text: If  is added to , the complex number is not changed, but this adds  to the argument of the th root, and provides a new th root. This can be done  times, and provides the  th roots of the complex number.
  It is usual to choose one of the  th root as the principal root. The common choice is to choose the th root for which  that is, the th root that has the largest real part, and, if there are two, the one with positive imaginary part. This makes the principal th root a continuous function in the whole complex plane, except for negative real values of the radicand. This function equals the usual th root for positive real radicands. For negative real radicands, and odd exponents, the principal th root is not real, although the usual th root is real. Analytic continuation shows that the principal th root is the unique complex differentiable function that extends the usual th root to the complex plane without the nonpositive real numbers.
  If the complex number is moved around zero by increasing its argument, after an increment of  the complex number comes back to its initial position, and its th roots are permuted circularly (they are multiplied by ). This shows that it is not possible to define a th root function that is continuous in the whole complex plane.