Square root any nonnegative number, then the principal square root of −"x" is The right side (as well as its negative) is indeed a square root of −"x", since For every non-zero complex number "z" there exist precisely two numbers "w" such that : the principal square root of "z" (defined below), and its negative. There are two complex numbers that square to a given arbitrary non-zero imaginary number formula_11 with real formula_12 where the two-digit pattern {3, 6} repeats over and over again in the partial denominators. Since , the above is also identical to the following generalized continued fractions: The
Square root of a matrix but if the process converges, the matrix formula_26 converges quadratically to a square root , while formula_27 converges to its inverse, . Yet another iterative method is obtained by taking the well-known formula of the Babylonian method for computing the square root of a real number, and applying it to matrices. Let "X" = "I", where "I" is the identity matrix. The iteration is defined by Again, convergence is not guaranteed, but if the process converges, the matrix formula_29 converges quadratically to a square root "A". Compared to Denman–Beavers iteration, an advantage of the Babylonian method is that only one