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MathGroup Archive 2004

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Re: Re: Re: Re: What is zero divided by zero?


I find your argument strange. I am a mathematician and I have published 
papers where I have introduced new definitons and new terminology, as 
has practically every research mathematician. I am free to introduce 
any new concept and name it anyway I like (though of course I can't 
force others to use my terminology) if it is self-consistent, useful 
and I make my meaning clear. The fact that "folks don't ordinarily 
speak of it" is relevant only until sombody chooses to do otherwise.
The set with one element with the obvious operations of addition and 
multiplication satisfies all the axioms of a field except the 
convention that 1 should be different form 0. It is perfectly well 
defined, it is useful for the purpose of this thread, and 1/0 =1 = 0 
holds in it. I chose to call it a "field" though I could equally well 
have called it a "desert"  but how does the name change anything?

Andrzej


On 7 Jun 2004, at 18:33, Murray Eisenberg wrote:

> I'm not sure what Zen world you refer to, but so far as I have met the
> term "field" in the actual mathematical world, the smallest field has 2
> elements, not 1.
>
> Thus, from http://mathworld.wolfram.com/Field.html:
>
> "Because the identity condition must be different for addition and
> multiplication, every field must have at least two elements."
>
> (I suppose you could say that, in the trivial ring consisting of just
> the 0 element, 0 is its own multiplicative inverse, since 0 * 0 = 0 and
> 0 is a multiplicative identity.  But folks don't ordinarily speak of
> multiplicative inverses, and hence don't speak of quotients, unless
> there's a multiplicative identity 1 =/= 0.)
>
>
> Andrzej Kozlowski wrote:
>
>> There is at least one mathematical context where it is perfectly well
>> defined: the Zen-like world of the field with one element, where
>> 0/0 = 0 = 1.
>> Andrzej
>
>
> -- 
> 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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