Fernando Q. Gouvêa reviews More Fallacies, Flaws and Flimflam by Edward J. Barbeau as part of MAA Reviews.
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
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
to prove that
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