Algebra & Geometry: An Introduction to University Mathematics

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Cover Half Title Title Page Copyright Page Dedication Contents Preface to the Second Edition Preface to the First Edition Prolegomena SECTION I: IDEAS CHAPTER 1: The Nature of Mathematics 1.1. MATHEMATICS IN HISTORY 1.2. MATHEMATICS TODAY 1.3. THE SCOPE OF MATHEMATICS 1.4. WHAT THEY (PROBABLY) DIDN’T TELL YOU IN SCHOOL 1.5. FURTHER READING CHAPTER 2: Proofs 2.1. MATHEMATICAL TRUTH 2.2. FUNDAMENTAL ASSUMPTIONS OF LOGIC 2.3. FIVE EASY PROOFS 2.4. AXIOMS 2.5. UN PETIT PEU DE PHILOSOPHIE 2.6. MATHEMATICAL CREATIVITY 2.7. PROVING SOMETHING FALSE 2.8. TERMINOLOGY 2.9. ADVICE ON PROOFS CHAPTER 3: Foundations 3.1. SETS 3.2. BOOLEAN OPERATIONS 3.3. RELATIONS 3.4. FUNCTIONS 3.5. EQUIVALENCE RELATIONS 3.6. ORDER RELATIONS 3.7. QUANTIFIERS 3.8. PROOF BY INDUCTION 3.9. COUNTING 3.10. INFINITE NUMBERS CHAPTER 4: Algebra Redux 4.1. RULES OF THE GAME 4.2. ALGEBRAIC AXIOMS FOR REAL NUMBERS 4.3. SOLVING QUADRATIC EQUATIONS 4.4. BINOMIAL THEOREM 4.5. BOOLEAN ALGEBRAS 4.6. CHARACTERIZING REAL NUMBERS SECTION II: THEORIES CHAPTER 5: Number Theory 5.1. REMAINDER THEOREM 5.2. GREATEST COMMON DIVISORS 5.3. FUNDAMENTAL THEOREM OF ARITHMETIC 5.4. MODULAR ARITHMETIC 5.5. CONTINUED FRACTIONS CHAPTER 6: Complex Numbers 6.1. COMPLEX NUMBER ARITHMETIC 6.2. COMPLEX NUMBER GEOMETRY 6.3. EULER’S FORMULA FOR COMPLEX NUMBERS 6.4. MAKING SENSE OF COMPLEX NUMBERS CHAPTER 7: Polynomials 7.1. TERMINOLOGY 7.2. THE REMAINDER THEOREM 7.3. ROOTS OF POLYNOMIALS 7.4. FUNDAMENTAL THEOREM OF ALGEBRA 7.5. ARBITRARY ROOTS OF COMPLEX NUMBERS 7.6. GREATEST COMMON DIVISORS OF POLYNOMIALS 7.7. IRREDUCIBLE POLYNOMIALS 7.8. PARTIAL FRACTIONS 7.9. RADICAL SOLUTIONS 7.10. ALGEBRAIC AND TRANSCENDENTAL NUMBERS 7.11. MODULAR ARITHMETIC WITH POLYNOMIALS CHAPTER 8: Matrices 8.1. MATRIX ARITHMETIC 8.2. MATRIX ALGEBRA 8.3. SOLVING SYSTEMS OF LINEAR EQUATIONS 8.4. DETERMINANTS 8.5. INVERTIBLE MATRICES 8.6. DIAGONALIZATION 8.7. BLANKINSHIP’S ALGORITHM CHAPTER 9: Vectors 9.1. VECTORS GEOMETRICALLY 9.2. VECTORS ALGEBRAICALLY 9.3. GEOMETRIC MEANING OF DETERMINANTS 9.4. GEOMETRY WITH VECTORS 9.5. LINEAR FUNCTIONS 9.6. ALGEBRAIC MEANING OF DETERMINANTS 9.7. QUATERNIONS CHAPTER 10: The Principal Axes Theorem 10.1. ORTHOGONAL MATRICES 10.2. ORTHOGONAL DIAGONALIZATION 10.3. CONICS AND QUADRICS CHAPTER 11: What are the Real Numbers? 11.1. THE PROPERTIES OF THE REAL NUMBERS 11.2. APPROXIMATING REAL NUMBERS BY RATIONAL NUMBERS 11.3. A CONSTRUCTION OF THE REAL NUMBERS Epilegomena Bibliography Index