Need help finding roots for a complex number using angles

[fishspawned]

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
and
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

 

[ehild]

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
and
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

(360 + 330)/3 = 230. It is the angle of the third root.

if i take (-30 - 360)/3 = -110 , it works.

(-30 - 360)/3 = -130, which is equivalent to 230, the previous result.

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

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°) .

 

[fishspawned]

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