Understanding Mathematica's Assignment Scheme for nth Roots of Unity?

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In summary, the Mathematica software assigns the various roots of x^6+1=0 to the various fractional powers of 1/6 and 5/6 according to standard convention.
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saltydog
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When I use Solve to solve for the roots of [itex]x^6+1=0[/itex], Mathematica returns the following roots:


[tex]i[/tex]
[tex]-i[/tex]
[tex](-1)^{\frac{1}{6}[/tex]
[tex]-(-1)^{\frac{1}{6}}[/tex]
[tex](-1)^{\frac{5}{6}}[/tex]
[tex]-(-1)^{\frac{5}{6}}[/tex]

I realize these are derived from the nth roots of unity but I don't understand how Mathematica is assigning the various roots to the various fractional powers of 1/6 and 5/6. For example, if I had used [itex]x^7[/itex], then Mathematica returns values of -1 raised to 4/7, 5/7, and 6/7. How do I know what n-th root is being assigned to the power of 4/7 for example. I know I can evaluate it via N[] but I'd like to know the assignment scheme.

Anyone know?
 
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  • #2
saltydog said:
When I use Solve to solve for the roots of [itex]x^6+1=0[/itex], Mathematica returns the following roots:


[tex]i[/tex]
[tex]-i[/tex]
[tex](-1)^{\frac{1}{6}[/tex]
[tex]-(-1)^{\frac{1}{6}}[/tex]
[tex](-1)^{\frac{5}{6}}[/tex]
[tex]-(-1)^{\frac{5}{6}}[/tex]

I realize these are derived from the nth roots of unity but I don't understand how Mathematica is assigning the various roots to the various fractional powers of 1/6 and 5/6. For example, if I had used [itex]x^7[/itex], then Mathematica returns values of -1 raised to 4/7, 5/7, and 6/7. How do I know what n-th root is being assigned to the power of 4/7 for example. I know I can evaluate it via N[] but I'd like to know the assignment scheme.

Anyone know?
they are assigned by principle value. Like sqrt(4) means 2 normally. It is easier to when looking at the numbers in polar form where the angle is restricted to (-pi,pi]
thus
z=|z|exp(iArctan(Im(z)/Re(z))=r*exp(i a)=(r<a)
[tex]i[/tex]
=(1<pi/2)
[tex]-i[/tex]
=(1<-pi/2)
[tex](-1)^{\frac{1}{6}[/tex]
=(1<pi)^(1/6)
=(1<pi/6)
[tex]-(-1)^{\frac{1}{6}}[/tex]
=-(1<pi/6)
=(1<-5pi/5)
[tex](-1)^{\frac{5}{6}}[/tex]
=(1<pi)^(5/6)
=(1<5pi/6)
[tex]-(-1)^{\frac{5}{6}}[/tex]
=-(1<5pi/6)
=(1<-pi/6)
in other words to find the principle value of z^x for z complex and x real
write z=(r<a) with -pi<a<=pi
z^x=(r^a<x*a)
where x*a is reduced back into (-pi,pi] if needed.

note this is not something mathematica does just because. It is the standard convention for principle values of roots. That's why have the high school teachers who say the principle value cube root of -1 is -1 should be dunked in rotten cabage.
 
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Mathematica's assignment scheme for nth roots of unity is based on the concept of complex numbers and the nth roots of 1. In general, the nth roots of unity are the solutions to the equation x^n = 1. This means that when you use Solve to solve for the roots of x^6+1=0, Mathematica is finding the solutions to x^6 = -1, which can also be written as x^6 = e^{i\pi}.

Now, to understand how Mathematica assigns the various roots to the fractional powers, we need to look at the concept of complex numbers. In complex numbers, we have two components - a real component and an imaginary component. The real component is denoted by the symbol "Re" and the imaginary component is denoted by the symbol "Im". These components are used to represent the coordinates of a point on the complex plane.

In the case of nth roots of unity, the real component is always 0, since the solutions lie on the unit circle which has a radius of 1. The imaginary component, on the other hand, can be any value between -1 and 1, depending on the power to which we raise -1. For example, when we raise -1 to the power of 1/6, we get a value of -0.866025 + 0.5i. This means that the point on the complex plane lies at a distance of 1 from the origin, and at an angle of 1/6 of a full rotation, which is equivalent to 60 degrees.

In general, when we raise -1 to the power of n/m, where n and m are integers, we get a point on the complex plane that lies at a distance of 1 from the origin, and at an angle of (n/m) of a full rotation. This is how Mathematica assigns the various roots to the various fractional powers of 1/6 and 5/6.

In your example, when you use x^7, Mathematica returns values of -1 raised to 4/7, 5/7, and 6/7. This means that the points on the complex plane for these solutions lie at a distance of 1 from the origin, and at angles of (4/7), (5/7), and (6/7) of a full rotation, respectively.

In summary, Mathematica's assignment scheme
 

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