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Math Help - Products of the unit circle

  1. #1
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    Products of the unit circle

    Here is the problem:
    Let P sub 0, ....., P sub n-1 be n equally spaced points on the unit circle. Compute the product of the distances from P sub 0 (naught) to all the remaining points.
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    Quote Originally Posted by chadlyter View Post
    Here is the problem:
    Let P sub 0, ....., P sub n-1 be n equally spaced points on the unit circle. Compute the product of the distances from P sub 0 (naught) to all the remaining points.
    I do not understand. What distances and what products.

    First we know that,
    P_k = exp((2pi/n)*ki)

    Now what?
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  3. #3
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    This is my understanding.
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  4. #4
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    Quote Originally Posted by Plato View Post
    This is my understanding.
    In that case the product is equal to "n" itself.
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  5. #5
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    Quote Originally Posted by ThePerfectHacker View Post
    In that case the product is equal to "n" itself.
    Correct! What is the proof?
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    Quote Originally Posted by Plato View Post
    Correct! What is the proof?
    Not an easy question to answer.
    But we can begin by simplifing.
    Attached Thumbnails Attached Thumbnails Products of the unit circle-picture21.gif  
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  7. #7
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    I am going in unit circles (get the pun) with this problem.

    But I managed to make this problem more complicated than it should be. Instead of working with cyclotonomic polynomials, as I should have. I reduced the problem to trigonometry which makes it more difficult.


    I am sure you know what I am about to say but let me just post it.

    A cyclotonomic polynomial is,
    \Phi_n(z) = 1+z+z^2+...+z^n

    We can factor them as,
    (z-z_1)(z-z_2)+...+(z-_n)
    Where,
    z_1,z_2,...,z_n
    Are points on the regular (n+1)-gon inscribed in the unit circle in the complex plane (except unity).

    Basically, I am saying this is the cyclotonomic polynomial of degree n evaluated at 1.
    Attached Thumbnails Attached Thumbnails Products of the unit circle-picture22.gif  
    Last edited by ThePerfectHacker; April 8th 2007 at 09:23 AM.
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