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

Richard Hennessy rich.hennessy at verizon.net
Sun Sep 12 17:18:31 CDT 2010

```So it's no wonder that we get wrong answers frequently from limit()
Rich

--------------------------------------------------
From: "Richard Hennessy" <rich.hennessy at verizon.net>
Sent: Sunday, September 12, 2010 6:06 PM
To: <maxima at math.utexas.edu>
Subject: Re: [Maxima] Extended real arithmetic (was Re: inf - inf = 0 ??)

> I get using real number mathematics that 0*x = 0 for all x.  I get from extended real arithmetic that und*0 # 0 or
> inf*0 # 0.  So I can prove that the answer is both zero and nonzero at the same time depending on the approach.  So
> extended real arithmetic is not self consistent.
>
> FWIW.
>
> --------------------------------------------------
> From: "Richard Hennessy" <rich.hennessy at verizon.net>
> Sent: Saturday, September 11, 2010 11:16 PM
> To: "Richard Fateman" <fateman at cs.berkeley.edu>
> Cc: "Maxima List" <maxima at math.utexas.edu>
> Subject: Re: [Maxima] Extended real arithmetic (was Re: inf - inf = 0 ??)
>
>> Rich,
>>
>> I have been reading your paper, there is more to this than I thought.  I have not finished reading the paper yet, it
>> is a lot to digest. I would have been satisfied with not allowing arithmetic on inf, minf, und or ind.  I suppose
>> that is not good enough.
>>
>> Rich
>>
>>
>> --------------------------------------------------
>> From: "Richard Fateman" <fateman at cs.berkeley.edu>
>> Sent: Saturday, September 11, 2010 10:07 AM
>> To: "Leo Butler" <l.butler at ed.ac.uk>
>> Cc: <maxima at math.utexas.edu>; "Andreas Eder" <andreas_eder at gmx.net>
>> Subject: Re: [Maxima] Extended real arithmetic (was Re: inf - inf = 0 ??)
>>
>>>  At the risk of repeating myself, I can refer you to a paper that suggests this fits within
>>> a framework of interval arithmetic.
>>> e.g.  interval(-inf, inf) means  ---  there is a real  number but we don't know anything about its value.
>>> this is perhaps indefinite (IND)
>>>
>>> an empty interval  (we can choose one to be canonical) that has nothing in its interior means
>>> --- there is no number that this could be, which is perhaps undefined (UND)
>>>
>>> http://www.cs.berkeley.edu/~fateman/papers/interval.pdf
>>>
>>> There are what I consider misuses of intervals regarding limits that are present in Mathematica and Maple,
>>> eg..
>>>
>>> limit(sin(x),x,inf) ?=?  interval(-1,1).
>>>
>>> This leads to either erroneous answers to limit problems and things related to limits, or to a substantially
>>> overloaded notion of "equality".   Like any interval that contains 0  is equal to 0.
>>>
>>> Whether this can be put together in a computer algebra system to be more consistent is possible, but
>>> consistency in general is tough, and certainly not achieved now.
>>> There are many examples in which, for some procedure f,
>>>
>>> f(a)  is   different from  subst(a,x,   simplify(f(x)) )
>>> where x is some arbitrary symbol and   a   is some particular value.
>>>
>>>
>>> RJF
>>>
>>>
>>>
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>>>
>>
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```