What is the strategy for finding the cube root of complex numbers?

Answer 1

Here's how you can find any power (fractions would be a root of a number) of any number (complex or real). A real number is a subset of the complex number set, with the imaginary part = 0. I'll refer you to a related link on Euler's formula for information about how this is derived. A complex number can be graphed on the Real-Imaginary plane, with reals on the horizontal axis, and imaginary on the vertical. Convert the complex number from x-y style coordinates in this plane to polar coordinates.

For a complex number a + bi, here's how you do that. We will end up with a magnitude and an angle. The magnitude is sqrt(a² + b²). The angle is found by tan-1(b/a). Now to find a power, apply the power to the magnitude (for cube root this is exponent of 1/3). Then multiply the angle by the power (in this case you divide by 3). Really for a cube root there will be 3 distinct roots. Since a the angle of a circle is 360° or 2pi radians, you can add 2pi radians to the angle of the original complex number, then divide by 3 to determine the second root. Add 4pi radians to the original angle and then divide by 3 to determine the 3rd root. Then convert back to x-y coordinates if you want to:

Magnitude*(cos(angle) + i*sin(angle)), for each of the 3 angles that you determined.
See the question: 'Strategy for finding the cube root of complex numbers'

Strategy_for_finding_the_cube_root_of_complex_numbers