At the very least, gcd(a,b) = m*a+n*b for some _integers_ m,n. The usual definition of gcd requires both m,n to be non-zero (actually gcd(m,n)=1). So gcd(a,0)=1*a+1*0. One could relax this to allow some kinds of algebraic integers -- e.g., Gaussian integers -- for Euclidean domains. But gcd(pi,e)=0=m*pi+n*e <=> m=n=0, so gcd(m,n)=0. At 06:21 PM 11/27/2013, Gareth McCaughan wrote:
Bill Gosper wrote:
If your developers are leery of GCD[1,Ï]â0, 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