Math CentralQuandaries & Queries


Question from dianah, a student:

how to find the square roots of complex number, z=3+4i

Two ways, depending on whether you like trigonometry or quadratic equations better.

  1. Let one of the roots, $w,$ be $x+iy.$ Square it; what do you get? Now set the real part of that to 3, and the imaginary part to 4.
    This gives a system of two second-order equations; turn it into a single quadratic by algebra. You should get two solutions; they are the square roots.

  2. Convert to polar form: $r = \sqrt(3^2 + 4^2), \theta = \arctan(4/3).$ The radius of the square root is the square root of the original radius; the argument (angle) is half the original angle. As the original angle theta can also be thought of as $\theta + 2 \pi, \large \frac{\theta}{2} + \pi$ is also a square root; this is always the negative of the first one you found. Now convert back.

Good Hunting!

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