# Mathematical Fallacies, Flaws, and Flimflam

### By Edward J. Barbeau

Catalog Code: FFL
Print ISBN: 978-0-88385-529-4
Electronic ISBN: 978-1-61444-518-0
152 pp., Paperbound, 2000
List Price: $29.95 MAA Member:$23.95
Series: Spectrum

This book is a collection of mathematical mistakes made by ordinary citizens, students, teachers, and occasionally seasoned researchers, along with an analysis for most of them. While all the material is for personal enlightenment and amusement, high school and college teachers may use the materials to illustrate important and subtle points in mathematics.

Newspapers are responsible for several of these mathematical mishaps, particularly in arithmetic (especially percentages) and probability. Quite a number of the “Fallacies” come from professional mathematicians. Some are the result of simple oversight, and others are deliberately crafted by the mathematician to drive home an important point to students.

1. Numbers
2. Algebra and Trigonometry
3. Geometry
4. Finite Mathematics
5. Probability
6. Calculus: Limits and Derivatives
7. Calculus: Integration and Differential Equations
8. Calculus: Multivariate and Applications
9. Linear and Modern Algebra
11. Parting Shots
References
Index of Topics
Index of Names

### Excerpt: Chapter 3: Geometry (p. 37)

1. The Impossibility of angle bisection

In a typical introductory course in abstract algebra, after you have proven the impossibility of trisecting an arbitrary angle using just straightedge and compasses, you sum up the argument as follows: “We have just shown that cos 20° is not constructible, and so we cannot construct a 20° angle either; thus we cannot trisect at 60° angle, and so we cannot trisect an arbitrary angle.”

You can often create some consternation by continuing: “Now the fact that we cannot construct a 20° angle also shows that we cannot bisect a 40° angle and so you cannot bisect an arbitrary angle with compasses and straightedge.” Since an angle bisection is possible with straightedge and compasses, all that has been shown is that an angle of 40° is not so constructible. If a 40° angle was given, it would have had to have been determined by some measuring device. A 60° angle is constructible, so if a trisection were possible, we would be able to obtain a 60° angle and then trisect it to obtain a 20° angle.

Contributed by Eric Chandler of Randolph-Macon Woman’s College in Lynchburg, VA.