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

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