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Re: Re: Re: Re: What is zero divided by zero?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg48630] Re: [mg48600] Re: [mg48585] Re: [mg48563] Re: What is zero divided by zero?
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Tue, 8 Jun 2004 00:48:31 -0400 (EDT)
*References*: <4xm5ym42r3vg@legacy> <wzog6i63na4c@legacy> <c9k4bo$fi9$1@smc.vnet.net> <c9pfb1$s4l$1@smc.vnet.net> <200406051119.HAA11743@smc.vnet.net> <200406052358.TAA28968@smc.vnet.net> <200406070933.FAA10935@smc.vnet.net> <6190E72C-B898-11D8-B3A8-000A95B4967A@mimuw.edu.pl> <40C495AE.2060607@math.umass.edu>
*Sender*: owner-wri-mathgroup at wolfram.com
> That it might be useful to form this "field", under some name, allow
> the multiplicative identiy to equal the additive identity, and thereby
> to see that 0^(-1) = 0, so that 0/0 = 0 in this "field", still doesn't
> provide a compelling argument to ME for saying 0/0 = 0 in other
> contexts!
But then nobody ever said that. In fact it was obvioulsy a joke, though
I guess it needs a certain kind of sense of humour to appreciate it. As
for Zen , well ... never mind.
Andrzej
On 8 Jun 2004, at 01:19, Murray Eisenberg wrote:
> Names have great stipulative and connotative power.
>
> Ceratinly the meanings of math names gets extended all the time, but
> one can ask whether breaking certain constraints on use of a name
> would, for purposes of communication and understanding, best result in
> a new name -- or at least clear warning to the reader that an
> unvonventional use of the term is being used.
>
> With your intended broadening of the term, one would of course
> immediately ask which established theorems about finite fields have to
> be restated (if any), which definitions relaxed, etc.
>
> That it might be useful to form this "field", under some name, allow
> the multiplicative identiy to equal the additive identity, and thereby
> to see that 0^(-1) = 0, so that 0/0 = 0 in this "field", still doesn't
> provide a compelling argument to ME for saying 0/0 = 0 in other
> contexts!
>
>
> Andrzej Kozlowski wrote:
>
>> 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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