcd of two real quantities is the largest quantity that goes into each a whole number of times.
[me:]
I'm unconvinced. In this context (multiplicative rather than additive) 0 is *larger* than everything else, not *smaller*, no?
[Bill:]
Well, we *are* looking for the "greatest".-) Do you propose a different answer, or deny there is one?
I deny there is one, or at least that there is any number that has a good claim.
And 0 emphatically doesn't "go into" anything "a whole number of times".
Touché. New wording: GCD(a,b):= the limit of the Euclidean process of iteratively subtracting the smaller from the larger.
The *limit* of doing this for commensurable quantities is zero. The gcd is the last thing you get immediately before 0. I agree that gcd(1,pi) can be thought of as a certain sort of limit of real numbers that tend to 0 -- but I don't think the usual sort of limit is the right one, because the metric it implicitly invokes isn't the right one. -- g