Re: Re: 0^0 = 1?

• To: mathgroup at smc.vnet.net
• Subject: [mg95724] Re: [mg95683] Re: 0^0 = 1?
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Sun, 25 Jan 2009 21:48:07 -0500 (EST)
• Organization: Mathematics & Statistics, Univ. of Mass./Amherst
• References: <gl7211\$c8r\$1@smc.vnet.net> <gl9mua\$ajr\$1@smc.vnet.net> <200901251148.GAA00455@smc.vnet.net>

```Let me repeat in a much more explicit form what I implied in an earlier
post about 0^0:  there are several possible definitions, each defensible
from a certain starting point.  Therefore there is NO single acceptable
definition.

except to elucidate what those various starting points are and what each
this or that being the value of 0^0.

Dave Seaman wrote:
> On Sat, 24 Jan 2009 11:17:16 +0000 (UTC), Daniel Lichtblau wrote:
>> Dave Seaman wrote:
>>> On Thu, 22 Jan 2009 11:56:58 +0000 (UTC), dh wrote:
>
>
>>>> Hi,
>
>>>> 0^0 means the limit if both base and exponent go to zero.
>
>>> No, that is not how 0^0 is defined.  Does 2+2 mean the limit as both
>>> summands go to 2?  The value may happen to be the same in that case, but
>>> that is not how 2+2 is defined.
>
>>> The value of x^y for cardinal numbers x and y is the cardinality of the
>>> set of mappings from y into x.  In the case where x and y are the empty
>>> set, there is exactly one such mapping.  Hence, 0^0 = 1.
>
>> That's a definition from set theory. I doubt it plays nice in the
>> complex plane. The definition of Power we go by is
>> ower[a,b] == Exp[Log[a]*b]
>> Among other advantages, it means branch cuts for Power are inherited
>> from Log.
>
> Branch cuts at 0 are not particularly relevant here, since Exp[-oo*0] =
> Exp[anything*0] = Exp[0] = 1.
>
>>> It's a theorem of ZF (as stated in Suppes, _Axiomatic_Set_Theory_) that
>>> m^0 = 1 for every cardinal number m.
>
>> I think the setting of cardinal numbers is not really a good choice for
>> symbolic computation.
>
>
> We define other operations (addition and multiplication, for example) by
> starting with the cardinals, then extending to the integers, the
> rationals, the reals, and then the complex numbers.  At each stage, we
> want the new definition to be consistent with the old.
>
>>> Another way is to notice that 0^0 represents an empty product, whose
>>> value is the identity element in the monoid of the integers (or the
>>> reals).
>
>>> 	In[1]:= Product[0,{k,0}]
>
>>> 	Out[1]= 1
>
>> This and the ZF result are arguments for making 0^0 equal to one. They
>> are not in any sense "proofs" that it must be one, given that Power
>> lives in the setting of functions of complex variables.
>
> The ZF result is indeed a proof in the context of the cardinal numbers.
> The empty product result applies to any monoid.  The complex numbers are
> a monoid.  The symbol "0" in Mathematica represents the complex number 0,
> and as you can see, Mathematica agrees that the empty product yields the
> complex number 1.
>
>>> One might also consider the series expansion for Exp[0], which reduces to
>
>>> 	1 = 0^0/0! + (lots of terms that all reduce to zero).
>
>> This point is questionable, since one does not in general encounter the
>> formula with a term x0^0 (where x0 is the point of expansion).
>
> It is not questionable.  The series expansion for Exp[x] is
>
> 	Sum[x^k/k!,{k,0,Infinity}]
>
> whether one normally encounters it in that form or not.
>
>>> Having x^y be discontinuous at (0,0) does not "cause problems" any more
>>> than having the Sign function be discontinuous at 0 causes problems.
>
>> That's a matter of opinion. Clearly we do not at this time agree with
>> you on this.
>
> But no one has presented an example of an actual "problem" that is caused
> by the definition.  Discontinuous functions exist.  That's the basic
> reason that we need to study limits in the first place.
>
>>> Anyone who works with limits should be aware that you can't just blindly
>>> assume continuity when evaluating limits.  You have to consider the
>>> actual definition of the limit.
>
>> True, but I don't see any relevance to this particular issue. The
>> question at hand is whether or not 0^0 should be assigned a concrete
>> value. If it is given a value, then since limits of x^y depend on how
>> x,y respectively approach 0, they might or might not equal that value
>> (so there is no underlying assumption of continuity). But that is
>> already the case, and the question at hand is whether the undefined
>> value should in fact be defined.
>
> We don't "assign" a value to 0^0.  We simply look at the definition and
> notice what it says for the case of 0^0.  The definition makes no
> statement at all about limits, and therefore thinking that the definition
> has implications regarding limits is a misconception.
>
>
>

--
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305

```

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