RE: 0^0 = ?

[Ted Harding]

Clearly, 0^0 is mathematically indeterminate.
Not does the limit lim[x->0,y->0] x^y exist.

However, as people have pointed out, if you want it to have
a "canonical" value, prime candidates are 0 and 1.

There seems to be a public preference for 1. However, there is
a democratic mathematical argument that perhaps 0 could be
seen as having more votes than 1.

Namely, consider going to the limit of x->0 and y->0 along a
line pointing to (0,0): let y = a*x with (say) a>0. Then

  x^y = x^(ax) = (x^x)^a

which, as x->0 from above, tends to 0 for all such values of a.
Only the solitary voice of a=0 gives a different result.

Unfortunate about the opposition party, who vote along the
line y = b*x with b<0, for whom

  x^y = (x^x)^b -> inf

as x->0 from above. Not to mention the complexity parties,
a coalition of fifth-columnists who like x to infiltrate 0 from
below.

I don't think we'll ever get them to agree!
Ted.


On 14-Nov-03 Boud Roukema wrote:
> 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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E-Mail: (Ted Harding) <address@hidden>
Fax-to-email: +44 (0)870 167 1972
Date: 14-Nov-03                                       Time: 16:03:43
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