Vector Representation of Complex Numbers

Complex numbers, described in Cartesian coordinates, can be depicted as vectors. These can be expressed in polar form, showcasing their magnitude and angle.

When complex numbers are inputted into a function, the result is another complex number. This highlights a key feature: the function's zero point, from which the vector representation can originate.

Consider a function defined as the complex factors' numerator product divided by the denominator's product.

Since each complex factor is essentially a vector, the function's magnitude at any point is calculated as the product of zero lengths divided by the product of pole lengths.

The function's angle is calculated as the sum of zero angles subtracted from the sum of pole angles. These angles are measured from the positive extension of the real axis of vectors drawn from the zero or pole of the function to the given point.

To solve a complex function at a specific point, vectors originating from the function's zeros and poles are calculated, ending at the chosen point. Substituting vector equations provides the magnitude and angle.