On 2009-Oct-22 17:11:45 -0700, William Stein <wst...@gmail.com> wrote: >On Thu, Oct 22, 2009 at 4:52 PM, John H Palmieri <jhpalmier...@gmail.com> >wrote: >> 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.
By co-incidence, the latest Parabola[1] arrived yesterday and it
includes a discussion by Michael Deakin[2] on this (along with "is 1
prime" and "is 0 a natural number". He states that traditionally,
Euler, Pfaff and Mobius held that 0^0 == 1, whilst Cauchy, Libri and
two anonymous authors held that 0^0 is undefined, with the latter camp
"winning". Starting in 1970, Vaughan and Knuth joined the debate in
the former camp and the article concludes that 0^0 == 1 is now the
preferred convention. (The article includes the mathematical
arguments the various proponents used).
>Well like it or not, it is a fact that 0.0^0.0 = 1 *is* the official
>ISO 99 standard. Note that ISO = "international standards
>organization".
Likewise the ISO APL standard also defines 0*0 as 1 ('*' is the
power operator in APL). I'm not sure what the FORTRAN standard
says.
These standards all relate to the power operator in specific
programming languages and don't necessarily reflect mathematical
conventions as programming languages have limitations that pure
mathematics doesn't. A review of the relevant working group
debates might shed some light on their reasoning.
[1] "Parabola incorporating Function, a mathematics magazine for
secondary schools", ISSN 1446-9723
http://www.maths.unsw.edu.au/highschool/parabola.html
[2] Adjunct Senior Research Fellow in Mathematics & Statistics at
Monash University.
--
Peter Jeremy
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