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Math Help - can any1 explain the roots of unity to me?

  1. #1
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    can any1 explain the roots of unity to me?

    i have my book obviously but its extremely unclear to me as to how to use the method to acquire roots other than guessing.

    for example

    y(8) + 8y(4) + 16y = 0


    how do i get the roots for my homogenous using roots of unity?

    the y(x)<- whatever inside parenthesis is to what prime it is. So for the first one it is y with the derivative taken 8x.
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  2. #2
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    The n roots of unity

    Suppose you want the n roots of -1

    e(int)= -1

    t= pi/n cos( pi/n)+isin(pi/n) is one root to get the others divide 2pi/n and succesively add multiples of 2pi/n to pi/n to get the n roots

    eg the 4 roots of -1 you use pi/4 ,3pi/4 etc in cos(t)+isin(t)

    To get the n roots of 1

    we have of course 0, then add 2pi/n to get the others

    eg the four roots of unity are 0,pi/2,pi,3pi/2, correspondingly 1,i,-1,-1
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  3. #3
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    Quote Originally Posted by p00ndawg View Post
    i have my book obviously but its extremely unclear to me as to how to use the method to acquire roots other than guessing.

    for example

    y(8) + 8y(4) + 16y = 0.

    how do i get the roots for my homogenous using roots of unity?

    the y(x)<- whatever inside parenthesis is to what prime it is. So for the first one it is y with the derivative taken 8x.
    The characteristic equation, then, is r^8+ r^4+ 16= 0. Let u= y^4 and that becomes u^2+ u+ 16= 0 which you can solve with the quadratic formula. After you have u, you need to solve y^4= u. That's where the "fourth roots of unity" are concerned.
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