COMPLEX NUMBERS. Consider the quadratic equation; x 2 1 ¡ 0. It has no solutions in the real number system since x 2 вг 1 or x де жзззз. 1 ий j j 1 ie. j2 1. (b) If z = a + ib is the complex number, then a and b are called real and Addition of complex numbers satisfies the following properties. One way of introducing the field C of complex numbers is via the arithmetic of 2 × 2 matrices. DEFINITION A complex number is a matrix of the form. [ x −y.

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After studying this chapter you should. • understand how quadratic equations lead to complex numbers and how to plot complex numbers on an Argand diagram. Abstract. This article discusses some introductory ideas associated with complex numbers, their algebra and geometry. This includes a look at their importance. Roots of unity. The relation- ship between exponential and trigonometric functions. The geometry of the Argand diagram. 1 The Need For Complex Numbers.

While linear equations are always solvable in real numbers, not all quadratic equations have this property. Until the 18th century, mathematicians avoided quadratic equations that were not solvable over R. This became one of the most useful symbols in mathematics. The study of complex numbers continues and has been enhanced in the last two and a half centuries; in fact, it is impossible to imagine modern mathematics without complex numbers. All mathematical domains make use of them in some way. This is true of other disciplines as well: for example, mechanics, theoretical physics, hydrodynamics, and chemistry. Our main goal is to introduce the reader to this fascinating subject. The book runs smoothly between key concepts and elementary results concerning complex numbers. The reader has the opportunity to learn how complex numbers can be employed in solving algebraic equations, and to understand the geometric interpretation of com- x Preface plex numbers and the operations involving them. In Chapters 3 and 4 we cover important applications in Euclidean geometry. The wealth of examples we provide, the presentation of many topics in a personal manner, the presence of numerous original problems, and the attention to detail in the solutions to selected exercises and problems are only some of the key features of this book. Among the techniques presented, for example, are those for the real and the complex product of complex numbers. In complex number language, these are the analogues of the scalar and cross products, respectively.

And like the complex numbers, the quaternions are surprisingly useful: They can be used to model the rotation of three-dimensional space, which makes them invaluable in rendering digital landscapes and spherical video, and in positioning and orienting objects like spaceships and cellphones in our three-dimensional world.

And just as with the quaternions, we need some special rules to govern how to multiply all the imaginary units. Like the quaternions, octonion multiplication is not commutative. But extending our idea of number out to the octonions costs us the associativity of multiplication as well.

When would anyone ever use that? Well, some physicists believe that the octonions may hold the key to describing how the strong, weak and electromagnetic forces act on quarks, leptons and their anti-particles.

If true, this could help resolve one of the great mysteries in modern physics. By repeatedly extending the real numbers to create larger systems — the complex numbers, the quaternions, the octonions — in which we can add, subtract, multiply and divide, we lose a little familiarity with each step.

Along the way, we may also lose touch with what we think of as real. But what we gain are new ways of thinking about the world.

And we can always find a use for that. In complex number language, these are the analogues of the scalar and cross products, respectively. After covering this part, the reader will appreciate the use of these techniques.

A special feature of the book is Chapter 5, an outstanding selection of genuine Olympiad and other important mathematical contest problems solved using the methods already presented.

This work does not cover all aspects pertaining to complex numbers. It distills a vast mathematical literature, most of which is unknown to the western public, capturing the essence of an abundant problem-solving culture.

Our work is partly based on a Romanian version, Numere complexe de la A la. Z, authored by D.

Andrica and N. The targeted audience includes high school students and their teachers, undergraduates, mathematics contestants such as those training for Olympiads or the W.

Putnam Mathematical Competition, their coaches, and any person interested in essential mathematics. This book might spawn courses such as Complex Numbers and Euclidean Geometry for prospective high school teachers, giving future educators ideas about things they could do with their brighter students or with a math club.

This would be quite a welcome development.

We Preface xi would also like to thank the referees who provided pertinent suggestions that directly contributed to the improvement of the text. The set R2 , together with the addition and multiplication operations, is called the set of complex numbers, denoted by C.

The claim holds due to the associativity of the addition of real numbers.

Algebraic Representation of Complex Numbers 3 The reader can easily prove the claims a , c and d. Algebraic Representation of Complex Numbers 1.