section_path: Complex roots/Square roots
section_name: Square roots
paragraph_text: Square roots
  
  The two square roots of a complex number are always negatives of each other. For example, the square roots of −4 are 2 i and −2 i, and the square roots of i are
  
  1 2 (1 + i) and − 1 2 (1 + i). {\displaystyle {\tfrac {1}{\sqrt {2}}}(1+i)\quad {\text{and}}\quad -{\tfrac {1}{\sqrt {2}}}(1+i).}
  
  If we express a complex number in polar form, then the square root can be obtained by taking the square root of the radius and halving the angle:
  
  r e i θ = ± r ⋅ e i θ / 2. {\displaystyle {\sqrt {re^{i\theta }}}=\pm {\sqrt {r}}\cdot e^{i\theta /2}.}
  
  A principal root of a complex number may be chosen in various ways, for example
  
  r e i θ = r ⋅ e i θ / 2 {\displaystyle {\sqrt {re^{i\theta }}}={\sqrt {r}}\cdot e^{i\theta /2}}
  
  which introduces a branch cut in the complex plane along the positive real axis with the condition 0 ≤ θ < 2 π, or along the negative real axis with − π < θ ≤ π.
  
  Using the first(last) branch cut the principal square root z {\displaystyle \scriptstyle {\sqrt {z}}} maps z {\displaystyle \scriptstyle z} to the half plane with non-negative imaginary(real) part. The last branch cut is presupposed in mathematical software like Matlab or Scilab.