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 is the unique positive square root of the positive operator "A""A", and "U" is defined by . If "A" is not invertible, then it still has a polar composition in which "P" is defined in the same way (and is unique). The unitary operator "U" is not unique. Rather it is possible to determine a "natural" unitary operator as follows: "AP" is a unitary operator from the range of "A" to itself, which can be extended by the identity on the kernel of "A". The resulting unitary operator "U" then yields the polar decomposition of "A". By Choi's result, a