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Math Help - Patterns from complex numbers (part A & B)

  1. #1
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    Exclamation Patterns from complex numbers (part A & B)

    Hi, I got my new internal assessment in math today and I found it quite complicated at some points. So, I'd appreciate if you could help me out.
    It goes like this:

    • Use de Moivre's theorem to obtain solutions to the equation z3-1=0
    • Use graphing software to plot these roots on an Argand diagram as well as a unit circle with centre origin.
    • Choose a root and draw line segments from this root to the other two roots.
    • Repeat those above for z4-1=0 and z5-1=0, comment your result and try to formulate a conjecture.
    • Prove your conjecture.


    Part B

    • Use de Moivre's theorem to obtain solutions to Zn=i
    • Represent each of these solutions on an Arganda diagram
    • Generalize and prove your result for Zn=a+bi, where /a+bi/ = 1
    • What happens when /a+bi/ is not equal to 1



    P.s. I posted the same thread in algebra forum too because I thought this question concerns both algebra and trigonometry.
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  2. #2
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    Re: Patterns from complex numbers (part A & B)

    You are doing just about everything wrong! If there is more than one sub-forum you feel a problem could be in, choose one. Do not double post. And you have shown no work of your own. I presume you would prefer hints to help YOU answer the question rather than just being given the answer- and we need to see where you have trouble to offer hints and help.

    What, exactly, does DeMoivres' theorem say and how is it relevant to this problem?
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  3. #3
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    Re: Patterns from complex numbers (part A & B)

    Quote Originally Posted by alireza1992 View Post
    Hi, I got my new internal assessment in math today and I found it quite complicated at some points. So, I'd appreciate if you could help me out.
    It goes like this:

    • Use de Moivre's theorem to obtain solutions to the equation z3-1=0
    • Use graphing software to plot these roots on an Argand diagram as well as a unit circle with centre origin.
    • Choose a root and draw line segments from this root to the other two roots.
    • Repeat those above for z4-1=0 and z5-1=0, comment your result and try to formulate a conjecture.
    • Prove your conjecture.


    Part B

    • Use de Moivre's theorem to obtain solutions to Zn=i
    • Represent each of these solutions on an Arganda diagram
    • Generalize and prove your result for Zn=a+bi, where /a+bi/ = 1
    • What happens when /a+bi/ is not equal to 1



    P.s. I posted the same thread in algebra forum too because I thought this question concerns both algebra and trigonometry.
    Let {\zeta _k} = \cos \left( {\frac{2k\pi }{n}} \right) + i\sin \left( {\frac{2k\pi }{n}} \right),\;k=0,1,\cdots,n-1~.
    Now each \zeta_k is a root of z^n-1=0~.
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