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Complex roots of unity

  1. Feb 22, 2007 #1
    1. The problem statement, all variables and given/known data

    I need to find the real and imaginary roots of z^4 = -1.

    3. The attempt at a solution

    The polar coordinates of -1 are at (-1, pi), (-1, 3pi) etc so if I assume the solutions take the form z = exp[i n theta] then

    n theta = pi + 2npi

    This dosen't seem to give the correct roots though, what am I doing wrong? I don't want the solution, just the method so I can work it through for myself if anyone can help.

  2. jcsd
  3. Feb 22, 2007 #2


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    Are you familiar with the fact that:

    Multiplying two complex numbers yields a complex number whose angle to the positive real axis is the sum of the factors' angles to the same axis (while the modulus/length of the complex number gained is the product of the facturs' moduli)?

    Thus, the fourth roots of a complex number must have one fourth the angle that complex number may have, as measured to the positive real axis.

    Remember that the number (-1) can be said to have the angles [itex]\pi,3\pi,5\pi,7\pi[/itex] to the positive x-axis.
    Ask yourself why these four angular representations are both necessary and sufficient to find ALL fourth roots of (-1)!
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