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Re: Sqrt of complex number

  • To: mathgroup at smc.vnet.net
  • Subject: [mg126691] Re: Sqrt of complex number
  • From: Andrzej Kozlowski <akozlowski at gmail.com>
  • Date: Thu, 31 May 2012 02:48:57 -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> <4FC6EB00.5070908@cs.berkeley.edu>

On 31 May 2012, at 05:52, Richard Fateman wrote:

> 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

Bravo. A very impressive and intellectual argument.


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