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Accessible to all students with a sound background in high school mathematics, A Concise Introduction to Pure Mathematics, Fourth Edition presents some of the most fundamental and beautiful ideas in pure mathematics. It covers not only standard material but also many interesting topics not usually encountered at this level, such as the theory of solving cubic equations; Euler’s formula for the numbers of corners, edges, and faces of a solid object and the five Platonic solids; the use of prime numbers to encode and decode secret information; the theory of how to compare the sizes of two infinite sets; and the rigorous theory of limits and continuous functions. New to the Fourth Edition Two new chapters that serve as an introduction to abstract algebra via the theory of groups, covering abstract reasoning as well as many examples and applications New material on inequalities, counting methods, the inclusion-exclusion principle, and Euler’s phi function Numerous new exercises, with solutions to the odd-numbered ones Through careful explanations and examples, this popular textbook illustrates the power and beauty of basic mathematical concepts in number theory, discrete mathematics, analysis, and abstract algebra. Written in a rigorous yet accessible style, it continues to provide a robust bridge between high school and higher-level mathematics, enabling students to study more advanced courses in abstract algebra and analysis. Cover 1 Dedication 6 Contents 8 Foreword 10 Preface 12 Chapter 1: Sets and Proofs 18 Chapter 2: Number Systems 30 Chapter 3: Decimals 38 Chapter 4: nth Roots and Rational Powers 44 Chapter 5: Inequalities 48 Chapter 6: Complex Numbers 56 Chapter 7: Polynomial Equations 68 Chapter 8: Induction 78 Chapter 9: Euler’s Formula and Platonic Solids 94 Chapter 10: The Integers 104 Chapter 11: Prime Factorization 112 Chapter 12: More on Prime Numbers 120 Chapter 13: Congruence of Integers 124 Chapter 14: More on Congruence 134 Chapter 15: Secret Codes 144 Chapter 16: Counting and Choosing 150 Chapter 17: More on Sets 164 Chapter 18: Equivalence Relations 174 Chapter 19: Functions 180 Chapter 20: Permutations 190 Chapter 21: Infinity 206 Chapter 22: Introduction to Analysis: Bounds 216 Chapter 23: More Analysis: Limits 224 Chapter 24: Yet More Analysis: Continuity 234 Chapter 25: Introduction to Abstract Algebra: Groups 242 Chapter 26: Introduction to Abstract Algebra: More on Groups 252 Solutions to Odd-Numbered Exercises 272 Further Reading 310 Index of Symbols 312