Understanding Roots of Unity: Proving Even Distribution with Math

[john951007]

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

 

[Mentallic]

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 n^th root of both sides. It then shouldn't be hard to notice how they're evenly spaced.