Roots of unity

  1. 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
     
  2. jcsd
  3. Mentallic

    Mentallic 3,751
    Homework Helper

    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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