Complex Numbers and Functions

Abstract

This chapter starts with the early discovery of complex numbers and their role in solving algebraic equations. Complex numbers have the algebraic form \(x+i\, y\), where x, y are real numbers, but they can also be geometrically represented as vectors (x, y) in the plane. Both representations have important advantages; the first one is well-suited for algebraic manipulations while the second provides significant geometric intuition. There is also a natural notion of distance between complex numbers that satisfies the familiar triangle inequality. Complex numbers also have a polar form \((r,\,\theta )\) based on their distance r to the origin and angle \(\theta \) from the positive real semi-axis. This alternative representation provides additional insight, both algebraic and geometric, and this is explicitly manifested even in simple operations, such as multiplication and division.