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Re: Sqrt of complex number
*To*: mathgroup at smc.vnet.net
*Subject*: [mg126689] Re: Sqrt of complex number
*From*: Richard Fateman <fateman at cs.berkeley.edu>
*Date*: Thu, 31 May 2012 02:48:15 -0400 (EDT)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
*References*: <jpspgr$hdj$1@smc.vnet.net> <201205290948.FAA06757@smc.vnet.net> <jq4klv$fsf$1@smc.vnet.net>
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 [...].
RJF
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