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1089 and All That: A Journey into Mathematics

David Acheson
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Teri J. Murphy and Henry J. Neeman
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1089 and All That is an instant classic.

Every mathematician has been confronted with the question, "What do you do?" The conversation goes in one of a few directions. Sometimes the person wants to explain that he or she is not very good at mathematics. Sometimes the person wants to engage in a discussion about the state of mathematics teaching. And sometimes the person is interested at a fairly deep level and perhaps even asks what mathematics is really about or what mathematicians do. This book is what should be on hand — dozens of copies of it — to give to people who fall into this last group — maybe even to some of the people in the other groups. (Readers will need a certain facility with symbolic notation, however.) The author's goal is to share with non-mathematicians some of the beauty, some of the "wonder" (his word), of mathematics.

This book packs a lifetime of wisdom and delight into sixteen brief chapters. A few of the chapters can be read independently from the rest of the book, but most rely on ideas introduced in earlier chapters. In the first chapter, the author sets the stage, explaining that the "1089 trick" caught his attention when he was a youngster. In the second chapter ("In Love with Geometrie"), he takes us from a visual proof of the Pythagorean Theorem into a brief introduction to topology. Most importantly, though, in this chapter he introduces the need for mathematical proof. Chapter 3 is about proof by contradiction (examples: the Königsberg Bridges, infinitely many primes, mention of Fermat's Last Theorem). Chapter 4 is about the usefulness of algebra — why don't we make this clear to students? — including the usefulness of combining algebra and geometry (e.g., to create a Cartesian coordinate system).

Some of the chapters are filled with mathematical ideas and representations, including excursions into ideas that show up calculus courses ("All Change!", "On Being as Small as Possible", and "'Are We Nearly There?'") and differential equations ("What is the Secret of All Life?"). Other chapters include more references to historical landmarks (especially "The Heavens in Motion"). There is a chapter about problems whose original "solutions" were wrong ("Great Mistakes"), an introduction to sine waves and harmonics ("Good Vibrations"), and a chapter about a surprising result ("Not Quite the Indian Rope Trick"). There is a chapter about pi, a chapter about e, and a chapter about i, including a grand finale showing the relationship among these three special numbers (eiπ = -1). Murphy's favorite chapter is "Chaos and Catastrophe", with its variety of examples (the three-body problem, weather forecasting).

The title of 1089 and All That is a take-off on a famous book on the history of England: 1066 and All That: a Memorable History of England Comprising All the Parts You Can Remember, Including 103 Good Things, 5 Bad Things, and 2 Genuine Dates, by W. C. Sellar and R. J. Yeatman. Like its namesake, 1089 is well illustrated, fun to read and easy to remember. It has been produced with great care: at more than one point, one notices how artfully figures were placed. For example, at least twice a surprising conclusion appears just as we turn the page.

The book takes only a couple of hours to read: it has 178 5" x 7" pages filled with a contagious enthusiasm. It is a must-read for any student majoring in any field based on mathematics (e.g., science, engineering, business). It might even be useful as one of the books in a general education mathematics course or a course for teachers. In fact, the person at OU who teaches one of the mathematics courses for pre-service secondary teachers read it and he plans to have his students read it. Murphy might even have her calculus students, most of whom are engineering majors, read it. It's an adorable, lovable, inspiring little masterpiece!

Teri J. Murphy ( is associate professor of mathematics at the University of Oklahoma. She has an M.S. in mathematics, an M.S. in applied mathematics, and a Ph.D. in mathematics education from the University of Illinois at Urbana-Champaign. Her research specialty is undergraduate mathematics education.

Henry J. Neeman is director of the OU Supercomputing Center for Education and Research, a visiting professor of Computer Science, and a research scientist at the Center for Analysis and Prediction of Storms at the University of Oklahoma. He has a Ph.D. in Computer Science from the University of Illinois at Urbana-Champaign. His research specialty is high performance computing.

1. 1089 and All That
2. In Love with Geometrie
3. But ... that's Absurd ...
4. The Trouble with Algebra
5. The Heavens in Motion
6. All Change!
7. On Being as Small as Possible
8. "Are We Nearly There?"
9. A Brief History of pi
10. Good Vibrations
11. Great Mistakes
12. What is the Secret of All Life?
13. e=2.718 ...
14. Chaos and Catastrophe
15. Not Quite the Indian Rope Trick
16. Real or Imaginary?