Re: Zero divided by a number...

• To: mathgroup at smc.vnet.net
• Subject: [mg51902] Re: Zero divided by a number...
• From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
• Date: Thu, 4 Nov 2004 01:51:11 -0500 (EST)
• References: <dz50uwrnio0u@legacy> <cm4qoc\$6j6\$1@smc.vnet.net> <200411020705.CAA21635@smc.vnet.net> <cm9vk0\$8ns\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```DrBob <drbob at bigfoot.com> wrote:
> > 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.
>
> You mean for any x, zero or not.

No. I meant precisely what I said. If x = 0, then x/0 is Indeterminate,
rather than ComplexInfinity.

> Let's not confuse Mathematica's result for an expression with a useful
> mathematical definition of it.

???
But Mathematica's result of ComplexInfinity _is_ correct mathematically (in
the extended complex number system, of course).

> x/0 is undefined, no matter WHAT
> Mathemematica does with the expression.

Wrong.

> Consider this:
>
> Simplify[x y/x]
> y
>
> Simplify[ComplexInfinity y/ComplexInfinity]
> Indeterminate
>
> As you can see, ComplexInfinity isn't a full-fledged member of the
> algebraic system.

Not a good argument. By precisely that same reasoning, 0 shouldn't be "a
full-fledged member of the algebraic system" either:

In[1]:= Simplify[0 x/0]

Out[1]= Indeterminate

> As another example, Gamma[-5] returns ComplexInfinity, but that doesn't
> mean defining Gamma that way (into the extended complex plane) removes
> the discontinuity -- which is what we'd like from a meaningful extension
> of Gamma.

In fact, defining Gamma[-5] in that way _does_ remove the discontinuity.
[As z -> -5, regardless of the path in the complex plane, |Gamma[z]|
increases without bound.]

David Cantrell

> On Tue, 2 Nov 2004 02:05:11 -0500 (EST), David W. Cantrell
> <DWCantrell at sigmaxi.org> 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.)
> >
> > David Cantrell

```

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