Phoebus J. Dhrymes - Mathematics for Econometrics

Let z 1, a complex number, be represented in Fig. by the point ( x 1, y 1), its coordinates .

It is easily verified that the length of the line from the origin to the point ( x 1, y 1) represents the modulus of z 1, which for convenience we denote by r 1. Let the angle described by this line and the abscissa be denoted by 1. As is well known from elementary trigonometry, we have We may thus write the complex number as Further, we may define the quantity and, consequently, write the complex number in the standard polar form In the representation above, r 1 is the modulus and 1 the argument of the complex number z 1. It may be shown that the quantity as defined in Eq.() has all the properties of real exponentials insofar as the operations of multiplication and division are concerned. If we confine the argument of a complex number to the range [0,2), we have a unique correspondence between the ( x , y ) coordinates of a complex number and the modulus and argument needed to specify its polar form. Thus, for any complex number z , the representations where are completely equivalent.

In polar form, multiplication and division of complex numbers are extremely simple operations. Thus, provided z 2 0.

We may extend our discussion to complex vectors , i.e. ordered n -tuples of complex numbers. Thus is a complex vector, where x and y are n -element (real) vectors (a concept to be defined immediately below). As in the scalar case, two complex vectors z 1, z 2 are equal if and only if