roots of unity


by john951007
Tags: roots, unity
john951007
john951007 is offline
#1
Feb4-14, 05:43 AM
P: 9
I don't understand why roots of unity are evenly distributed? Every time when we calculate roots of unity, we get one result and then plus the difference in degree, but I think this follows the rule of even distribution and I don't understand that, it is easy to be trapped in a reasoning cycle.
how to prove it using mathematics?

Thank you
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Mentallic
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#2
Feb4-14, 06:01 AM
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Quote Quote by john951007 View Post
Every time when we calculate roots of unity, we get one result and then plus the difference in degree
Are you asking if you have a complex root with some argument [itex]\theta[/itex] then why would you also have a corresponding root with argument [itex]-\theta[/itex]?
If that is the case then what you're noticing are complex conjugates, and it's very important to remember that every real polynomial that has a complex root will also have a complex conjugate root.

But if you're actually looking for a reason why the roots of unity are all evenly spaced around the unit circle in the complex plane, then read up about De Moivre's theorem and notice that if

[tex]z^n=1[/tex]

where
[tex]1=e^{2\pi k i}[/tex] with k being any integer, or if you're working with the trigonometric form,
[tex]1=\cos({2\pi k})+i\sin({2\pi k})[/tex]

and now just take the nth root of both sides. It then shouldn't be hard to notice how they're evenly spaced.


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