It is impossible to imagine modern mathematics without complex numbers. "Complex Numbers from A to ...Z" introduces the reader to this fascinating subject which, from the time of L. Euler, has become one of the most utilized ideas in mathematics.
The exposition concentrates on key concepts and then elementary results concerning these numbers. The reader learns how complex numbers can be used to solve algebraic equations and to understand the geometric interpretation of complex numbers and the operations involving them.
The theoretical parts of the book are augmented with rich exercises and problems at various levels of difficulty. A special feature of the book is the last chapter, a selection of real outstanding Olympiad and other important mathematical contest problems solved by employing the methods already presented.
The book reflects the unique experience of the authors. It distills a vast mathematical literature, most of which is unknown to the western public, and captures the essence of an abundant problem culture.
Author(s): Titu Andreescu, Dorin Andrica
Edition: 1
Publisher: Birkhäuser Boston
Year: 2005
Language: English
Pages: 332
Cover
......Page 1
About the Authors......Page 2
Complex Numbers from A to. . . Z......Page 3
Copyright
......Page 4
Contents......Page 6
Preface......Page 10
Notation......Page 13
1.1.1 Definition of complex numbers......Page 14
1.1.2 Properties concerning addition......Page 15
1.1.3 Properties concerning multiplication......Page 16
1.1.4 Complex numbers in algebraic form......Page 18
1.1.5 Powers of the number i......Page 20
1.1.6 Conjugate of a complex number......Page 21
1.1.7 Modulus of a complex number......Page 22
1.1.8 Solving quadratic equations......Page 28
1.1.9 Problems......Page 31
1.2.1 Geometric interpretation of a complex number......Page 34
1.2.2 Geometric interpretation of the modulus......Page 36
1.2.3 Geometric interpretation of the algebraic operations......Page 37
1.2.4 Problems......Page 40
2.1.1 Polar coordinates in the plane......Page 41
2.1.2 Polar representation of a complex number......Page 43
2.1.3 Operations with complex numbers in polar representation......Page 48
2.1.5 Problems......Page 51
2.2.1 Defining the n^{th} roots of a complex number......Page 53
2.2.2 The n^{th} roots of unity......Page 55
2.2.3 Binomial equations......Page 63
2.2.4 Problems......Page 64
3.1.1 The distance between two points......Page 65
3.1.2 Segments, rays and lines......Page 66
3.1.3 Dividing a segment into a given ratio......Page 69
3.1.4 Measure of an angle......Page 70
3.1.6 Rotation of a point......Page 73
3.2 Conditions for Collinearity, Orthogonality and Concyclicity......Page 77
3.3 Similar Triangles......Page 80
3.4 Equilateral Triangles......Page 82
3.5.1 Equation of a line......Page 88
3.5.2 Equation of a line determined by two points......Page 90
3.5.3 The area of a triangle......Page 91
3.5.4 Equation of a line determined by a point and a direction......Page 94
3.5.6 Distance from a point to a line......Page 95
3.6.1 Equation of a circle......Page 96
3.6.3 Angle between two circles......Page 98
4.1 The Real Product of Two Complex Numbers......Page 101
4.2 The Complex Product of Two Complex Numbers......Page 108
4.3 The Area of a Convex Polygon......Page 112
4.4 Intersecting Cevians and Some Important Points in a Triangle......Page 115
4.5 The Nine-Point Circle of Euler......Page 118
4.6.1 Fundamental invariants of a triangle......Page 122
4.6.2 The distance OI......Page 124
4.6.3 The distance ON......Page 125
4.6.4 The distance OH......Page 126
4.7.1 Barycentric coordinates......Page 127
4.7.2 Distance between two points in barycentric coordinates......Page 129
4.8 The Area of a Triangle in Barycentric Coordinates......Page 131
4.9.1 The Simson–Wallance line and the pedal triangle......Page 137
4.9.2 Necessary and sufficient conditions for orthopolarity......Page 144
4.10 Area of the Antipedal Triangle......Page 148
4.11 Lagrange’s Theorem and Applications......Page 152
4.12 Euler’s Center of an Inscribed Polygon......Page 160
4.13.1 Translation......Page 163
4.13.3 Reflection in a point......Page 164
4.13.5 Isometric transformation of the complex plane......Page 165
4.13.6 Morley’s theorem......Page 167
4.13.7 Homothecy......Page 170
4.13.8 Problems......Page 172
5.1 Problems Involving Moduli and Conjugates......Page 173
5.2 Algebraic Equations and Polynomials......Page 189
5.3 From Algebraic Identities to Geometric Properties......Page 193
5.4 Solving Geometric Problems......Page 202
5.5 Solving Trigonometric Problems......Page 226
5.6 More on the n^{th} Roots of Unity......Page 232
5.7 Problems Involving Polygons......Page 241
5.8 Complex Numbers and Combinatorics......Page 249
5.9 Miscellaneous Problems......Page 258
6.1.1 Complex numbers in algebraic representation (pp. 18–21)......Page 265
6.1.3 Polar representation of complex numbers (pp. 39–41)......Page 270
6.1.4 The n^{th} roots of unity (p. 52)......Page 272
6.1.5 Some geometric transformations of the complex plane (p. 160)......Page 273
6.2.1 Problems involving moduli and conjugates (pp. 175–176)......Page 274
6.2.2 Algebraic equations and polynomials (p. 181)......Page 281
6.2.3 From algebraic identities to geometric properties (p. 190)......Page 284
6.2.4 Solving geometric problems (pp. 211–213)......Page 286
6.2.5 Solving trigonometric problems (p. 220)......Page 299
6.2.6 More on the n^{th} roots of unity (pp. 228–229)......Page 301
6.2.7 Problems involving polygons (p. 237)......Page 304
6.2.8 Complex numbers and combinatorics (p. 245)......Page 310
6.2.9 Miscellaneous problems (p. 252)......Page 314
Glossary......Page 318
References......Page 324
Index of Authors......Page 328
Subject Index......Page 330
