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

• To: mathgroup at smc.vnet.net
• Subject: [mg48600] Re: [mg48585] Re: [mg48563] Re: What is zero divided by zero?
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Mon, 7 Jun 2004 05:33:26 -0400 (EDT)
• Organization: Mathematics & Statistics, Univ. of Mass./Amherst
• 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>
• Sender: owner-wri-mathgroup at wolfram.com

```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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