A Bridge to Advanced Mathematics: From Natural to Complex Numbers

Description

Most introduction to proofs textbooks focus on the structure of rigorous mathematical language and only use mathematical topics incidentally as illustrations and exercises. In contrast, this book gives students practice in proof writing while simultaneously providing a rigorous introduction to number systems and their properties. Understanding the properties of these systems is necessary throughout higher mathematics. The book is an ideal introduction to mathematical reasoning and proof techniques, building on familiar content to ensure comprehension of more advanced topics in abstract algebra and real analysis with over 700 exercises as well as many examples throughout. Readers will learn and practice writing proofs related to new abstract concepts while learning new mathematical content. The first task is analogous to practicing soccer while the second is akin to playing soccer in a real match. The authors believe that all students should practice and play mathematics. The book is written for students who already have some familiarity with formal proof writing but would like to have some extra preparation before taking higher mathematics courses like abstract algebra and real analysis. Cover Title page Copyright Contents Preface 1. The Content of the Book 2. How to Use This Book? Chapter 1. Natural Numbers N 1.1. Basic Properties 1.2. The Principle of Induction 1.3. Arithmetic and Geometric Progressions 1.4. The Least Element Principle 1.5. There are 10 Kinds of People in the World 1.6. Divisibility 1.7. Counting and Binomial Formula 1.8. More Exercises Chapter 2. Integer Numbers Z 2.1. Basic Properties 2.2. Integer Division 2.3. Euclidean Algorithm Revisited 2.4. Congruences and Modular Arithmetic 2.5. Modular Equations 2.6. The Chinese Remainder Theorem 2.7. Fermat and Euler Theorems 2.8. More Exercises Chapter 3. Rational Numbers Q 3.1. Basic Properties 3.2. Not Everything Is Rational 3.3. Fractions and Decimal Representations 3.4. Finite Continued Fractions 3.5. Farey Sequences and Pick’s Formula 3.6. Ford Circles and Stern–Brocot Trees 3.7. Egyptian Fractions 3.8. More Exercises Chapter 4. Real Numbers R 4.1. Basic Properties 4.2. The Real Numbers Form a Field 4.3. Order and Absolute Value 4.4. Completeness 4.5. Supremum and Infimum of a Set 4.6. Roots and Powers 4.7. Expansion of Real Numbers 4.8. More Exercises Chapter 5. Sequences of Real Numbers 5.1. Basic Properties 5.2. Convergent and Divergent Sequences 5.3. The Monotone Convergence Theorem and Its Applications 5.4. Subsequences 5.5. Cauchy Sequences 5.6. Infinite Series 5.7. Infinite Continued Fractions 5.8. More Exercises Chapter 6. Complex Numbers C 6.1. Basic Properties 6.2. The Conjugate and the Absolute Value 6.3. Polar Representation of Complex Numbers 6.4. Roots of Complex Numbers 6.5. Geometric Applications 6.6. Sequences of Complex Numbers 6.7. Infinite Series of Complex Numbers 6.8. More Exercises Epilogue Appendix. Sets, Functions, and Relations A.1. Logic A.2. Sets A.3. Functions A.4. Cardinality of Sets A.5. Relations A.6. Proofs A.7. Peano’s Axioms and the Construction of Integers A.8. More Exercises Bibliography Index Back Cover