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

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