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Math Help - complex numbers

  1. #1
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    complex numbers

    not sure if this is the right forum so sorry if i got it wrong. i'm having alot of trouble with this question. the numbers in brackets in front of the w's are meant to be subscripts

    if w_n = \exp\left(2\pi i/n\right) show w_n^n = 1 and that

    1 + w_n +w_n^2 + w_n^3 + \dots +  w_n^{n-1} = 0

    and show

    \left(x + y\cdot w_3 + z\cdot w_3^2\right)\left(x + y\cdot w_3^2 + z\cdot w_3\right) = x^2 + y^2 +z^2 - xy -yz -zx

    sorry for the lack of tex in it. thanks so much for any help given!!

    ------------------------------------------------------------

    Edit by Chris L T521: Reformatted question with \text{\LaTeX}. Please inform me if I have misinterpreted anything.
    Last edited by Chris L T521; August 13th 2009 at 04:18 PM. Reason: reformatted question with LaTeX.
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  2. #2
    Senior Member pankaj's Avatar
    Joined
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    New Delhi(India)
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    w^k_{n}=e^{\frac{2k\pi}{n}}=\left(e^{\frac{2\pi}{n  }}\right)^k

    If k=n,then, w^n_{n}=\cos 2\pi+i\sin 2\pi=1+i.0=1

    1+w_{n}+w^2_{n}+w^3_{n}+.....+w^{n-1}_{n}=\frac{1-w^n_{n}}{1-w_{n}}=\frac{1-1}{1-w_{n}}=0

    For, n=3,anove results are, w^3_{3}=1 and 1+w_{3}+w^2_{3}=0

    x^2+y^2w^3_{3}+z^2_{3}+xy(w_{3}+w^2_{3})+yz(w^2_{3  }+w^4_{3})+zx((w_{3}+w^2_{3})

    =x^2+y^2+z^2+xy(-1)+yz(w^2_{3}+w_{3})+zx(-1)

    =x^2+y^2+z^2-xy-yz-zx
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