Re: Re: Re: Reduction of Radicals

*To*: mathgroup at smc.vnet.net*Subject*: [mg71973] Re: [mg71948] Re: [mg71932] Re: [mg71902] Reduction of Radicals*From*: Murray Eisenberg <murray at math.umass.edu>*Date*: Wed, 6 Dec 2006 06:04:06 -0500 (EST)*Organization*: Mathematics & Statistics, Univ. of Mass./Amherst*References*: <200612031126.GAA08075@smc.vnet.net> <200612041139.GAA02996@smc.vnet.net> <200612051104.GAA27240@smc.vnet.net> <AD2FBE40-E3EB-4294-9448-E967363EADFD@mimuw.edu.pl>*Reply-to*: murray at math.umass.edu

No disagreement! Andrzej Kozlowski wrote: > *This message was transferred with a trial version of CommuniGate(tm) Pro* > Of course this is completely true. But perhaps it has somewhat obscured > the point I was trying to make, which was meant to answer Dimitris's > question: "why is Mathematica's convention about principal parts in > variance with those of classical algebraist's?" It seems to me that the > answer is simply that they (the classical algebraists and perhaps some > of their modern successors) did not care about the relation with > "principal values" of multivalued functions in analysis and did not even > know about them (Vieta, for one, live a whole century before Euler). > Even in certain modern texts on algebra the relationship between > radicals and logarithms is not relevant, so it is not surprising that > some authors even today may find it convenient even today to use > different "principal values' when working in a purely algebraic setting. > The point I was making about computer algebra programs like Mathematica > is that with their universal scope why have to be consistent across both > algebra and analysis in a way that a book on, say, Galois theory, need > not be. > > Andrzej Kozlowski > > On 5 Dec 2006, at 20:04, Murray Eisenberg wrote: > >> That Mathematica gives value I Pi for Log[-1] is consistent with the >> most common convention is that the principal argument, Arg, of a nonzero >> complex number z satisfies -Pi < Arg[z] <= Pi. >> >> Then the usual definition of the principal logarithm, Log, is >> >> Log[z] = Log[Abs[z]] + I Arg[z], >> >> and the multi-valued argument function, arg, would be given as: >> >> arg[z] = set of all Arg[z] + n 2 Pi I (n an integer) >> >> The multi-valued logarithm, log, would be given as >> >> log[z] = Log[Abs[z]] + I arg[z]. >> >> In this case one can define >> >> z^w = Exp[w log[z]] >> >> and the principal value of this as Exp[w Log[z]]. >> >> Andrzej Kozlowski wrote: >>> ... >>> The issue of which branch of a multivalued function should be chosen >>> as the so called "principal branch" is, of course, a matter of >>> convention. Since Mathematica defines Power[x,y] as Exp[y Log[x]], >>> the issue of what is (-1)^(1/3) is equivalent to choosing the value >>> of Log[-1]. Mathematica chooses the value >>> >>> Log[-1] >>> >>> I*Pi >>> >>> Of course once that is decided, everything else follows: >>> >>> FullSimplify[(-1)^(1/3) - Exp[(1/3)*I*Pi]] >>> 0 >>> >>> >>> It seems to me (though it is not something that lies within the scope >>> of my "professional" interest), that before the advent of computer >>> algebra there no need was felt for a uniform way of choosing >>> principal values for various multivalued functions that occur in >>> algebra and analysis. In other words, the relation >>> >>> x^y = Exp[y,Log[x]] >>> >>> was not treated as the definition of x^y, but as a relation that held >>> only up to the choice of branches of the multivalued functions >>> involved. So it seems to me that it was always thought that the >>> natural choice for Log[-1] is I Pi, but before computer algebra >>> systems appeared it was not necessarily felt that the "principal >>> value" of x^(1/3) is the one that makes x^y = Exp[y,Log[x]] hold. >>> Note that to keep this relation true and to have the principal value >>> of the cube root of -1 equal to -1, one would have do choose 3 I Pi >>> as the principal value of Log[-1], which does not seem very natural. >> >> --Murray Eisenberg murray at math.umass.edu >> Mathematics & Statistics Dept. >> Lederle Graduate Research Tower phone 413 549-1020 (H) >> University of Massachusetts 413 545-2859 (W) >> 710 North Pleasant Street fax 413 545-1801 >> Amherst, MA 01003-9305 >> > > -- Murray Eisenberg murray at math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2859 (W) 710 North Pleasant Street fax 413 545-1801 Amherst, MA 01003-9305

**References**:**Reduction of Radicals***From:*"dimitris" <dimmechan@yahoo.com>

**Re: Reduction of Radicals***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**Re: Re: Reduction of Radicals***From:*Murray Eisenberg <murray@math.umass.edu>