Mathematical Fallacies, Flaws, and Flimflam

MAA, 15. juuni 2000 - 167 pages
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.

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Contents

 Numbers 1 Whose real world? 2 United in purpose 3 A case of black and whitebut not so much black 4 Effects of changing temperature 5 Distributing addition over multiplication 6 An exponential mess 7 A divisibility property 8
 Where the grass is greener 78 How to make a million 80 A problem of Lewis Carroll 81 Nontransitive dice 82 Three coins in the fountain 83 Getting black balls 84 The car and goats and other problems 86 Your lucky number is in Pi 90

 Why Wiles proof of the Fermat Conjecture is false 9 A quick ? proof of irrationality 10 How the factorial works 11 Algebra and Trignometry 15 A superficial volume problem 16 How to solve a quadratic equation 17 An old method for solving a cubic 18 An exponential equation 19 The multiplication rules for logarithms 20 An application of the CauchySchwarz Inequality 21 Factoring homogeneous polynomials 22 The zero polynomial 23 An inductive fallacy 24 On not identifying equations and identities 25 A surd equation 27 The disappearing solution 28 Solving an inequality 31 An appearance of finite geometric sequences 32 Floored by an Olympiad problem 33 A New Identity for the Ceiling Function 34 Geometry 37 A luney way to square a circle 39 The SteinerLehmus Theorem 41 A geometry problem 42 A case of irregularity 43 A counterexample to Morleys Theorem 45 Going for the stars 46 The speeders delight 48 A solution to problem 480 50 Tangency by double roots 51 A puzzling graph 52 The wilting lines 54 The height of a trapezoid 55 Forces with a given resultant 56 A linear pythagorean theorem 57 The surface area of a sphere 59 Volume of a tin can 60 The Puptent Problem 61 The spirit is willing but the ham is rotten 62 Finite Mathematics 63 All positive integers are equal 64 Four weighings suffice 65 There is a unique positive integer 66 Doggedly bisexual 67 Equal unions 68 Surjective functions 69 Spoiled for choice 70 A full house 71 Which balls are actually there? 72 Red and blue hats 73 An invalid argument 74 A logical paradox 75 Probability 77
 Limits and Derivatives 91 The shortest distance from a point to a parabola 92 A foot by any other name 93 A degree of differentiation 94 The derivative of the sum is the sum of the derivatives 95 Double exponential 96 Calculation of a limit 98 Which is the correct asymptote? 99 Every derivative is continuous 101 Telescoping series 102 Integration and Differential Equations 103 The integral of log sin x 104 Evaluation of a sum 105 Integrals of products 106 LHopitals Rule under the integral sign 107 Why integrate? 108 The disappearing factor 109 A positive vanishing integral 110 Average chord length 111 Area of an ellipse 112 An Euler equation 113 Solving a secondorder differential equation 114 Power series thinning 115 Multivariate and Applications 117 Polar paradox? 118 Evaluating double integrals 119 The converse to Eulers theorem on homogeneous functions 120 The wrong logarithm 121 The conservation of energy according to Escher 122 Calculating the average speed 123 Hanging oneself with a minimum of rope 124 Generalizing an approach to the radius of curvature 126 Linear and Modern Algebra 129 The SchwarzCauchy Inequality 130 An entrance examination question 131 An inversion conundrum 132 The CayleyHamilton Theorem 133 All groups are simple 134 Groups with separate identities 135 The number of conjugates of a group element 136 Even and odd permutations 137 How large is the set of degenerate real symmetric matrices? 139 Advanced Undergraduate Mathematics 141 The countability of the reals 142 A consequence of the nearness of rationals to reals 143 A universal property of real subsets 144 A topological spoof 145 Is there a nonmeasurable set? 146 The continuum hypothesis 147 A heavyduty proof that 10 148 Parting Shots 151 References 161 Copyright

References to this book

 The Teaching and Learning of Mathematics at University Level: An ICMI StudyDerek HoltonLimited preview - 2001
 Analysis for Applied MathematicsWard CheneyLimited preview - 2001