Roots of unity

by john951007
Tags: roots, unity
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 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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 Quote by john951007 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 $\theta$ then why would you also have a corresponding root with argument $-\theta$?
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

$$z^n=1$$

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

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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