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MathGroup Archive 2004

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Re: Zero divided by a number...

  • To: mathgroup at smc.vnet.net
  • Subject: [mg51897] Re: Zero divided by a number...
  • From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
  • Date: Thu, 4 Nov 2004 01:50:55 -0500 (EST)
  • References: <cm9ut5$8ii$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
> Everything Richard wrote is correct. He only forgot to say that that
> all these statements are true as statements about *complex numbers*.
> Thus instead of saying "x/0 is undefined ..." he should have said "is
> undefined as a complex number" or "is not a complex number" etc. The
> word "number" is ambiguous, and there are some strange people, even
> some mathematicians, who call things like Infinity "numbers" but I have
> never heard of anyone refer to them as "complex numbers'.

Perhaps you would be amused to know that Bertrand Russell, in constructing
what we might nowadays call the positive extended reals from the positive
rationals, refers to "the real number infinity". But that is merely of
historical interest. The real and complex number systems, as we know them
today, contain no infinite elements, of course.

> ("Complex" of course includes "real").
> (Besides, I don't believe that there is anyone, including yourself, who
> really did not understand what Richard meant.)

I thought and I still think that I understood exactly what Richard meant.

Not only is this a _Mathematica_ newsgroup, but Richard himself _supplied
context_. He said "Mathematica handles 0 appropriately.  x/0 is undefined
for any number x." So it seems clear to me that he thought, incorrectly,
that _in Mathematica_ x/0 is always undefined.

Indeed, I would not be at all surprised if Richard -- after realizing that
1/0 is ComplexInfinity, rather than undefined, in Mathematica -- now thinks
that Mathematica does _not_ handle division by 0 appropriately.

David Cantrell


> On 2 Nov 2004, at 16:05, David W. Cantrell wrote:
> >
> > rwprogrammer at hotmail.com (Richard) wrote:
> > [snip]
> >> Mathematica handles 0 appropriately.  x/0 is undefined for any number
> >> x.
> >
> > In Mathematica, it is _not_ true that "x/0 is undefined for any number
> > x." Rather, for any nonzero x, x/0 is defined as ComplexInfinity.
> >
> >> This is extremely simple to see if only you view division as the
> >> opposite of multipication.
> >
> > That view of division is simply inadequate in number systems (such as
> > the extended complex numbers) in which division of nonzero quantities
> > by zero is defined.
> >
> >> A/B = C  implies that C * B = A.
> >>
> >> 12/4 = 3 because 3*4 = 12.
> >> 0/7  = 0 because 0*7 = 0.
> >> 7/0 is undefined because x*0 does not equal 7 for any number x.
> >> Therefore it has no answer (except undefined).
> >
> > In Mathematica, 7/0 yields ComplexInfinity, but that certainly does not
> > imply that 0 * ComplexInfinity = 7. (In fact, 0 * ComplexInfinity is
> > Indeterminate in Mathematica.)


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