# [Maxima] Extended real arithmetic (was Re: inf - inf = 0 ??)

Stavros Macrakis macrakis at alum.mit.edu
Fri Sep 10 15:26:22 CDT 2010

```Not clear what your point is.  Is it that:

a) the current Maxima behavior is incorrect?  -- which we all agree with.
In particular, UND*expr should be UND for all exprs.
or
b) there is no possible correct behavior?
or
c) that manipulating IND and UND cannot give the correct result for many
limit problems?  -- which we also all agree with.  Clearly
limit(sin(x),x,inf)=>IND and IND-IND must be IND, not 0.

In terms of Maxima's current behavior, limit( sqrt(x+1) - sqrt(x), x, inf)
=> 0 and limit(sin(x),x,inf) => IND (not UND), and 0*IND => 0, which is
correct.

Along the same lines, limit( tan(x), x, inf) => UND and 0*UND should be UND
(which it is not currently), which is also correct.

It is perfectly well-founded to extend numerical arithmetic to include
things like UND*0 => UND, IND*IND => IND, etc. as long as it is clear what
UND means.  It does not (and cannot) mean that the answer is *known* to be
undefined; it means only that the answer is not known to be defined.

-s

On Fri, Sep 10, 2010 at 15:47, Jean Pellegri <jean.pellegri at alice.it> wrote:

>  I too am skeptical
>
>
>
> Example :
>
>
>
> (sqrt(x+1)-sqrt(x))*cos(x)   ========è   0 X UND = 0
>
>
>
>
>
> (sqrt(x+1)-sqrt(x))*tan(x)   ========è   0 X UND = UND
>
>
>
> jean
>
>
>
>
>
> *De :* maxima-bounces at math.utexas.edu [mailto:
> maxima-bounces at math.utexas.edu] *De la part de* Richard Hennessy
>
> *Envoyé :* vendredi 10 septembre 2010 20:48
> *À :* Stavros Macrakis; Barton Willis
> *Cc :* maxima List
> *Objet :* Re: [Maxima] Extended real arithmetic (was Re: inf - inf = 0 ??)
>
>
>
> I am a little skeptical about it.  My biggest concern is that the axioms of
> extended real or complex arithmetic may not be self consistent.  If that is
> the case then you could prove that an answer is correct and incorrect at the
> same time by using different methods perhaps.  If they are in fact self
> consistent and complete than I am all for it.
>
>
>
> Rich
>
>
>
>
>
> *From:* Stavros Macrakis <macrakis at alum.mit.edu>
>
> *Sent:* Friday, September 10, 2010 10:03 AM
>
> *To:* Barton Willis <willisb at unk.edu>
>
> *Cc:* maxima List <maxima at math.utexas.edu>
>
> *Subject:* Re: [Maxima] Extended real arithmetic (was Re: inf - inf = 0
> ??)
>
>
>
> Especially disappointing because extended real arithmetic can in fact solve
> this correctly:
>
>
>
>    p1: sqrt(x+1)-sqrt(x)\$
>
>    p2: cos(x)\$
>
>
>
> limit(p1,x,inf) => 0
>
> limit(p2,x,inf) => IND
>
>
>
> from which Maxima should be able using extended real arithmetic that
> limit(p1*p2) => 0
>
> On Fri, Sep 10, 2010 at 08:52, Barton Willis <willisb at unk.edu> wrote:
>
> (((sqrt(x+1)-sqrt(x)))*cos(x),x,inf);
>
>
>  ------------------------------
>
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