Complex Numbers and Exponentials

A complex number is nothing more than a point in the xy–plane. The first component, x, of the complex number (x, y) is called its real part and the second component, y, is called its imaginary part, even though there is nothing imaginary about it. The sum and product of two complex numbers (x1, y1) and (x2, y2) is defined by (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2) (x1, y1) (x2, y2) = (x1x2 − y1y2, x1y2 + x2y1) respectively. We’ll get an effective memory aid for the definition of multiplication shortly. It is conventional to use the notation x + iy (or in electrical engineering country x + jy) to stand for the complex number (x, y). In other words, it is conventional to write x in place of (x, 0) and i in place of (0, 1). In this notation, the sum and product of two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 is given by