Complex Numbers from A to... Z

Transcription

1 Titu Andreescu Dorin Andrica Complex Numbers from A to... Z Birkhauser Boston Basel Berlin

2 Contents Preface Notation ix xiii 1 Complex Numbers in Algebraic Form Algebraic Representation of Complex Numbers Definition of complex numbers Properties concerning addition Properties concerning multiplication Complex numbers in algebraic form Powers of the number i Conjugate of a complex number Modulus of a complex number Solving quadratic equations Problems Geometric Interpretation of the Algebraic Operations Geometric interpretation of a complex number Geometric interpretation of the modulus Geometric interpretation of the algebraic operations Problems 27

3 vi Contents 2 Complex Numbers in Trigonometric Form Polar Representation of Complex Numbers Polar coordinates in the plane Polar representation of a complex number Operations with complex numbers in polar representation Geometric interpretation of multiplication Problems Then* Roots of Unity Defining the n th roots of a complex number The n th roots of unity Binomial equations Problems 52 3 Complex Numbers and Geometry Some Simple Geometric Notions and Properties The distance between two points Segments, rays and lines Dividing a segment into a given ratio Measure of an angle Angle between two lines Rotation of a point Conditions for Collinearity, Orthogonality and Concyclicity Similar Triangles Equilateral Triangles Some Analytic Geometry in the Complex Plane Equation of a line Equation of a line determined by two points The area of a triangle Equation of a line determined by a point and a direction The foot of a perpendicular from a point to a line Distance from a point to a line The Circle Equation of a circle The power of a point with respect to a circle, Angle between two circles 86

4 Contents vii 4 More on Complex Numbers and Geometry The Real Product of Two Complex Numbers The Complex Product of Two Complex Numbers The Area of a Convex Polygon Intersecting Cevians and Some Important Points in a Triangle The Nine-Point Circle of Euler Some Important Distances in ajriangle Fundamental invariants of a triangle The distance OI The distance ON The distance OH Distance between Two Points in the Plane of a Triangle Barycentric coordinates Distance between two points in barycentric coordinates The Area of a Triangle in Barycentric Coordinates Orthopolar Triangles The Simson-Wallance line and the pedal triangle Necessary and sufficient conditions for orthopolarity Area of the Antipedal Triangle Lagrange's Theorem and Applications Euler's Center of an Inscribed Polygon Some Geometric Transformations of the Complex Plane Translation Reflection in the real axis Reflection in a point Rotation Isometric transformation of the complex plane Morley's theorem Homothecy Problems Olympiad-Caliber Problems Problems Involving Moduli and Conjugates Algebraic Equations and Polynomials From Algebraic Identities to Geometric Properties Solving Geometric Problems Solving Trigonometric Problems More on then* Roots of Unity 220

5 viii Contents 5.7 Problems Involving Polygons Complex Numbers and Combinatorics Miscellaneous Problems Answers, Hints and Solutions to Proposed Problems Answers, Hints and Solutions to Routine Problems Complex numbers in algebraic representation (pp ) Geometric interpretation of the algebraic operations (p. 27) Polar representation of complex numbers (pp ) Then* roots of unity (p. 52) Some geometric transformations of the complex plane (p. 160) Solutions to the Olympiad-Caliber Problems Problems involving moduli and conjugates (pp ) Algebraic equations and polynomials (p. 181) From algebraic identities to geometric properties (p. 190) Solving geometric problems (pp ) Solving trigonometric problems (p. 220) More on then* roots of unity (pp ) Problems involving polygons (p. 237) Complex numbers and combinatorics (p. 245) Miscellaneous problems (p. 252) 302 Glossary 307 References 313 Index of Authors 317 Subject Index 319