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Re: Re: Why can't Mathematica tell when something is algebraically zero?

• To: mathgroup at smc.vnet.net
• Subject: [mg108187] Re: [mg108127] Re: Why can't Mathematica tell when something is algebraically zero?
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 10 Mar 2010 01:44:14 -0500 (EST)
• References: <hn2ltj$3kt$1@smc.vnet.net> <201003091119.GAA06777@smc.vnet.net> <AF2BE895-2E51-4F9C-9410-8CB5CB418899@mimuw.edu.pl>

In fact, I was wrong, and I forgot of something I once new (or assumed 
that it had been "fixed"). Assuming has no effect on PowerExpand and the 
answer returned is completely misleading.  For example:

Assuming[Im[r] != 0,
 PowerExpand[r^2 Sqrt[(r^3 + r + 2)/r] - Sqrt[r^3 (r^3 + r + 2)]]]
0

which is of course wrong.

PowerExpand only works with Assumptions passed view the Assumptions -> 
... method and then the answer is very complicated:

PowerExpand[
 r^2*Sqrt[(r^3 + r + 2)/r] - Sqrt[r^3*(r^3 + r + 2)],
   Assumptions -> Element[r, Reals]]

r^(3/2)*Sqrt[r^3 + r + 2]*
     E^(I*Pi*Floor[Arg[r]/(2*Pi) - Arg[r^3 + r + 2]/
                (2*Pi) + 1/2]) - r^(3/2)*Sqrt[r^3 + r + 2]*
     E^(I*Pi*Floor[-((3*Arg[r])/(2*Pi)) -
              Arg[r^3 + r + 2]/(2*Pi) + 1/2])

But, what is rather strange, this works:

PowerExpand[r^2 Sqrt[(r^3 + r + 2)/r] - Sqrt[r^3 (r^3 + r + 2)],
 Assumptions -> r <= 0]

even though it does not work with Simplify and Reduce

Reduce[r^2 Sqrt[(r^3 + r + 2)/r] - Sqrt[r^3 (r^3 + r + 2)] == 0 &&
  r < 0, r]
Reduce::cpow: Reduce was unable to prove that a radical of an expression 
containing only real variables and parameters is real valued. If you are 
interested only in solutions for which all radicals contained in the 
input are real valued, use Reduce with domain argument Reals. >>

The question is then, does PowerExpand really do that correctly? It 
would be reassuring if someone from WRI would confirm it. (Getting the 
right answer with PowerExpand itself is not very reassuring...).

I remember a while ago there was already a discussion here concerning 
the fact that PowerExpand is probably the only function that admits 
Assumptions only in the form Assumptions ->  and not via Assuming. I 
though that was going to be changed but it remains as it used to be. If 
changing this behaviour is really impossible I think there ought to be 
some sort of warning message that appears when one attempts to use
Assuming[something,PowerExpand[expr]], for obvious reasons.

Andrzej Kozlowski


On 9 Mar 2010, at 14:22, Andrzej Kozlowski wrote:

> It seems better to use:
>
> Assuming[Element[r, Reals],
> PowerExpand[r^2 Sqrt[(r^3 + r + 2)/r] - Sqrt[r^3 (r^3 + r + 2)]]]
>
> 0
>
> PowerExpand with Assumptions ought to be as reliable as Simplify 
(which does not work in this case), whereas if you use it without 
assumptions it can return answers that will be false for some values of 
the variables (it makes use of the maximal assumptions that permit the 
expansion). The fact that Simplify works here with the assumption r>=0 
but not r<0 appears to be a bug.
>
> Andrzej Kozlowski
>
>
>
>
> On 9 Mar 2010, at 12:19, dh wrote:
>
>> Hi,
>> Sqrt is a multivalued function. Therefore, your expression is not
>> necessarily zero. If you are sure that all your variables are real 
and
>> that you only want the main branch of Power, then you can use 
PowerExpand:
>> r^2 Sqrt[(r^3 + r + 2)/r] - Sqrt[r^3 (r^3 + r + 2)] // PowerExpand
>> Daniel
>>
>> On 08.03.2010 12:09, mmdanziger wrote:
>>> This isn't the first time that I've encountered something like this 
in
>>> Mathematica but in my calculations I got a term like this:
>>>
>>> r^2 Sqrt[(r^3 + r + 2)/r] - Sqrt[r^3 (r^3 + r + 2)]
>>>
>>> Which is obviously identically zero.  For some reason Simplify or 
even
>>> FullSimplify can't figure this out.  Once you get dependent on
>>> Mathematica these things are pretty disturbing...you forget about 
your
>>> own knowledge because the program tells you that things are
>>> different.  Then you sit there like an idiot checking an algebraic
>>> identity that any beginning precalc student should be able to solve 
no
>>> problem.
>>>
>>> Is there any way to get Mathematica to "wake up" to these things?  
It
>>> has such a powerful algebraic engine for most things, why can't it 
see
>>> something simple like the above?  Do you really have to manually
>>> override and tell the program when things should be zero?
>>>
>>> For the time being I'll just sift through and test things by hand 
but
>>> I can't believe that there isn't a better way.
>>>
>>> Best,
>>> md
>>>
>>
>>
>> --
>>
>> Daniel Huber
>> Metrohm Ltd.
>> Oberdorfstr. 68
>> CH-9100 Herisau
>> Tel. +41 71 353 8585, Fax +41 71 353 8907
>> E-Mail:<mailto:dh at metrohm.com>
>> Internet:<http://www.metrohm.com>
>>
>>
>

• References:
  • Re: Why can't Mathematica tell when something is algebraically zero?
    • From: dh <dh@metrohm.com>