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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>