its square root have the same sign. The real part of the principal value of the square root is always nonnegative. For example, the principal square roots of are given by: Notes In the following, the complex z and w may be expressed as: where and . Because of the discontinuous nature of the square root function in the complex plane, the following laws are not true in general. Counterexample for the principal square root: and This equality is valid only when Counterexample for the principal square root: and This equality is valid only when Counterexample for the principal square root: )This equality is valid only when A similar problem appears with other complex functions with branch cuts, e.g., the complex logarithm and the relations or which are not true in general. Wrongly assuming one of these laws underlies several faulty "proofs", for instance the following one showing that : The third equality cannot be justified (see invalid proof). It can be made to hold by changing the meaning of √ so that this no longer represents the principal square root (see above) but selects a branch for the square root that contains The left-hand side becomes either if the branch includes +i or if the branch includes −i, while the right-hand side becomes where the last equality, is a consequence of the choice of branch in the redefinition of √. Nth roots and polynomial roots The definition of a square root of as a number such that has been generalized in the following way. A cube root of is a number such that ; it is denoted If is an integer greater than two, a th root of is a number such that ; it is denoted Given any polynomial , a root of is a number such that . For example, the th roots of are the roots of the polynomial (in ) Abel–Ruffini theorem states that, in general, the roots of a polynomial of degree five or higher cannot be expressed in terms of th roots. Square roots of matrices and operators If A is a positive-definite matrix or operator, then there exists precisely one positive definite matrix or operator B with ; we then define . In general matrices may have multiple square roots or even an infinitude of them. For example, the identity matrix has an infinity of square roots, though only one of them