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.

**References**:**Re: Sqrt of complex number***From:*Richard Fateman <fateman@cs.berkeley.edu>