A Concise Introduction to Pure Mathematics

Citation

Liebeck, M. W. (2015). A Concise Introduction to Pure Mathematics (4th ed.). CRC Press.

Chapter Summary

Context and Argument

Martin Liebeck, of Imperial College London, wrote this book as a first look at university mathematics for students who have a sound high-school background. The publisher describes it as presenting some of the most fundamental and beautiful ideas in pure mathematics: it covers standard material and also topics not usually met at this level, including the solution of cubic equations, Euler's formula for the corners, edges and faces of a solid and the five Platonic solids, the use of prime numbers to encode and decode secret information, the comparison of sizes of infinite sets, and the rigorous theory of limits and continuous functions.

The fourth edition adds two chapters introducing abstract algebra through group theory, new material on inequalities, counting methods, the inclusion-exclusion principle and Euler's phi function, and many new exercises, with solutions to the odd-numbered ones given at the back. There are twenty-six short chapters, which make the book a series of self-contained mathematical episodes rather than a single sustained argument.

Chapter 1: Sets and Proofs

The opening chapter introduces the language of sets and the basic ways of constructing mathematical arguments, which the rest of the book uses.

Chapters 2–7: Numbers and Equations

Chapter 2 ("Number Systems") looks at the integers, rational numbers and real numbers and their properties. Chapter 3 ("Decimals") treats decimal representations, including the link between periodic decimals and rational numbers. Chapter 4 covers _n_th roots and rational powers, Chapter 5 inequalities, Chapter 6 complex numbers and Chapter 7 polynomial equations, where the theory of solving cubic equations is a feature of the book.

Chapters 8–9: Induction; Euler's Formula and Platonic Solids

Chapter 8 develops proof by induction. Chapter 9 uses Euler's formula, which relates the numbers of corners, edges and faces of a solid, to show that there are exactly five Platonic solids.

Chapters 10–15: The Integers and Secret Codes

Chapter 10 returns to the integers and their divisibility properties. Chapters 11 and 12 treat prime factorization and further results on prime numbers, and Chapters 13 and 14 treat congruence of integers. Chapter 15 ("Secret Codes") applies primes and congruences to encoding and decoding secret information.

Chapters 16–21: Counting, Sets, Functions and Infinity

Chapter 16 ("Counting and Choosing") introduces counting methods. Chapter 17 returns to sets, and Chapter 18 introduces equivalence relations. Chapter 19 treats functions and Chapter 20 permutations. Chapter 21 ("Infinity") explains how sizes of infinite sets are compared.

Chapters 22–24: Analysis

These three chapters give a first rigorous treatment of analysis: bounds (Chapter 22), limits (Chapter 23) and continuity (Chapter 24).

Chapters 25–26: Groups

The two chapters added in the fourth edition introduce abstract algebra through groups, with examples that show how abstract reasoning works.

Back Matter

The book ends with solutions to odd-numbered exercises, a list of further reading and an index of symbols.

Key Concepts

Chapters 1–9

• Proof and sets: the language and methods on which later chapters depend.
• Number systems: integers, rationals, reals and complex numbers.
• Polynomial equations: including the solution of cubics.
• Induction: proof of statements about all natural numbers.
• Euler's formula: a relation between corners, edges and faces that restricts the possible Platonic solids.

Chapters 10–21

• Prime factorization: the decomposition of integers into primes.
• Congruence: arithmetic with remainders, used in the chapter on secret codes.
• Counting: methods for counting arrangements and selections, including the inclusion-exclusion principle.
• Equivalence relations and functions: the structures that organise sets.
• Comparing infinite sets: a way of saying when two infinite sets have the same size.

Chapters 22–26

• Bounds, limits and continuity: the foundations of analysis.
• Groups: the basic structure of abstract algebra.

Critical Analysis

Strengths

• Accessible entry point. The publisher states that the book is accessible to all students with a sound background in high-school mathematics, so it suits the transition to university-level pure mathematics.
• A wide survey with unusual topics. The book covers numbers, algebra, geometry, counting, infinity, analysis and groups in one volume, and includes topics not usually seen at this level, such as cubic equations and Platonic solids.
• Short chapters. Twenty-six chapters, averaging roughly a dozen pages each in a book of about 320, let a reader take the material in manageable pieces and in several orders.
• Exercises with solutions. The fourth edition has numerous new exercises and gives solutions to the odd-numbered ones, which suits self-study.

Limitations

• Breadth over depth. With so many topics in one book, none is treated at length. The analysis section has three chapters and the algebra only two, so a reader who wants a full course in either will need another book.
• Not a book about proof methods alone. Chapter 1 and the chapters on induction and sets introduce proof, but the book is a tour of mathematical topics, not a systematic treatise on logic and proof techniques.
• Little on applications. The book contains a chapter on secret codes, but it is mainly about pure mathematics for its own sake.

Where It Fits in the Course

Among the books on logic, sets and proof, Liebeck's is the broad tour of pure mathematics. Hammack's Book of Proof is a free introduction to proof methods, and Velleman's How to Prove It develops the logic of statements and quantifiers and the language of sets, relations and functions. Either of these is more systematic on proof than Liebeck, who sets it within a survey of topics. Eccles's An Introduction to Mathematical Reasoning and Allenby's Numbers and Proofs are also works on mathematical reasoning and number systems, and can be used alongside Liebeck for more on those themes.

Leary and Kristiansen's A Friendly Introduction to Mathematical Logic and Enderton's A Mathematical Introduction to Logic are books on formal logic. They go beyond Liebeck's first chapter and are for readers who want to study logic itself, once comfortable with mathematical proof.

How to Use This Book

The book is meant for first-year undergraduates and for motivated students with a sound high-school background who want to see what university pure mathematics looks like. It is also suitable for self-study, since solutions to odd-numbered exercises are included.

A good plan is to read in order through Chapters 1–9, which set up proof, numbers and induction, and then to choose among the later blocks according to need: Chapters 10–15 for number theory and codes, Chapters 16–21 for counting, functions and infinity, Chapters 22–24 for analysis and Chapters 25–26 for groups. Readers heading into a first analysis or algebra course should read the relevant block before the course starts.

Work the exercises as you go, check the odd-numbered ones against the solutions, and attempt the even-numbered ones without help. The list of further reading at the end suggests where to go next once a topic has caught your interest.