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Re: How Mathematica select one of the roots?

To: mathgroup@smc.vnet.net
Subject: [mg12291] Re: [mg12237] How Mathematica select one of the roots?
From: Daniel Lichtblau <danl@wolfram.com>
Date: Thu, 7 May 1998 18:51:27 -0400
References: <199805050729.DAA17059@smc.vnet.net.>

msm@smol.carrier.kiev.ua wrote:
> 
> Dear All!
> 
> I have some question about Mathematica 3.0.
> 
> When Mathematica 3.0 calculate
> (-1.)^(1/3)
> it generate complex value result
> 0.5 + 0.8660254037844385*I.
> 
> But when I try to Solve equation
> x^3+1==0
> it generates three different roots.
> 
> So, here is the question:
> 
> "How Mathematica 3.0 select one of the roots in the first example?"
> 
> Thanks in advance for any "real" answer!
> 
> SY
> S. Molyavko

We use principal roots in Mathematica. So (-1.)^(1/3) is defined to be
Exp[(1/3)*Log[-1.]] which of course give the result you saw. This is
because the principal value for Log[-1] is I*Pi.

Note that if we did not follow a convention of this sort then radicals
of numbers would not have unique values, hence N[(-1)^(1/3)] would be
meaningless. This would render such expression rather of limited use.

Daniel Lichtblau
Wolfram Research

References:
- How Mathematica select one of the roots?
  - From: msm@smol.carrier.kiev.ua

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