| Chapter 1. Heuristics 1.1. Search for a Pattern 1.2. Draw a Figure 1.3. Formulate an Equivalent Problem 1.4. Modify the Problem 1.5. Choose Effective Notation 1.6. Exploit Symmetry 1.7. Divide into Cases 1.8. Work Backward 1.9. Argue by Contradiction 1.10. Pursue Parity l.ll. Consider Extreme Cases 1.12. Generalize Chapter 2. Two Important Principles: Induction and Pigeonhole 2.1. Induction: Build on P(k) 2.2. Induction: Set Up P(k + 1) 2.3. Strong Induction 2.4. Induction and Generalization 2.5. Recursion 2.6. Pigeonhole Principle Chapter 3. Arithmetic 3.1. Greatest Common Divisor 3.2. Modular Arithmetic 3.3. Unique Fac'torization 3.4. Positional Notation 3.5. Arithmetic of Complex Numbers Chapter 4. Algebra 4.1. Algebraic Identities 4.2. Unique Factorization of Polynomials 4.3. The Identity Theorem 4.4. Abstract Algebra Chapter 5. Summation of Series 5.1. Binomial Coefficients 5.2. Geometric Series 5.3. Telescoping Series 5.4. Power Series Chapter 6. Intermediate Real Analysis 6.1. Continuous Functions 6.2. The Intermediate-Value Theorem 6.3. The Derivative 6.4. The Extreme-Value Theorem 6.5. Rolle's Theorem 6.6. The Mean Value Theorem 6.7. L'Hopital's Rule 6.8. The Integral 6.9. The Fundamental Theorem Chapter 7. Inequalities 7.1. Basic Inequality Properties 7.2. Arithmetic-Mean-Geometric-Mean Inequa 7.3. Cauchy-Schwarz Inequality 7.4. Functional Considerations 7.5. Inequalities by Series 7.6. The Squeeze Principle Chapter 8. Geometry 8.1. Classical Plane Geometry 8.2. Analytic Geometry 8.3. Vector Geometry 8.4. Complex Numbers in Geometry Contents Glosary of Symbols and Definitions Sources Index |
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