Something that perhaps could be added is that leaving  0 `gcd` 0  undefined has two obvious annoying consequences: gcd is no longer idempotent (i.e. we don't have a `gcd` a = a, for all a), and it is no longer associative ((a `gcd` 0) `gcd` 0 is well-defined whilst a `gcd` (0 `gcd` 0) is not).

(We actually wrote something about this on a recent paper. If you're interested, see  http://www.joaoff.com/publications/2009/euclid-alg )

Regards,
Joao

2009/5/3 Nathan Bloomfield <nbloomf@gmail.com>
> This, to defend myself, was not how it was explained in high school.

No worries. I didn't realize this myself until college; most nonspecialist teachers just don't know any better. Nor did, it appears, the original authors of the Haskell Prelude. :)

BTW, this definition of gcd makes it possible to consider gcds in rings that otherwise have no natural order- such as rings of polynomials in several variables, group rings, et cetera.

Nathan Bloomfield


On Sun, May 3, 2009 at 11:16 AM, Achim Schneider <barsoap@web.de> wrote:
Nathan Bloomfield <nbloomf@gmail.com> wrote:

> The "greatest" in gcd is not w.r.t. the canonical ordering on the
> naturals; rather w.r.t. the partial order given by the divides
> relation.
>
This, to defend myself, was not how it was explained in high school.

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