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Re: Reduction of Radicals
*To*: mathgroup at smc.vnet.net
*Subject*: [mg71932] Re: [mg71902] Reduction of Radicals
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Mon, 4 Dec 2006 06:39:21 -0500 (EST)
*References*: <200612031126.GAA08075@smc.vnet.net>
On 3 Dec 2006, at 20:26, dimitris wrote:
> Based on this reference
>
> Cardan Polynomials and the Reduction of Radicals (by T. Osler)
>
> (see also references therein)
>
> (you can download the paper here:
> http://www.jstor.org/view/0025570x/di021218/02p0059q/0?
> currentResult=0025570x%2bdi021218%2b02p0059q%2b0%2c03&searchUrl=http
> %3A%2F%2Fwww.jstor.org%2Fsearch%2FBasicResults%3Fhp%3D25%26so%
> 3DNewestFirst%26si%3D1%26Query%3DOsler
> )
>
> the following expression can be reduced to 1
>
> z = (2 + Sqrt[5])^(1/3) + (2 - Sqrt[5])^(1/3)
>
> Mathematica gives
>
> N[%]
> 1.9270509831248424 + 0.535233134659635*I
>
> This is because by default it returns a complex number for the cube
> root of a negative number
>
> List @@ z
> N[%]
>
> {(2 - Sqrt[5])^(1/3), (2 + Sqrt[5])^(1/3)}
> {0.30901699437494756 + 0.535233134659635*I, 1.618033988749895}
>
> However defining
>
> mycuberoot[x_] := Block[{w}, w = w /. Solve[w^3 == 1][[3]]; If[Re[x] <
> 0, w*x^(1/3), x^(1/3)]]
>
> Then
>
> {2 - Sqrt[5], 2 + Sqrt[5]}
> mycuberoot /@ %
> FullSimplify[%]
> Together[Plus @@ %]
>
> {2 - Sqrt[5], 2 + Sqrt[5]}
> {(-1)^(2/3)*(2 - Sqrt[5])^(1/3), (2 + Sqrt[5])^(1/3)}
> {(1/2)*(1 - Sqrt[5]), (1/2)*(1 + Sqrt[5])}
> 1
>
> Is there a particular reason why by default Mathematicas returns a
> complex number for the cube root of a negative number or it is a
> matter
> of choise?
>
> Following the same procedure I prove that
>
> (10 + 6*Sqrt[3])^(1/3) + (10 - 6*Sqrt[3])^(1/3)
>
> is equal to 2. Indeed
>
> {10 + 6*Sqrt[3], 10 - 6*Sqrt[3]}
> mycuberoot /@ %
> FullSimplify[%]
> Together[Plus @@ %]
>
> {10 + 6*Sqrt[3], 10 - 6*Sqrt[3]}
> {(10 + 6*Sqrt[3])^(1/3), (-1)^(2/3)*(10 - 6*Sqrt[3])^(1/3)}
> {1 + Sqrt[3], 1 - Sqrt[3]}
> 2
>
> This behavior of Mathematica does not affect simplifications by e.g.
> RootReduce?
>
> I must admit that I have gaps on my knowledge in these symbolic
> aspects
>
> (I start to be interested in after I try to solve the the secular
> Rayleigh equation)
> so more experienced members of the forum may forgive any possible
> mistakes of mine!
>
> Anyway I don't understand this difference in treating nested radicals
> between literature and Mathematica.
>
> I really appreciate any kind of insight/guideness/comments.
>
> Regards
> Dimitris
>
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.
RootReduce is somewhat different. Given an algebraic number it first
tries to find its minimal polynomial. This does not depend on any
particular representation of the algebraic number. After that, it
expresses the answer in terms of roots of that minimal polynomial,
using root isolation and the associated ordering. This is something
that can be done in several ways, with Mathematica alone having two
different ordering methods. Other systems use other methods, so the
the order of roots may not be the same for different systems
(although the real roots will always correspond).
It is only when algebraic numbers are expressed in terms of radicals
that conventions about "principal branches" come into play. I expect
that all symbolic algebra systems use the same convention based on
the definition x^y = Exp[y,Log[x]], but I have not tried to check
this. My own view is that radical representations of algebraic
numbers should now be considered as belonging only to the history of
mathematics.
Best regards
Andrzej
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