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.
Sets and Proofs. Number Systems. Decimals. nth Roots and Rational Powers. Inequalities. Complex Numbers. Polynomial Equations. Induction. Euler's Formula and Platonic Solids. The Integers. Prime Factorization. More on Prime Numbers. Congruence of Integers. More on Congruence. Secret Codes. Counting and Choosing. More on Sets. Equivalence Relations. Functions. Permutations. Infinity. Introduction to Analysis: Bounds. More Analysis: Limits. Yet More Analysis: Continuity. Introduction to Abstract Algebra: Groups. Introduction to Abstract Algebra: More on Groups. Solutions to Odd-Numbered Exercises. Further Reading. Index of Symbols. Index.