represents electric current) and called the imaginary unit, which is defined such that . Using this notation, we can think of i as the square root of −1, but we also have and so −i is also a square root of −1. By convention, the principal square root of −1 is i, or more generally, if x is 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. Principal square root of a complex number To find a definition for the square root that allows us to consistently choose a single value, called the principal value, we start by observing that any complex number can be viewed as a point in the plane, expressed using Cartesian coordinates. The same point may be reinterpreted using polar coordinates as the pair where is the distance of the point from the origin, and is the angle that the line from the origin to the point makes with the positive real () axis. In complex analysis, the location of this point is conventionally written If then the of is defined to be the following: The principal square root function is thus defined using the nonpositive real axis as a branch cut. If is a non-negative real number (which happens if and only if ) then the principal square root of is in other words, the principal square root of a non-negative real number is just the usual non-negative square root. It is important that because if, for example, (so ) then the principal square root is but using would instead produce the other square root The principal square root function is holomorphic everywhere except on the set of non-positive real numbers (on strictly negative reals it is not even continuous). The above Taylor series for remains valid for complex numbers with The above can also be expressed in terms of trigonometric functions: Algebraic formula When the number is expressed using its real and imaginary parts, the following formula can be used for the principal square root: where is the sign of (except that, here, sgn(0) = 1). In particular, the imaginary parts of the original number and the principal value of