The point of an absolute value is that |x| is the distance from x to 0. Of course, a "distance" must be anon-negative real number. For a complex number, we can represent the number x+ iy by the point (x,y) in the "complex plane". The distance form (x, y) to (0, 0), by the Pythagorean theorem: .

That isnotbut itiswhere is the "complex conjugate" of z: the complex conjugate of z= x+ iy is which, in the case that z is real, z= x+ 0i, reduces to x so that .