Dear all, I have even problems with this in a sense of topological terms. `Reals' come with a notion of approximation, so a/<very small>=<very big> and one does not want it to be zero _suddenly_, which would make division non continuous (on the open set reals minus 0). The pseudo inverses I know (in linear algebra) provide partial inverses, that is, if A is a linear operator having a null space, then you may want to invert it on the complement of the null space where it acts non trivially. If that is the point, it is surely not `a/0=0' what one does want. This looks to me much like multiplying by zero, which tends to transform wrong statements into true ones:
Let a<>b, a=b is false, but 0 = (a+b)/0 = a/0 + b/0 = 0 + 0 = 0 is true. Hence such an operation destroys the equational representation of the algebraic structure. I don't think this has to do with the difference between FLOAT and (exact) reals... If you would represent reals exactly as streams of digits, then a/0 is a process which would not terminate (truth value would be `bottom' of `undefined', neither true nor false). To do exact real number arithmetics one does need topology. Cheers BF. On Thu, Mar 24, 2011 at 2:32 PM, Bill Page <[email protected]> wrote: > Thanks Ralf, that makes a bit more sense. What exactly (formally) does > one get if one requires x/0 = 0? It is not a field or even a division > algebra, right? > > On Thu, Mar 24, 2011 at 2:49 AM, Ralf Hemmecke <[email protected]> wrote: >>> I don't know whether it is in fricas or not, but I have something on my >>> wish list for fricas - namely the option for a / 0 = 0 for the field of >>> real numbers, as is standard in Isabelle/HOL, and implicit for the >>> pseudo-inverse in linear algebra. >> >> When you speak of "field of real numbers", which domain in FriCAS do you >> mean exactly? "Float" is certainly not that domain. >> >> Do you really mean that division by zero should give zero instead of a >> division-by-zero error? >> >> Maybe related to this? >> >> http://comments.gmane.org/gmane.comp.mathematics.maxima.general/33656 >> http://www.math.utexas.edu/pipermail/maxima/2011/024297.html >> >> Ralf >> >> -- >> You received this message because you are subscribed to the Google Groups >> "FriCAS - computer algebra system" group. >> To post to this group, send email to [email protected]. >> To unsubscribe from this group, send email to >> [email protected]. >> For more options, visit this group at >> http://groups.google.com/group/fricas-devel?hl=en. >> >> > > -- > You received this message because you are subscribed to the Google Groups > "FriCAS - computer algebra system" group. > To post to this group, send email to [email protected]. > To unsubscribe from this group, send email to > [email protected]. > For more options, visit this group at > http://groups.google.com/group/fricas-devel?hl=en. > > -- % PD Dr Bertfried Fauser % Research Fellow, School of Computer Science, Univ. of Birmingham % Honorary Associate, University of Tasmania % Privat Docent: University of Konstanz, Physics Dept <http://www.uni-konstanz.de> % contact |-> URL : http://www.cs.bham.ac.uk/~fauserb/ % Phone : +44-121-41-42795 -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/fricas-devel?hl=en.
