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More Fallacies, Flaws and Flimflam

Edward J. Barbeau
Mathematical Association of America
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
, on

Some fifteen years ago, I reviewed Mathematical Fallacies, Flaws and Flimflam, the first book collecting the best of Edward J. Barbeau’s regular column in the College Mathematics Journal. Much of what I said then applies here as well: this is an entertaining book that can also be useful in the classroom.

The typical FFF item gives an example of a mathematical argument that is wrong but tricky. Sometimes the problem is that the argument, while visibly (even extravagantly) incorrect, gives the right answer. Other times, the argument contains a subtle error, or uses a method that is correct for unexpected reasons.

For example, here’s a method (section 1.4) for adding two fractions with the same single-digit denominator and single-digit numerators, such as 5/8 and 1/8. First you juxtapose the two denominators to get 88. Then you juxtapose the numerators, getting 51, but of course addition is commutative, so you should also juxtapose them in the other order, getting 15. So

\[ \frac{5}{8} + \frac{1}{8} = \frac{51+15}{88} = \frac{66}{88} = \frac{6}{8}.\]

Neat, and perhaps an interesting one to try on students. A more advanced example is found in section 4.6, where L’Hospital’s rule is applied to

\[ \lim_{x\to\infty} \frac{x-\sin x^2}{x+\sin x^2}\]

to prove that \(-1=1\). In section 6.9, a long computation shows that

\[ \int_0^1 \frac{x^4+1}{x^6+1}\,dx = 0.\]

I also enjoyed a student's solution of the following problem, from section 10.5:

Let \(M\) be an \(n\times n\) matrix for which \(M^2=\alpha M\) for some scalar \(\alpha\). Evalutate the determinant \(\det(I+M)\), where \(I\) is the identity matrix.

The solution proceeds from the identity \(M(I+M)=(1+\alpha M)\) by taking determinants to get \(\det(I+M)=(1+\alpha)^n\), which Barbeau invites his readers to test on 

\[ M = \begin{bmatrix} -1 & 1\\1 & -1\end{bmatrix}.\]

Very cool. In some cases, the flimflam is not in the argument, but in the question. For example, Lewis Carroll asks in one of his “pillow problems”

Three points are taken at random on an infinite Plane. Find the chance of their being the vertices of an obtuse-angled Triangle.

As section 8.5 points out, it’s not at all clear that this problem has an answer. Carroll claimed that the probability is

\[ \frac{3\pi}{8\pi-6\sqrt{3}},\]

but we have it on the authority of Richard Guy that “There are three times as many obtuse-angled triangles as there are acute-angled ones,” which would suggest the correct answer is 3/4.

More Fallacies, Flaws & Flimflam is fun to read and might even teach you something.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. His students are not usually as creative as some of the students featured in this book.

1. Arithmetic
2. School Algebra
3. Geometry
4. Limits, Sequences and Series
5. Differential Calculus
6. Integral Calculus
7. Combinatorics
8. Probability and Statistics
9. Complex Analysis
10. Linear and Modern Algebra
11. Miscellaneous
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