RE: 0^0 = ?

[Boud Roukema]

On Thu, 13 Nov 2003, John W. Eaton wrote:

> On 13-Nov-2003, address@hidden <address@hidden> wrote:
>
> | I know that by L'Htpital's Rule you should get:
> |
> | ln(y)=x*ln(x) = ln(x)/(1/x)  so
> |
> | Lim x->0+  ln(x)/(1/x) = ( 1/x )/( -1/( x^2)) =
> |
> | Lim x->0+ (-x) = 0
> | so  ln(y) = 0 and then y=1.
> |
> | Maybe this is the reason for the behavior?
>
> The 0^0 == 1 behavior is part of the IEEE 754 standard for floating
> point arithmetic.  The paper "What every computer scientist should
> know about floating point arithmetic" by David Goldberg provides a
> rationale for the behavior that is a bit different than above (it's at
> the end of a section titled "ambiguity").  To start with, I think you
> need to look at this as y^x, not x^x.  A quick google search should
> turn up a copy of the paper if you want the details.

Sorry, to me it seems undecided in IEEE 754:

http://grouper.ieee.org/groups/754/
p217

"One definition [of the FORTRAN standard] might be to use the method
shown in section 'Infinity' on page 199". For example, to determine the
value of a^b, consider non-constant analytic functions f and g with the
property that f(x) -> a and g(x) -> b as x -> 0. If f(x)^g(x) always
approaches the same limit, then this should be the value of a^b. ..."

This gives 0^0=1.
But it's only a suggestion of how the FORTRAN standard
might be implemented. It doesn't seem to be part of the IEEE 754 standard.

Argument for 0^0=NaN:
Footnote 1: "The conclusion that 0^0=1 depends on the restriction that
f be nonconstant. ... " This "gives 0 as a possible value, and so 0^0 would
have to be defined as a NAN."

Arugment for 0^0=1:
Footnote 2: reference to "Concrete Mathematics" by Graham, Knuth and Patashnik
saying that the say it's useful for the binomial theorem.

BTW, i couldn't find an online version of the IEEE 754 standard itself - only
how to buy a copy - did anyone find it?

boud





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