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Math Help - complex no

  1. #1
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    complex no

    If a=e^(i*8pi/11) then find re(a+a^2+a^3+a^4+a^5) I used demoivres theorem but it is getting too lenghty
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    Quote Originally Posted by prasum View Post
    If a=e^(i*8pi/11) then find re(a+a^2+a^3+a^4+a^5) I used demoivres theorem but it is getting too lenghty
    Let \omega = e^{2\pi i/11}, so that a = \omega^4. The roots of the equation z^{11}=1 are z=\omega^k for 0\leqslant k\leqslant10. The sum of all these roots is 0. If we exclude the root z=1 then the sum of the remaining roots is 1 (and therefore so is the sum of the real parts of these roots). If you draw a picture of these 10 complex roots on the unit circle in the complex plane, you will see that they consist of a,\ a^2,\ a^3,\ a^4 and a^5, together with their complex conjugates. It follows that the real part of a+a^2+a^3+a^4+a^5 must be 1/2.
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    how do you know that re(a+a^2+a^3+a^4+a^5)=-1/2
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    It was said above that -1=w^1+\dots+w^{10}=a^1+\dots+a^5+\overline{a^1+\do  ts+a^5} (here the bar denotes complex conjugation). Also note that the real part of z+\bar{z} is twice the real part of z.
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