## Mathematical Fallacies, Flaws, and FlimflamThrough hard experience mathematicians have learned to subject even the most 'evident' assertions to rigorous scrutiny, as intuition and facile reasoning can often be misleading. However, errors can slip past the most watchful eye, they are often subtle and difficult to detect; but when found they can teach us a lot and can present a real challenge to straighten out. This book collects together a mass of such errors, drawn from the work of students, textbooks, and the media, as well as from professional mathematicians themselves. Each of these items is carefully analysed and the source of the error is exposed. All serious students of mathematics will find this book both enlightening and entertaining. |

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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 |

161 | |

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### Common terms and phrases

Abel's theorem algebra American Mathematical Monthly angle argument assume balls calculus calculus of variations Cayley-Hamilton theorem circle coins computation conics constant contains Contributed by David convergent countable cubic curve defined degree derivative determine differential equal equation example exponential factor Fermat's last theorem Figure finite Flaws Flimflam function geometry given graph Hence homogeneous hyperbolic geometry hypothesis identity induction infinite integer integral integrand intersection points interval invertible length Marilyn Vos Savant Martin Gardner Math Mathematical Fallacies matrix Metropolitan Toronto obtain pair paradox permutation polynomial positive integer probability problem proof rabbits radius Randolph-Macon Woman's College rational reader real numbers result Richard Askey satisfy selected sequence side sinx solution solving square subset Suppose switch theorem Toronto triangle trisect Underwood Dudley University vanishes variable vertex vertices whence yields zero

### References to this book

The Teaching and Learning of Mathematics at University Level: An ICMI Study Derek Holton Limited preview - 2001 |