**Spectrum**

Volume: 25;
2000;
167 pp;
Softcover

Print ISBN: 978-0-88385-529-4

Product Code: SPEC/25

List Price: $25.00

AMS Member Price: $18.75

MAA member Price: $18.75

**Electronic ISBN: 978-1-61444-518-0
Product Code: SPEC/25.E**

List Price: $25.00

AMS Member Price: $18.75

MAA member Price: $18.75

# Mathematical Fallacies, Flaws, and Flimflam

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*Edward J. Barbeau*

MAA Press: An Imprint of the American Mathematical Society

Through hard experience, mathematicians have learned to subject even the most evident assertions to rigorous scrutiny, as intuition and facile reasoning can often lead them astray. However, the impossibility and impracticality of completely watertight arguments make it possible for errors to slip by the most watchful eye. They are often subtle and difficult of detection. When found, they can teach us a lot and can present a real challenge to straighten out. Presenting students with faulty arguments to troubleshoot can be an effective way of helping them critically understand material, and it is for this reason that I began to compile fallacies and publish them first in the Notes of the Canadian Mathematical Society and later in the College Mathematics Journal in the Fallacies, Flaws and Flimflam section. I hoped to challenge and amuse readers as well as to provide them with material suitable for teaching and student assignments. This book collects the items from the first eleven years of publishing in the CMJ. One source of such errors is the work of students. Occasionally, a text book will weigh in with a specious result or solution. Nonprofessional sources, such as newspapers, are responsible for a goodly number of mishaps, particularly in arithmetic (especially percentages) and probability; their use in classrooms may help students become critical readers and listeners of the media. Quite a few items come from professional mathematicians. The reader will find in this book some items that are not erroneous but seem to be. These need a fuller analysis to clarify the situation. All the items are presented for your entertainment and use. The mathematical topics covered include algebra, trigonometry, geometry, probability, calculus, linear algebra, and modern algebra.

#### Reviews & Endorsements

For several years now, Ed Barbeau has been editing a regular column in the CMJ called 'Fallacies, Flaws, and Flimflam.' The column collects interesting examples of mistakes, fallacies, and other mathematical howlers. Some come from students, some from publications, others from contributors who have invented their own swindles and fallacies. This book (which acquired a 'Mathematical' in the title to accomodate readers unfamiliar with the column) is a kind of 'Best of FFF.' It collects over 150 articles from the column, organized by topic into sections ranging from 'Numbers' to 'Advanced Undergraduate Mathematics,' with some 'Parting Shots' thrown in for good measure. There are some wild errors and some quite subtle ones; some things we recognize from many past student productions, others that are quite original...Collected together, they make a book of the sort one reads compulsively from cover to cover. I suspect it can be very useful in our teaching, since some of these creative mistakes would make great discussion starters...'Mathematical Fallacies, Flaws, and Flimflam' is definitely worth your time.

-- Fernando Q. Gouvéa, MAA Reviews

# Table of Contents

## Mathematical Fallacies, Flaws, and Flimflam

- front cover cover11
- copyright page ii3
- title page iii4
- FOREWORD ix10
- Contents xi12
- 1 NUMBERS 118
- 1. How to get drunk and rich at the same time 118
- 2. Fifty per cent more for fifty per cent less 118
- 3. Whose real world? 219
- 4. United in purpose 320
- 5. A case of black and white---but not so much black 421
- 6. Effects of changing temperature 522
- 7. To those that have shall be given 522
- 8. Distributing addition over multiplication 623
- 9. Distributing exponents over sums 623
- 10. An exponential mess 724
- 11. A product of logarithms 724
- 12. A divisibility property 825
- 13. All perfect numbers are even 825
- 14. Why Wiles' proof of the Fermat Conjecture is false 926
- 15. A quick (?) proof of irrationality 1027
- 16. A rational combination of two transcendentals 1027
- 17. How the factorial works 1128
- Dollars and sense 1128

- 2 ALGEBRA AND TRIGONOMETRY 1532
- 1. Do you know how to split the atom? 1532
- 2. The number of tickets 1532
- 3. A superficial volume problem 1633
- 4. The end justifies the means 1633
- 5. How to solve a quadratic equation 1734
- 6. A new method for solving a cubic 1734
- 7. An old method for solving a cubic 1835
- 8. An exponential equation 1936
- 9. Logarithms distribute over sums 1936
- 10. The multiplication rules for logarithms 2037
- 11. A lack of technical unanimity 2037
- 12. A straightforward cancellation 2037
- 13. An application of the Cauchy-Schwarz Inequality 2138
- 14. Surprising symmetry 2138
- 15. Factoring homogeneous polynomials 2239
- 16. Polynomial detection 2239
- 17. The remainder theorem 2239
- 18. The zero polynomial 2340
- 19. An inductive fallacy 2441
- 20. On not identifying equations and identities 2542
- 21. A surd equation 2744
- 22. The disappearing solution 2845
- 23. Solving an inequality 3148
- 24. An appearance of finite geometric sequences 3249
- 25. Glide-reflecting the sine curve 3249
- 26. A trigonometric identity 3249
- 27. Floored by an Olympiad problem 3350
- 28. A New Identity for the Ceiling Function 3451

- 3 GEOMETRY 3754
- 1. The impossibility of angle bisection 3754
- 2. Trisecting an angle with ruler and compasses 3754
- 3. A luney way to square a circle 3956
- 4. The Steiner-Lehmus Theorem 4158
- 5. A geometry problem 4259
- 6. A case of irregularity 4360
- 7. A counterexample to Morley's Theorem 4562
- 8. Going for the stars 4663
- 9. Identifying the angle 4663
- 10. The speeder's delight 4865
- 11. A solution to problem 480 5067
- 12. Tangency by double roots 5168
- 13. A puzzling graph 5269
- 14. The wilting lines 5471
- 15. The height of a trapezoid 5572
- 16. Forces with a given resultant 5673
- 17. A linear pythagorean theorem 5774
- 18. The surface area of a sphere 5976
- 19. Drenching a sphere 5976
- 20. Volume of a tin can 6077
- 21. The Puptent Problem 6178
- 22. The spirit is willing but the ham is rotten 6279

- 4 FINITE MATHEMATICS 6380
- 1. Rabbits reproduce; integers don't 6380
- 2. All positive integers are equal 6481
- 3. Every second square is the same 6481
- 4. Four weighings suffice 6582
- 5. Perron's paradox 6582
- 6. There is a unique positive integer 6683
- 7. A criterion for a cyclic graph 6683
- 8. Doggedly bisexual 6784
- 9. Equal unions 6885
- 10. Surjective functions 6986
- 11. Hockey ranking 6986
- 12. Spoiled for choice 7087
- 13. Arranging a collection 7087
- 14. A full house 7188
- 15. Which balls are actually there? 7289
- 16. Red and blue hats 7390
- 17. An invalid argument 7491
- 18. A logical paradox 7592

- 5 PROBABILITY 7794
- 1. Meeting in a knockout tournament 7794
- 2. Where the grass is greener 7895
- 3. How to make a million 8097
- 4. A problem of Lewis Carroll 8198
- 5. Nontransitive dice 8299
- 6. Three coins in the fountain 83100
- 7. Getting black balls 84101
- 8. An encounter in the cafeteria 84101
- 9. The car and goats and other problems 86103
- 10. Your lucky number is in Pi 90107

- 6 CALCULUS: LIMITS AND DERIVATIVES 91108
- 1. All powers of x are constant. 91108
- 3. 3 equals 2 91108
- 4. The shortest distance from a point to a parabola 92109
- 5. A foot by any other name 93110
- 6. A degree of differentiation 94111
- 7. The derivative of the sum is the sum of the derivatives 95112
- 8. Differentiating x^x 95112
- 9. Double exponential 96113
- 10. Iterated exponential 96113
- 11. Calculation of a limit 98115
- 12. Which is the correct asymptote? 99116
- 13. Every derivative is continuous 101118
- 14. Telescoping series 102119

- 7 CALCULUS: INTEGRATION ANDDIFFERENTIAL EQUATIONS 103120
- 1. A new way to obtain the logarithm 103120
- 2. The constant of integration 103120
- 3. The integral of log sin x 104121
- 4. Evaluation of a sum 105122
- 5. Integrals of products 106123
- 6. L'Hopital's Rule under the integral sign 107124
- 7. A power series representation 107124
- 8. More fun with series 107124
- 9. Why integrate? 108125
- 10. The disappearing factor 109126
- 11. Cauchy's negative definite integral 109126
- 12. A positive vanishing integral 110127
- 13. Blowing up the integrand 110127
- 14. Average chord length 111128
- 15. Area of an ellipse 112129
- 16. Infinite area but a finite volume 112129
- 17. An Euler equation 113130
- 18. Solving a second-order differential equation 114131
- 19. Power series thinning 115132

- 8 CALCULUS: MULTIVARIATE ANDAPPLICATIONS 117134
- 1. Variable results with partial differentiation 117134
- 2. Polar paradox? 118135
- 3. Polar increment of area 118135
- 4. Evaluating double integrals 119136
- 5. One-step double integration 119136
- 6. The converse to Euler's theorem on homogeneous functions 120137
- 7. The wrong logarithm 121138
- 8. The conservation of energy according to Escher 122139
- 9. Calculating the average speed 123140
- 10. Maximizing a subtended angle 123140
- 11. Hanging oneself with a minimum of rope 124141
- 12. Throwing another fallacy out the window 124141
- 13. Generalizing an approach to the radius of curvature 126143
- 14. The lopsided uniform rod 126143

- 9 LINEAR AND MODERN ALGEBRA 129146
- 1. A proof that 0 = 1 129146
- 2. Matrices and the TI-81 graphics calculator 129146
- 3. The Schwarz-Cauchy Inequality 130147
- 4. An entrance examination question 131148
- 5. Nonsquare invertible matrices 131148
- 6. An inversion conundrum 132149
- 7. The Cayley-Hamilton Theorem 133150
- 8. All groups are simple 134151
- 9. Groups with separate identities 135152
- 10. The least common multiple order 135152
- 11. The number of conjugates of a group element 136153
- 12. Even and odd permutations 137154
- How large is the set of degenerate real symmetric matrices? 139156

- 10 ADVANCED UNDERGRADUATE MATHEMATICS 141158
- 1. Troublemakers 141158
- 2. The countability of the reals 142159
- 3. The plane constitutes an uncountable set 142159
- 4. A consequence of the nearness of rationals to reals 143160
- 5. A universal property of real subsets 144161
- 6. A topological spoof 145162
- 7. Is there a nonmeasurable set? 145162
- 8. Is there a nonmeasurable set? 146163
- 9. Is there a function continuous only on the rationals? 146163
- 10. The continuum hypothesis 147164
- 11. A heavy-duty proof that 1 = 0 148165
- 12. All complex numbers are real 148165
- 13. Opening the floodgates 148165

- 11 PARTING SHOTS 151168
- References 161178
- Index of Topics 163180
- Index of Names 165182