Introduction to the Geometry of Complex Numbers

Roland Deaux, Howard Eves

• Publisher: Dover Publications
• Year: 2013
• Print ISBN: 9780486466293
• eBook ISBN: 9780486158044
• Topic: Mathematics
• Subtopic: Geometry
• Index: Mathematics

Geared toward readers unfamiliar with complex numbers, this text explains how to solve the kinds of problems that frequently arise in the applied sciences, especially electrical studies. To assure an easy and complete understanding, it develops topics from the beginning, with emphasis on constructions related to algebraic operations.
The three-part treatment begins with geometric representations of complex numbers and proceeds to an in-depth survey of elements of analytic geometry. Readers are assured of a variety of perspectives, which include references to algebra, to the classical notions of analytic geometry, to modern plane geometry, and to results furnished by kinematics. The third chapter, on circular transformations, revives in a slightly modified form the essentials of the projective geometry of real binary forms. Numerous exercises appear throughout the text.

CHAPTER ONE

GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS

I. FUNDAMENTAL OPERATIONS

1. Complex coordinate. Consider the complex number

x + iy (x and y real)

which we denote by z. Draw, in a plane, two perpendicular coordinate axes Ox, Oy. The point Z having for abscissa the real part x of the number z and for ordinate the coefficient y of i is called the representative point, or the image, of the number z. Conversely, each real point Z₁, of the plane is the image of a unique complex number z₁ equal to the abscissa of Z₁ increased by the product with i of the ordinate of this point. The number z₁ is called the complex coordinate, or the affix, of the point Z₁.

A plane in which each real point is considered as the image of a complex number is called the Gauss plane, the Cauchy plane, or the plane of the complex variable.

We shall denote a point of the Gauss plane by an upper case letter, and its affix by the corresponding lower case letter.

Corollaries. 1° The Ox axis is the locus of the images of the real numbers. The Oy axis is the locus of the images of the pure imaginary numbers. This is why Ox and Oy are sometimes called the real axis and the imaginary axis of the Gauss plane.

2° The number—z is the affix of the symmetric of point Z with respect to the origin O.

2. Conjugate coordinates. The complex number conjugate to

z = x + iy

will be designated by the notation

z = x – iy

which is read, “ z bar,” and which will never appear as the written representation of a vector. A vector will always be indicated by the juxtapos...

Table of contents

• DOVER BOOKS ON MATHEMATICS
• Title Page
• Copyright Page
• PREFACE TO THE AMERICAN EDITION
• FOREWORD
• Table of Contents
• CHAPTER ONE - GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS
• CHAPTER TWO - ELEMENTS OF ANALYTIC GEOMETRY IN COMPLEX NUMBERS
• CHAPTER THREE - CIRCULAR TRANSFORMATIONS
• INDEX - References are to articles or to exercises (E)