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)
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- 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 >