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 2 by 2 matrix denominator "t" be zero. The formula above fails completely if "δ" and "τ" are both zero; that is, if "D" = −"A" and "A" = −"BC", so that both the trace and the determinant of the matrix are zero. In this case, if "M" is the null matrix (with "A" = "B" = "C" = "D" = 0), then the null matrix is also a square root of "M", as are for any real or complex values of "b" and "c". Otherwise "M" has no square root. If "M" is diagonal (that is, "B" = "C" = 0), one can