Pure and Applied Undergraduate Texts
Volume: 61; 2023; 442 pp
MSC: Primary 00; 03; 05; 11; 97; Secondary 68;
Lighten up about mathematics! Have fun. If you read this book, you will have to endure bad math puns and jokes and out-of-date pop culture references. You'll learn some really cool mathematics to boot. In the process, you will immerse yourself in living, thinking, and breathing logical reasoning. We like to call this proofs, which to some is a bogey word, but to us it is a boogie word. You will learn how to solve problems, real and imagined. After all, math is a game where, although the rules are pretty much set, we are left to our imaginations to create. Think of this book as blueprints, but you are the architect of what structures you want to build. Make sure you lay a good foundation, for otherwise your buildings might fall down. To help you through this, we guide you to think and plan carefully. Our playground consists of basic math, with a loving emphasis on number theory. We will encounter the known and the unknown. Ancient and modern inquirers left us with elementary-sounding mathematical puzzles that are unsolved to this day. You will learn induction, logic, set theory, arithmetic, and algebra, and you may one day solve one of these puzzles.
Readership
Appropriate for a course serving as a transition to advanced mathematics and for undergraduate students interested in mathematical reasoning and an introduction to proofs.
Author(s): Bennett Chow
Series: Pure and Applied Undergraduate Texts, 61
Edition: 1
Publisher: American Mathematical Society
Year: 2023
Language: English
Commentary: 2020 Mathematics Subject Classification. Primary 00-XX, 03-XX, 05-XX, 11-XX, 97-XX; Secondary 68-XX.
Pages: 442
City: Rhode Island
Tags: Proof Theory; Number Theory; Mathematical Logic; Mathematical Induction; Sets; Functions; Modular Arithmetic; Counting Finite Sets; Class Arithmetic; Groups; Fields
Cover
Title page
Contents
Preface
Philosophy about learning and teaching
Content of this book
Style of this book
Problem solving
LaTeX
Origins
Further reading
Acknowledgments
Notations and Symbols
Chapter 1. Evens, Odds, and Primes: A Taste of Number Theory
1.1. A first excursion into prime numbers
1.2. Even and odd integers
1.3. Calculating primes and the sieve of Eratosthenes
1.4. Division
1.5. Greatest common divisor
1.6. Statement of prime factorization
1.7*. Perfect numbers
1.8*. One of the Mersenne conjectures
1.9*. Twin primes: An excursion into the unknown
1.10*. Goldbach’s conjecture
1.11. Hints and partial solutions for the exercises
Chapter 2. Mathematical Induction
2.1. Mathematical induction
2.2. Rates of growth of functions
2.3. Sums of powers of the first ? positive integers
2.4. Strong mathematical induction
2.5. Fibonacci numbers
2.6. Recursive definitions
2.7. Arithmetic and algebraic equalities and inequalities
2.8. Hints and partial solutions for the exercises
Chapter 3. Logic: Implications, Contrapositives, Contradictions, and Quantifiers
3.1. The need for rigor
3.2. Statements
3.3. Truth teller and liar riddle: Asking the right question
3.4*. Logic puzzles
3.5. Logical connectives
3.6. Implications
3.7. Contrapositive
3.8. Proof by contradiction
3.9. Pythagorean triples
3.10. Quantifiers
3.11. Hints and partial solutions for the exercises
Chapter 4. The Euclidean Algorithm and Its Consequences
4.1. The Division Theorem
4.2. There are an infinite number of primes
4.3. The Euclidean algorithm
4.4. Consequences of the Division Theorem
4.5. Solving linear Diophantine equations
4.6. “Practical” applications of solving linear Diophantine equations (wink \smiley)
4.7*. (Polynomial) Diophantine equations
4.8. The Fundamental Theorem of Arithmetic
4.9. The least common multiple
4.10. Residues modulo an odd prime
4.11. Appendix
4.12. Hints and partial solutions for the exercises
Chapter 5. Sets and Functions
5.1. Basics of set theory
5.2. Cartesian products of sets
5.3. Functions and their properties
5.4. Types of functions: Injections, surjections, and bijections
5.5. Arbitrary unions, intersections, and cartesian products
5.6*. Universal properties of surjections and injections
5.7. Hints and partial solutions for the exercises
Chapter 6. Modular Arithmetic
6.1. Multiples of 3 and 9 and the digits of a number in base 10
6.2. Congruence modulo ?
6.3. Inverses, coprimeness, and congruence
6.4. Congruence and multiplicative cancellation
6.5*. Fun congruence facts
6.6. Solving linear congruence equations
6.7*. The Chinese Remainder Theorem
6.8. Quadratic residues
6.9. Fermat’s Little Theorem
6.10*. Euler’s totient function and Euler’s Theorem
6.11*. An application of Fermat’s Little Theorem: The RSA algorithm
6.12*. The Euclid–Euler Theorem characterizing even perfect numbers
6.13*. Twin prime pairs
6.14. Chameleons roaming around in a zoo
6.15. Hints and partial solutions for the exercises
Chapter 7. Counting Finite Sets
7.1. The addition principle
7.2. Cartesian products and the multiplication principle
7.3. The inclusion-exclusion principle
7.4. Binomial coefficients and the Binomial Theorem
7.5. Counting functions
7.6. Counting problems
7.7*. Using the idea of a bijection
7.8. Hints and partial solutions for the exercises
Chapter 8. Congruence Class Arithmetic, Groups, and Fields
8.1. Congruence classes modulo ?
8.2. Inverses of congruence classes
8.3. Reprise of the proof of Fermat’s Little Theorem
8.4. Equivalence relations, equivalence classes, and partitions
8.5. Elementary abstract algebra
8.6. Rings, principal ideal domains, and all that
8.7. Fields
8.8*. Quadratic residues and the law of quadratic reciprocity
8.9. Hints and partial solutions for the exercises
Bibliography
Index
Back Cover
