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Arithmetical Wonderland

Andy Liu
MAA Press
Publication Date: 
Number of Pages: 
MAA Press Classroom Resource Materials
[Reviewed by
Tom Schulte
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Characters from Lewis Carroll’s Alice’s Adventures in Wonderland and Through the Looking-Glass are a common thread in this unorthodox textbook in elementary number theory. The narrative mostly takes the form of dialogues between Alice and the twins Tweedledum and Tweedledee. Others, appropriately including the Red Queen, set up motivation for Alice to explain more of the fundamentals of “The Queen of Mathematics.”

The survey of arithmetic begins with counting numbers, basic operations, and such properties as associativity, commutativity, and distributivity. With such elementary material, character humor, and less than two hundred pages of main text, this book may appear at first to be a breezy review of topics de rigueur in grade school and possibly high school. Actually, the scope here is much more ambitious and the slim size results from an economical and engaging presentation. By the end of the first six pages in Chapter Zero, the reader is proving the uniqueness of the identity element for binary operations.

Exploring the topic of division, the authors in Chapter One introduce proofs, for example of divisibility tests and of the inability of the S-skew polyomino to divide any rectangle. It is at this point that we see that the topics are ambitious for most high school courses. We also realize how efficient the development is, especially considering that half the content of the preceding forty pages is Alice trying out some jokey maieutics on fantasy characters. The density of \(\mathbb{Q}\), irrationality proofs, and an introduction to Newton’s Method for cube root calculation to arbitrary accuracy are some of the topics in this book. Many students will not encounter this material until college (if then), but here it can be understood years ahead of one’s peers. I am also glad to see, and I am sure Richard Feynman would judge it wise, that the exploration of other bases and conversion between them is held off until the final chapter and also gamely handled.

There are a few things I feel are awkwardly presented and some missed opportunities. I hope these will be approached differently in a new edition sure to follow the success of this one. First, enlisting a typography of underlines and right parentheses to display the multi-step greatest common divisor makes for a muddle. Given the theme, use the acceptability of cartoons for a more fluid depiction! Further, pressing into service \(\Delta\) and \(\nabla\) as new binary operators for gcd and lcm is unnecessary and detracts from the book’s value as an adjunct to more mainstream texts, which is a natural role for this work to fill. Finally, showing only a single Sieve of Eratosthenes without commentary or display of different choices for columns or even a prime spiral is a missed opportunity in a witty work so rich in engaging detail.

Appropriately, tribute is paid to Martin Gardner, author of The Annotated Alice, over a few pages before the solutions to the odd-numbered exercises.

Tom Schulte, mathematics instructor at Oakland Community College, wishes he had met this book three decades ago.

Preface to a Preliminary Edition

0. Review of Arithmetic
0.1 Counting Numbers
0.2 Integers
0.3 Inequalities
0.4 Extras

1. Divisibility
1.1 Basic Properties of Divisibility
1.2 The Arithmetic of Divisibility
1.3 Divisibility Problems
1.4 Extras

2. Congruence
2.1 The Division Algorithm
2.2 Basic Properties and Arithmetic of Congruence
2.3 Congruence and Divisibility
2.4 Extras

3. Common Divisors and Multiples
3.1 Greatest Common Divisors and the Euclidean Algorithm
3.2 Relatively Prime Numbers
3.3 Least Common Multiples
3. 4 Extras

4. Linear Diophantine Equations
4.1 Bézoutain Algorithm
4.2 Homogeneous and Non-homogeneous Equations
4.3 Linear Diophantine Problems
4.4 Extras

5. Prime Factorizations
5.1 Prime and Composite Numbers
5.2 Fundamental Theorem of Arithmetic
5.3 Applications of the Fundamental Theorem of Arithmetic
5.4 Extras

6. Rational and Irrational Numbers
6.1 Fractions
6.2 Decimals
6.3 Real Numbers
6.4 Extras

7. Numeration Systems
7.1 Arithmetic in Other Bases
7.2 Conversion between Bases
7.3 Applications of Other Bases
7.4 Extras

Appendix: A Legacy of Martin Gardner
Solution to Odd-numbered Exercises
About the Author