1 - Complex Numbers

Summary

You, have you really understood all that stuff?

What?

The story of imaginary numbers?

Robert Musil, The Confusions of Young Törless (1907)

Complex numbers lie at the very core of quantum mechanics and are therefore absolutely essential to a basic understanding of quantum computation. In this chapter we present this important system of numbers from both the algebraic and the geometric standpoints. Section 1.1 presents some motivation and the basic definitions. The algebraic structure and operations on complex numbers are given in Section 1.2. The chapter concludes with Section 1.3, where complex numbers are presented from a geometric point of view and advanced topics are discussed. Our hope is that this chapter will help you get a little closer to what Sir Roger Penrose has very aptly called the “magic of complex numbers” (Penrose, 2005).

Reader Tip. Many readers will find that they are already familiar with some of the material presented in this chapter. The reader who feels confident in her comprehension of the fundamental knowledge of complex numbers, the basic operations, and their properties can safely move on to later chapters. We suggest, though, that you at least skim through the following pages to see what topics are covered. Return to Chapter 1 as a reference when needed (using the index to find specific topics).

BASIC DEFINITIONS

The original motivation for the introduction of complex numbers was the theory of algebraic equations, the part of algebra that seeks solutions of polynomial equations.