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Math Help - Roots of unity and the length from one root to the other roots

  1. #1
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    Roots of unity and the length from one root to the other roots

    Hello,


    I had an assignment that required me to solve for the roots of unity for various equations of the form . Then , I was asked to represent the roots of unity for each equation on an argand diagram in the form of a regular polygon.

    I did all of that , however, i have a question:

    is there a relation ship between the power n and the length from one root to the other roots?

    When n = 3,
    The roots are and 1. The length from one root to the other roots are both 1.7321 (sqrt of 3)

    When n = 4,

    The roots are . The length from one root to the other roots are (sqrt 2, sqrt and 2)
    Roots of unity and the length from one root to the other roots-image001-1.png



    I am wondering , for the equation , is there a formula relating the power of z to the length from one roots to the other roots?
    Thanks.
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  2. #2
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    Re: Roots of unity and the length from one root to the other roots

    The nth roots of unity lie at the vertices of a polygon with n sides, each vertex having distance from the center of 1. Drawing a line from the center to each vertex divides the polygon into n isosceles triangles, each having two sides of length 1, vertex angle of 360/n degrees and you want to find the length of the third side. If you draw a line from the vertex of such a triangle to the center of the base, you get two right triangles with hypotenuse length 1 and one angle of 180/n. The opposite side of that triangle has length 1(sin(180)/n) so the side of the polygon is twice that :
    2sin(180/n).
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  3. #3
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    Re: Roots of unity and the length from one root to the other roots

    Hi, thank you very much for answering the question first.

    However, i have already got the conjecture that the length of the polygon would be 2 sin (180/n), but this conjecture cannot be proven algebraically or MI that this statement would be true for any value of n, can it?

    I am just wondering is there any other relationships between the length from one root to the other roots? (Lets say, if you draw a line from one root (any root) to the other roots (not only the adjacent root)

    n = 3, length is sqrt 3 and sqrt 3
    n = 4, length is sqrt 2 , 2 and sqrt 2
    n = 5, length is not an exact value.. but what i have got is 1.1756, 1.9021, 1.9021 and 1.1756
    n = 6, length is 1, sqrt 3, 2, sqrt 3, 1
    n = 7, length is 0.86, 1.56, 1.94, 1.94, 1.56, 0.86

    seems like theres a pattern here, but i can't seem to figure it out... would help a lot if someone could tell me if theres a conjecture for this or not, thank you very much.
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