Chapter 1 From the Real Numbers to the Complex Numbers 1. Introduction Many problems throughout mathematics and physics illustrate an amazing princi- ple: ideas expressed within the realm of real numbers find their most elegant ex- pression through the unexpected intervention of complex numbers. Many of these delightful interventions arise in elementary, recreational mathematics. On the other hand most college students either never see complex numbers in action or they wait until the junior or senior year in college, at which time the sophisticated courses have little time for the elementary applications. Hence too few students witness the beauty and elegance of complex numbers. This book aims to present a variety of elegant applications of complex analysis and geometry in an accessible but precise fashion. We begin at the beginning, by recalling various number systems such as the integers Z, the rational numbers Q, and the real numbers R, before even defin- ing the complex numbers C. We then provide three possible equivalent definitions. Throughout we strive for as much geometric reasoning as possible. 2. Number systems The ancients were well aware of the so-called natural numbers, written 1, 2, 3,.... Mathematicians write N for the collection of natural numbers together with the usual operations of addition and multiplication. Partly because subtraction is not always possible, but also because negative numbers arise in many settings such as financial debts, it is natural to expand the natural number system to the larger system Z of integers. We assume that the reader has some understanding of the integers the set Z is equipped with two distinguished members, written 1 and 0, and two operations, called addition (+) and multiplication (*), satisfying familiar laws. These operations make Z into what mathematicians call a commutative ring with unit 1. The integer 0 is special. We note that each n in Z has an additive 1

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