Re: Re: Re: More /.{I->-1} craziness. Schools

*To*: mathgroup at smc.vnet.net*Subject*: [mg106925] Re: [mg106656] Re: [mg106882] Re: More /.{I->-1} craziness. Schools*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Thu, 28 Jan 2010 02:45:33 -0500 (EST)*References*: <hjbvc0$2tp$1@smc.vnet.net> <hjeqh1$g3c$1@smc.vnet.net>*Reply-to*: drmajorbob at yahoo.com

OK... so numbers are allowed to have NEGATIVE precision? LESS than zero digits of accuracy? Really? Whatever for? Bobby On Wed, 27 Jan 2010 18:23:32 -0600, Daniel Lichtblau <danl at wolfram.com> wrote: > DrMajorBob wrote: >> 0 and 1 are not "fuzzballs", so what interval could be >= 1 and also 0.? >> Bobby > > I had in mind the spoiler answer Richard Fateman provided in his first > post mentioning this particular tangent, err, example. > > http://forums.wolfram.com/mathgroup/archive/2010/Jan/msg00638.html > > At the bottom we find: > --- > I would especially avoid .nb objects, and most especially on topics of > numerical analysis, where the design flaws are, in my opinion, so > fundamental. Example (mathematica 7.0): > {x >= 1, x > 1, x > 0, x} > evaluates to > {True, False, False, 0.} > > can you construct x? > > RJF > > One possible answer, below.... > > x=0``-.5 > --- > > The point is that with Mathematica's version of significance arithmetic, > equality, I believe, is effectively treated as having a nontrivial an > intersection (of the implicit intervals defining two numbers). If > neither has any fuzz (i.e. both are exact), then Equal allows for no > fuzz, so this is only a subtlety if at least one of the values is > approximate. > > One implication is that a "zero" of sufficiently low (as in bad) > accuracy can be regarded as 1, or -1, or Pi, if those values happen to > fall within the accuracy (which I refer to as fuzz). > > The other inequalities follow from the preservation of trichotomy. For > explicitly real values we regard that as important. mathematica makes no > pretense that Equal is transitive and I do not see any way to do that > and also have useful approximate arithmetic. > > There has been some amount of communication off-line on this topic, > which is why some of us (well, me, at least) sometimes forget the > examples are not universally obvious to those who have not memorized the > enitre thread. > > Daniel > > >> On Wed, 27 Jan 2010 00:44:22 -0600, Daniel Lichtblau <danl at wolfram.com> >> wrote: >> >>> Richard Fateman wrote: >>>> [...] >>>> If all of Mathematica functionality were available in the free player >>>> version, WRI would need to drastically change its business model. And >>>> even it it were free, we still have behavior like this: (..for some >>>> values of zero) >>>> >>>> {x >== 1, x > 0, x} evaluates to {True, False, 0.} >>>> >>>> RJF >>> >>> Let's take simple intervals, that is, intervals that are segments. >>> Define less and greater in the obvious ways, that is, one segment lies >>> strictly below the other (right endpoint of lesser is less than left >>> endpoint of larger). Let us further define two intervals to be equal >>> whenever they have nonempty intersection. >>> >>> With these definitions, which I think are sensible, the behavior you >>> describe above is consistent with arithmetic on intervals. As the >>> numbers involved, at least some of them, are fuzzballs, this strikes me >>> as an appropriate behavior. >>> >>> Daniel Lichtblau >>> Wolfram Research > -- DrMajorBob at yahoo.com

**Follow-Ups**:**Re: Re: Re: Re: More /.{I->-1} craziness. Schools***From:*Andrzej Kozlowski <akozlowski@gmail.com>