The complex structure of a point x = x_1, x_2 in the plane is defined by the linear map J:R^2->R^2 J(x_1, x_2) = (-x_2, x_1), and corresponds to a counterclockwise rotation by π/2. This map satisfies J^2 | = | -I (J x)·(J y) | = | x·y (J x)·x | = | 0, where I is the identity map.