On Oct 22, 4:15 pm, Fredrik Johansson <fredrik.johans...@gmail.com>
wrote:
> On Fri, Oct 23, 2009 at 12:51 AM, John H Palmieri
>
>
>
>
>
> <jhpalmier...@gmail.com> wrote:
>
> > On Oct 22, 2:14 pm, William Stein <wst...@gmail.com> wrote:
> >> On Thu, Oct 22, 2009 at 2:02 PM, John H Palmieri <jhpalmier...@gmail.com>
> >> wrote:
>
> >> > On Oct 22, 8:57 am, William Stein <wst...@gmail.com> wrote:
> >> >> On Thu, Oct 22, 2009 at 8:11 AM, John H Palmieri
> >> >> <jhpalmier...@gmail.com> wrote:
>
> >> >> > Anyway, 0^0 is undefined in mathematics, so it's good that it's
> >> >> > undefined in Sage.
>
> >> >> It's defined for Sage *integers*:
>
> >> >> sage: 0^0
> >> >> 1
>
> >> > What about:
>
> >> > sage: 0.000^0.000
> >> > 1.00000000000000
>
> >> > Shouldn't this be undefined?
>
> >> > John
>
> >> Sage's behavior for 0.0^0.0 is determined by MPFR's, and MPFR follows
> >> "the ISO C99 standard for the pow function" as explained here:
>
> >> http://www.mpfr.org/mpfr-current/mpfr.html
>
> >> In particular, see the rule that "pow(x, ±0) returns 1 for any x, even
> >> a NaN." Indeed:
>
> >> sage: RR('NaN')^0
> >> 1.00000000000000
>
> > Wow, I thought Sage did math. The mathematical standard for 0^0 (for
> > real numbers) is that it doesn't exist, right? Or did I miss a memo
> > somewhere? What about these:
>
> What do you mean by "mathematical standard"?
First, it's what I've always been taught, and I trust my teachers and
professors -- if they were doing something unusual or something about
which there was some controversy, they would mention it. For some
more evidence, for example, in the article cited by Francis Clarke,
Knuth notes on p. 407 that Cauchy said that 0^0 was undefined. There
were a few flawed attempts to explain that 0^0 = 1, after which, "The
debate stopped there [in 1834], apparently with the conclusion that
0^0 should be undefined." I interpret this as saying that, based on
the historical record, 0^0 is undefined. Knuth then proceeds to argue
that this is the wrong convention, but the whole reason he has to work
so hard is because 0^0 being undefined *is* the standard.
I can see that there are reasons for changing this, but I think it's a
little dangerous to do so. (For example, it will cause endless
trouble in calculus classes because students don't understand that
anything might be discontinuous, so if their calculator and computer
tell them that 0^0 = 1, they won't really understand why it's an
indeterminate form -- they'll just resentfully view it as another
random, arbitrary thing that mathematicians do and that they (the
students) have to memorize. In actuality, it's setting 0^0 to be 1
that's arbitrary, at least when you have real arguments, because of
the discontinuity of x^y.)
John
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