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

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

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.