1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Need help finding roots for a complex number using angles

  1. Nov 22, 2016 #1
    • Member warned that the homework template is required
    so i am starting with the equation x3 = √(3) - i

    first : change to a vector

    magnitude = √[ (√(3))2 + 12] = 2
    and angle = tan-1( 1/√(3) ) = 30 degrees
    (in fourth quadrant)

    so i have a vector of 2 ∠ - 30

    so i plot the vector on the graph and consider that :

    1. the fundamental theorum of algebra tells me i must have three roots.
    2. multiplying two complex roots written as vectors will result in the magnitudes multiplied and the angles added together

    so to get the root, i simply find the cube root of the magnitude and divide the angle by three

    this gets me two initial answers:

    3√(2) ∠ - 10
    3√(2) ∠ 110

    the question now comes up with the third root, because I am assuming there should be only one more. But i'm getting MANY more. Since the magnitude remains as 3√(2) then i simply need to find an angle that can be divided into three to get -30 (or 330) degrees.

    well : if i take (360 + 330)/3 = 130, it works
    if i take (-30 - 360)/3 = -110 , it works.
    in fact any addition of 360 works, so I can only guess that these are not really roots. But they do produce entirely different complex numbers - all of them.

    I have been assuming there are only three possible roots. Where is my logic going wrong here? Please help. I would greatly appreciate a little enlightenment
  2. jcsd
  3. Nov 22, 2016 #2


    User Avatar
    Homework Helper

    (360 + 330)/3 = 230. It is the angle of the third root.
    (-30 - 360)/3 = -130, which is equivalent to 230, the previous result.
    If you add any times 360° to the angle of the vector, you get the same vector. So you have only three different roots, and usually they are chosen form the interval [0, 360°) .
  4. Nov 22, 2016 #3
    ha! a simple addition mistake threw me completely off. thank you so much. everything makes sense now and i can walk away from this .content
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted