Numbers and Proofs

Citation

Allenby, R. B. J. T. (1997). Numbers and Proofs. Butterworth-Heinemann.

Chapter Summary

The book is a gentle introduction to proof for first- and second-year mathematics undergraduates. Rather than teaching logic in the abstract, it illustrates the useful methods of proof on problems about numbers (integers, rational, real and complex), and the publisher says each proof is preceded by a discussion of the thoughts a reader might have before attempting it. The aim is that readers learn to decipher other people's proofs and to construct their own. The style is described as entertaining and informal, and the book is meant to be a bridge to more demanding undergraduate mathematics. It has fourteen chapters, followed by a solutions appendix, answers to exercises and references.

Chapter 1: The Need for Proof

The opening chapter makes the case for proof, in line with the book's stated stress on backing truths found by experiment with logically sound arguments.

Chapter 2: Statements and Connectives

The language of statements and the connectives that join them, the logical grammar that later proofs rely on.

Chapter 3: True or False?

Deciding whether statements are true or false, continuing the logical groundwork of the previous chapter.

Chapter 4: Sets, Negations, Notations and Functions

The vocabulary that proofs are written in: sets, negations of statements, notation and functions.

Chapters 5–6: Proofs . . . For All; There Exists . . . Proofs

Two chapters on the two basic shapes of argument. The first deals with proving that something holds for all members of a collection, and the second with proving that something exists.

Chapter 7: The Principle of Mathematical Induction

Induction as a method of proof, following the "for all" and "there exists" chapters.

Chapters 8–10: The Integers and Rational Numbers; The Rational Numbers and the Real Numbers; Real Numbers and Complex Numbers

The methods are applied to the number systems, moving up from the integers and rationals through the real numbers to the complex numbers. These are the chapters most directly about the numbers of the title.

Chapters 11–12: Guessing, Analogy and Transformation; Generalisation and Specialisation

Chapters about how to find a proof: guessing, using analogy and transformation, and generalising or specialising a problem.

Chapters 13–14: Fallacies and Paradoxes; A Mixed Bag

A chapter on fallacies and paradoxes, which shows what goes wrong in faulty arguments, and a final mixed collection of problems.

Key Concepts

Chapters 1–4: Logic and Language

• Statements and connectives: how statements are combined, and what makes a compound statement true or false.
• Sets, negations and functions: the notation and the negation of statements that proofs depend on.

Chapters 5–7: Methods of Proof

• Proof for all and proof of existence: the two basic forms that a proof takes.
• Mathematical induction: proving a statement for all natural numbers from a base case and an inductive step.

Chapters 8–14: Numbers and Problem Solving

• Number systems: the integers, rationals, reals and complex numbers as the setting for proof.
• Analogy, generalisation and specialisation: strategies for finding a proof.
• Fallacies and paradoxes: arguments that appear correct but are not.

Critical Analysis

Strengths

• Thinking before proof. The publisher stresses that a discussion comes before each proof, showing what thoughts one might have before attempting it, so the reader sees how a proof is found as well as how it is checked.
• A single setting for proof. Because the methods are taught on problems about the number systems, the examples are familiar to a student arriving from school mathematics. The publisher describes the book as an ideal bridge to more complex undergraduate mathematics.
• Beyond routine logic. The final chapters on guessing, analogy, generalisation and specialisation, and on fallacies and paradoxes, treat discovery and error, not just verification. The book also stresses the importance of backing truths found by experiment with watertight arguments.
• Support for self-study. The contents list both a solutions appendix and answers to exercises, so a reader can check work independently.

Limitations

• Narrow range of examples. The contents are organised around numbers; the contents list no separate chapter on relations, cardinality or algebraic structures such as groups. Students heading toward algebra will want an additional text.
• Logic as a working tool only. Two chapters cover statements, connectives and truth and falsity, but the book is a gentle introduction and does not set out to develop formal logic; readers wanting a precise treatment of first-order languages should turn to a logic text.
• An informal voice. The publisher highlights the author's entertaining and informal style. Readers who prefer a terse, formal presentation may find another book more to their taste.

Where It Fits in the Course

Allenby's book is an introduction to proof that teaches through numbers. Richard Hammack's Book of Proof is a free text for students moving from computational to abstract mathematics, working through sets, logic, direct proof, contrapositive, contradiction and induction, and then relations, functions and cardinality; it covers a wider range of topics, while Allenby is closer to a single running theme. Daniel Velleman's How to Prove It starts from the logic of statements and quantifiers and shows how the logical form of a theorem suggests a proof strategy, and it is the more systematic of the two on logic. Peter Eccles's An Introduction to Mathematical Reasoning: Numbers, Sets and Functions is the nearest relative in subject matter, and Martin Liebeck's A Concise Introduction to Pure Mathematics surveys several areas of pure mathematics. Enderton's A Mathematical Introduction to Logic and Leary and Kristiansen's A Friendly Introduction to Mathematical Logic are formal treatments of logic that come after, not instead of, a book like this one.

How to Use This Book

The book is for first- and second-year undergraduates, including those whose mathematics has been mainly computational. School mathematics is enough preparation.

Read Chapters 1 to 7 in order, since they supply the logical vocabulary and the three main techniques (for all, there exists, induction). Chapters 8 to 10 reward careful work: attempt each proof yourself after reading the discussion that precedes it and before reading the book's version. Chapters 11 to 14 are well suited to a second pass or to browsing.

Do the exercises and check them against the solutions appendix and the answers. Pair the book with Hammack or Velleman if you want more exercises on a particular technique, and move on to a first course in analysis or algebra when the methods feel natural.