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Tools of the Trade: Introduction to Advanced Mathematics

Paul J. Sally, Jr.
American Mathematical Society
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This is a dull book by an exciting teacher. Its premise is faulty: that we should study the tools of mathematics in isolation from the problems.

The book is intended for a bridge or transition course. Each chapter starts with a flurry of definitions, then proves some (usually easy) theorems based on the definitions, then we are done and go to the next chapter. We never actually use the tools on anything.

Most bridge courses focus on discrete math, but this one is slanted towards analysis (it also has some linear algebra and number theory). Each chapter ends with a series of Projects, which are advanced investigations into the subject of that chapter. I thought these were interesting but not very well integrated into the rest of the book. The Introduction claims that these projects can be used for Inquiry Based Learning, but I am skeptical that this will be successful — there's too big a jump from the hand-holding of the rest of the chapter.

Let's contrast this book with a couple of others and see how it might be done better.

Ed Burger's Extending the Frontiers of Mathematics is a bridge book based on discrete math. It is made up of numerous short chapters, and each starts with one or more interesting problems. Then we devise methods to solve the problems. The one is a through-and-through Inquiry Based Learning book, and is a good choice if that is your preference.

Ralph P. Boas Jr.'s A Primer of Real Functions is not usually thought of as a bridge course, but it covers a lot of same material as Sally's book and can come at the same place in the curriculum. It is not really a problem-oriented book, but it brings up a lot of issues that the student will already have seen in calculus, and studies them in more detail and explores some extensions. It takes a relaxed approach to proof, sometimes very rigorous and sometimes loose, which I think is a good way to edge students into proof.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.



Chapter 1. Sets, Functions, and Other Basic Ideas
1. Sets and Elements
2. Equality, Inclusion, and Notation
3. The Algebra of Sets
4. Cartesian Products, Counting, and Power Sets
5. Some Sets of Numbers
6. Equivalence Relations and the Construction of Q
7. Functions
8. Countability and Other Basic Ideas
9. Axiom of Choice
1O. Independent Projects

Chapter 2. Linear Algebra
1. Fundamentals of Linear Algebra
2. Linear Transformations
3. Linear Transformations and Matrices
4. Determinants
5. Geometric Linear Algebra
6. Independent Projects

Chapter 3. The Construction of the Real and Complex Numbers
1. The Least Upper Bound Property and the Real Numbers
2. Consequences of the Least Upper Bound Property
3. Rational Approximation
4. Intervals
5. The Construction of the Real Numbers
6. Convergence in R
7. Automorphisms of Fields
8. Construction of the Complex Numbers
9. Convergence in C
10. Independent Projects

Chapter 4. Metric and Euclidean Spaces
l. Introduction
2. Definition and Basic Properties of Metric Spaces
3. Topology of Metric Spaces
4. Limits and Continuous Functions
5. Compactness, Completeness and Connectedness
6. Independent Projects

Chapter 5. Complete Metric Spaces and the p-adic Completion of Q
l. The Contraction Mapping Theorem and Its Applications
2. The Baire Category Theorem and Its Applications
3. The Stone-Weierstrass Theorem
4. The p-adic Completion of Q
5. Challenge Problems