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Re: Simplify and Abs in version 6.0

• To: mathgroup at smc.vnet.net
• Subject: [mg78115] Re: Simplify and Abs in version 6.0
• From: Michael <mcauxeu at gmail.com>
• Date: Sat, 23 Jun 2007 07:09:13 -0400 (EDT)
• References: <200706210958.FAA27636@smc.vnet.net><f5gape$gc1$1@smc.vnet.net>

On Jun 22, 4:11 am, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote:
> It is a good think that mathemaica does not return the answer that  
> you suggest:
>
> {-\[ImaginaryI] a, \[ImaginaryI] a}
>
> because it would be completely wrong as in Mathematica {a,b} is an  
> ordered pair and not a set. Mathematica has no built in structures  
> corresponding to sets and has no way of expressing the answer you  
> (seem to) have in mind. You can get the answer with Sqrt[a^2] more  
> directly by:
>
>   ComplexExpand[{I*Abs[a], (-I)*Abs[a]}, TargetFunctions -> {Re, Im}]
>
> {I*Sqrt[a^2], (-I)*Sqrt[a^2]}
>
> This, of course, can't be reduced any further without additional  
> information about a the sign of a.
>
> Andrzej Kozlowski
>
> On 21 Jun 2007, at 18:58, Michael wrote:
>
>
>
> > Hi,
>
> > In Mathematica version 6.0, I'm having difficulty trying to coax the
> > input
>
> > In[1]:=  FullSimplify[{\[ImaginaryI] Abs[a], -\[ImaginaryI] Abs[a]},
> > Element[a, Reals]]
>
> > to produce the set {-\[ImaginaryI] a, \[ImaginaryI] a}.
> > Interestingly, this *does* result for
>
> > In[2]:=  FullSimplify[{\[ImaginaryI] Abs[a], -\[ImaginaryI] Abs[a]}, a
> >> = 0]
>
> > and
>
> > In[3]:=  FullSimplify[{Sqrt[-1] Abs[a], -Sqrt[-1] Abs[a]}, a < 0]
>
> > Am I missing something here?  I've tried Allan Hayes' suggestion in
> > 2003, with
>
> > In[4]:=  FullSimplify[{\[ImaginaryI] Abs[a], -\[ImaginaryI] Abs[a]},
> > Element[a,Reals],ComplexityFunction -> ((Count[#, _Abs, Infinity]) &)]
> > Out[4]=  {\[ImaginaryI] Sqrt[a^2], -\[ImaginaryI] Sqrt[a^2]}
>
> > to no avail; of course, I could just add a PowerExand@ to the above
> > expression, but this seems like a lot to do, especially when
> > Mathematica already has been explicitly told that "a" is a real
> > number.
>
> > Any tricks or hints would be greatly appreciated!
>
> > Regards,
>
> > Michael- Hide quoted text -
>
> - Show quoted text -

Hi,

My apologies for assuming that my input was a set, and not an ordered
pair.  Everything now makes more sense, and my sincere thanks to
Daniel, Andrzej, and Murray for their help and patience!

Regards,

Michael

• References:
  • Simplify and Abs in version 6.0
    • From: Michael <mcauxeu@gmail.com>