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MathGroup Archive 2004

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Re: Re: Zero divided by a number...

  • To: mathgroup at smc.vnet.net
  • Subject: [mg51832] Re: [mg51789] Re: Zero divided by a number...
  • From: DrBob <drbob at bigfoot.com>
  • Date: Wed, 3 Nov 2004 01:24:23 -0500 (EST)
  • References: <dz50uwrnio0u@legacy> <cm4qoc$6j6$1@smc.vnet.net> <200411020705.CAA21635@smc.vnet.net>
  • Reply-to: drbob at bigfoot.com
  • Sender: owner-wri-mathgroup at wolfram.com

> 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.

Let's not confuse Mathematica's result for an expression with a useful mathematical definition of it. x/0 is undefined, no matter WHAT Mathemematica does with the expression. 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. The same conclusion follows from the fact that 0*ComplexInfinity isn't zero. (Zero times ANYTHING meaningful is zero.)

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.

Bobby

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
>
>
>
>



-- 
DrBob at bigfoot.com
www.eclecticdreams.net


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