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Re: Sqrt of complex number

  • To: mathgroup at
  • Subject: [mg126689] Re: Sqrt of complex number
  • From: Richard Fateman <fateman at>
  • Date: Thu, 31 May 2012 02:48:15 -0400 (EDT)
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  • References: <jpspgr$hdj$> <> <jq4klv$fsf$>

On 5/30/2012 1:10 AM, Andrzej Kozlowski wrote:
> On 29 May 2012, at 11:48, Richard Fateman wrote:
>> What you ask for is an expression that includes all answers: the
>> algebraic solution to the equation x^2-(3-4 I)=0.
>> Notice that you can create the set of the 2 roots of a quadratic, that
>> is, both square roots, by typing this.
>> Table[Root[#^2 - (3 - 4 I)&, n], {n, 1, 2}]
>> but this is a list, not "an algebraic number".
>> If you want to manipulate "an arbitrary root", that is -2+I OR 2-I
>> without specifying which one, it seems that Mathematica could provide
>> this facility by allowing you to type, for a symbol n,
>> y = Root[#^2- (3-4I)&,n]
>> (actually,  Root[x^2-3+4I,n]  might do just a well and be less obscure).
> Objects of the kind  Root[#^2-3+4I&,n], where n is a symbol would be mathematically meaningless

certainly not meaningless.  y^2 would be -3+4I.
  and very unlikely to be of any serious use.

How can you predict what serious use can be made of this?

> The reason is that there is no canonical ordering of the complex roots of a polynomial and thus

> there is nothing "mathematical" that can be said about the "n-th root of a polynomial".

Clearly false; I just said something mathematical.  I can say more, like 
the sum of n different roots is defined, assuming you can pick
roots n, n+1, ... n+k of a degree k polynomial.

>> For example,  we would know the unambiguous single value for y^2, and
>> we could perhaps compute  y*Conjugate[y].
> For example:
> Reduce[
>   ForAll[y, y^2 - (3 - 4 I) == 0,
>    Element[a, Reals]&&  a == y*Conjugate[y]]]
> a == 5

<sarcasm> thats clear </sarcasm>
> Reduce[ForAll[y, y^2 - (3 - 4 I) == 0, a == y^2]]
> a == 3 - 4 I
>> Unfortunately, Mathematica's designers/programmers do not allow you to
>> write Root[x^2-3+4I,n]  unless n is a specific integer, namely 1 or 2.
> Because when you evaluate Root[#^2-3+4I&,1]  or Root[#^2-3+4I&,2]
> Mathematica isolates the roots of the equation x^2-3+4I==0.

>When you write Root[#^2-3+4I&,n] there is nothing for Mathematica to do.

Right.  It should leave it alone, until you do something with it.
Like there is nothing for Mathematica to do with Sin[n].

> You are thinking of using root ordering as a just a dumb notation,
No, you are insisting that you can do something useful
only by using root ordering.

> which it is not. If it were meant to me just notation it would be a very clumsy one.

Not as clumsy as your Reduce [...].

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