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 may have complex entries even if "M" is a real matrix; this will be the case, in particular, if the determinant "δ" is negative. When a matrix can be expressed as real multiple of the exponent of a matrix, formula_5 then the square root is formula_6 In this case the square root is real, and can be interpreted as the square root of a type of complex number. If "M" is an idempotent matrix, meaning that "MM" = "M", then if it is not the identity matrix its determinant is zero, and its trace equals its rank which (excluding the