Roots in the complex plane

Real numbers alone do not suffice for algebra. To obtain elegant, general results, we extend the real field by adjoining so-called imaginary numbers, introduced by Cardano and Bombelli as early as the 16^th century. Why? To solve polynomial equations with real coefficients in complete generality. We do this by introducing the imaginary number "i", defined by the curious property i² = –1. The number system is then extended to two dimensions by considering expressions of the form a + bi (a and b are real numbers), called complex numbers. We can add, multiply, and divide these numbers without difficulty, and the complex numbers form a field (to use the technical term). From the 19^th century onward, the geometry of these numbers was studied extensively, notably by Gauss and Cauchy.

Visualizing addition and multiplication ------------------------------------------

The complex number z = a + bi can be represented by the point with Cartesian coordinates (a, b) in the plane. Switching to polar coordinates allows us to write this same point in the form

$(r\cos \theta ,r\sin \theta )(r\; \backslash cos\backslash theta,\; r\; \backslash sin\backslash theta)$,