A simple geometric method for finding the square roots of a complex number

Question

Consider the following method for finding the $2$ square roots of a complex number:

• Draw the number on an $XY$ plane, as a vector starting from $(0,0)$

• Let $L$ denote the length of that vector

• Let $A$ denote the angle between that vector and the positive side of the $X$ axis

• The square roots are:

  • A vector starting from $(0,0)$, with length $\sqrt[2]{L}$ and angle $\frac{1}{2}A$
  • A vector starting from $(0,0)$, stretching to the same length in the opposite direction

Is this method generally correct for any given complex number?

Thanks

Answer 1

Yes it is correct. If you write the complex number in polar form, $z=L \exp(iA)$ you are computing $\sqrt L \exp (i\frac A 2)$ and $\sqrt L \exp (i(\frac A 2+ \pi ))$ which are just what you want.