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).<br>
<br>(We actually wrote something about this on a recent paper. If you're interested, see <a href="http://www.joaoff.com/publications/2009/euclid-alg">http://www.joaoff.com/publications/2009/euclid-alg</a> )<br><br>Regards,<br>
Joao<br><br><div class="gmail_quote">2009/5/3 Nathan Bloomfield <span dir="ltr"><<a href="mailto:nbloomf@gmail.com">nbloomf@gmail.com</a>></span><br><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
<div class="im">> This, to defend myself, was not how it was explained in high school.<br><br></div>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. :)<br>
<br>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.<br><font color="#888888"><br>Nathan Bloomfield</font><div>
<div></div><div class="h5"><br><br>
<div class="gmail_quote">On Sun, May 3, 2009 at 11:16 AM, Achim Schneider <span dir="ltr"><<a href="mailto:barsoap@web.de" target="_blank">barsoap@web.de</a>></span> wrote:<br><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
<div>Nathan Bloomfield <<a href="mailto:nbloomf@gmail.com" target="_blank">nbloomf@gmail.com</a>> wrote:<br>
<br>
> The "greatest" in gcd is not w.r.t. the canonical ordering on the<br>
> naturals; rather w.r.t. the partial order given by the divides<br>
> relation.<br>
><br>
</div>This, to defend myself, was not how it was explained in high school.<br>
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