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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.

**Sets and Proofs **

**Number Systems **

**Decimals **

**n**^{th} **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**

### Biography

**Martin Liebeck** is a professor of pure mathematics at Imperial College London. He earned his B.A., M.Sc., and D.Phil. in mathematics from the University of Oxford. Dr. Liebeck has published over 130 research articles and seven books. His research interests encompass algebraic groups, finite simple groups, probabilistic group theory, permutation groups, and algebraic combinatorics.

"In addition to preparing students to go on into mathematics, it is also a wonderful choice for a student who will not necessarily go on in mathematics but wants a gentle but fascinating introduction into the culture of mathematics. … This book will give a student the understanding to go on in further courses in abstract algebra and analysis. The notion of a proof will no longer be foreign, but also mathematics will not be viewed as some abstract black box. At the very least, the student will have an appreciation of mathematics. As usual, Liebeck’s writing style is clear and easy to read. This is a book that could be read by a student on his or her own. … While there is a difference in mathematical education between the U.K. and the U.S., this book will serve both groups of students extremely well."

—From the Foreword by Robert Guralnick, University of Southern California, Los Angeles, USA"Liebeck’s book stands out from the crowd of similar books by being short (as the title says, it is concise) and by trying to expose students to mathematical ideas beyond the basics of sets and logic. In addition to the pre-Analysis and pre-Algebra chapters, there are chapters on complex numbers, inequalities, some number theory and combinatorics, and the Platonic solids. Students are taught how to understand and create proofs, but they are also given a glimpse of what it is all for…applied mathematicians also need to know about proofs, counting, inequalities, bounds, and even groups — and this book could help them learn all that."

—Fernando Q. Gouvêa,MAA Reviews,June 2016

Praise for Previous Editions:It would in fact be difficult to find in this excellent book three consecutive pages that do not contain material useful to students or practitioners. … A diligent, active reader of this outstanding book will have the best foundation at minimum cost for making meaningful contributions to mathematics, science, or engineering."

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—Computing Reviews, November 2011"Now in an updated and expanded third edition,

A Concise Introduction to Pure Mathematicsprovides an informed and informative presentation into a representative selection of fundamental ideas in mathematics … . Of special note is the inclusion of solutions to all of the odd-numbered exercises. An ideal, accessible, elegant, student-friendly, and highly recommended choice for classroom textbooks for high school and college-level mathematics curriculums,A Concise Introduction to Pure Mathematicsis further enhanced with a selective bibliography, an index of symbols, and a comprehensive index."

—Library Bookwatch, December 2010"This book displays a unique combination of lightness and rigor, leavened with the right dose of humor. When I used it for a course, students could not get enough, and I have been recommending independent study from it to students wishing to take a core course in analysis without having taken the prerequisite course. The material is very well chosen and arranged, and teaching from Liebeck’s book has in many different ways been among my most rewarding teaching experiences during the last decades."

—Boris Hasselblatt, Tufts University, Medford, Massachusetts, USA"The book will continue to serve well as a transitional course to rigorous mathematics and as an introduction to the mathematical world … ."

—Gerald A. Heuer,Zentralblatt MATH, 2009"… a pleasure to read … a very welcome and highly accessible book."

—Michael Ward,The Mathematical Gazette, March 2007