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.
A glimpse at the Table of Contents offers examples from number theory, algebra and trigonometry, geometry finite mathematics, probability, calculus, linear algebra and advanced undergraduate mathematics.
Table of Contents
2. Algebra and Trigonometry
4. Finite Mathematics
6. Calculus: Limits and Derivatives
7. Calculus: Integration and Differential Equations
8. Calculus: Multivariate and Applications
9. Linear and Modern Algebra
10. Advanced Undergraduate Mathematics
11. Parting Shots
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.
About the Author
Edward J. Barbeau received his Ph.D. from the University of Newcastle-upon-Tyne, England. He was a postdoctoral Fellow at Yale University and has taught mathematics at the University of Toronto since 1967. He was named Fellow of the Ontario Institute for Studies in Education (1989), and has received the David Hilbert Award (1991) from the World Federation of National Mathematics Competitions and the Adrien Pouliot Award (1995) from the Canadian Mathematical Society.
His other books include: Five Hundred Mathematical Challenges with M.S. Klamkin and W.O Moser (published by The Mathematical Association of America), Polynomials, and After Math. He has been a frequent speaker and was invited on two occasions to give lectures by the Royal Canadian Institute, devoted to disseminating science to the layman. He also gave a series of three radio broadcasts, “Proof and Truth in Mathematics,” over the Canadian Broadcasting Corporation Network.