Algebra & Geometry: An Introduction to University Mathematics

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