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

**References**:**Re: Zero divided by a number...***From:*"David W. Cantrell" <DWCantrell@sigmaxi.org>