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

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  • Subject: [mg107055] Re: More /.{I->-1} craziness. Schools
  • From: David Bailey <dave at>
  • Date: Tue, 2 Feb 2010 03:23:51 -0500 (EST)
  • References: <hjbvc0$2tp$> <hjeqh1$g3c$> <hjh877$r4r$> <> <> <op.u67uiib3tgfoz2@bobbys-imac.local> <hjrf98$n1e$>

Daniel Lichtblau wrote:

> I had in mind the spoiler answer Richard Fateman provided in his first 
> post mentioning this particular tangent, err, example.
> 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?
> 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

Maybe an notebook option to flag numbers of extremely low precision with 
a colour might be useful. I guess this might be useful more generally in 
numerical analysis.

David Bailey

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