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Need help figuring out what this Maths question means

  1. Mar 6, 2005 #1
    Hi everybody, this is my first post here.

    I got this question, but I don't know what it means:


    Fix n ≥ 1. If the nth roots of 1 are w0, w1, w2, . . . , wn−1, show that they satisfy:

    (z − w0)(z − w1)(z - w2) · · · (z − wn−1) = z^n − 1

    (z and wn are all complex numbers)

    What I don't understand is, what does it mean by "nth roots of 1"? :confused:
    I think by "roots" it means polynomial roots, but what does it mean to have a root of a number in this context?

    Any help would be appreciated.
     
  2. jcsd
  3. Mar 6, 2005 #2
    try this:
    let a^n=1
    =cis(0+2k*pi) for k=integer
    by de-moivres
    a=cis(2(k/n)*pi)

    there you have the roots, the bth root is probably k=b
     
  4. Mar 6, 2005 #3

    cepheid

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Square root of one: [tex] \sqrt{1} = 1^{\frac{1}{2}} [/tex]

    Fourth root of one: [tex] \sqrt[4]{1} = 1^{\frac{1}{4}} [/tex]

    .
    .
    .

    etc.

    .
    .
    .

    nth root of one: [tex] \sqrt[n]{1} = 1^{\frac{1}{n}} [/tex]

    There is more than one nth root i.e. more than one number (call them [itex] z [/itex]) that satisfies the equation:

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

    In fact there are exactly "n" of them, just as there are two square roots of one, and four fourth roots of one. Do you see why?
     
  5. Mar 6, 2005 #4
    Oops. Thanks, I shouldn't have missed that. I'll put it down to being the beginning of the semester. :redface:
     
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