Mathematical Fallacies, Flaws and Flimflam

by Barbeau, Edward J.

• ISBN13: 9780883855294
• ISBN10: 0883855291
• Format: Paperback
• Copyright: 2000-01-01
• Publisher: Mathematical Assn of Amer

Summary

Through 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.

Table of Contents

Numbers	1	(14)
How to get drunk and rich at the same time	1	(1)
Fifty per cent more for fifty per cent less	1	(1)
Whose real world?	2	(1)
United in purpose	3	(1)
A case of black and white---but not so much black	4	(1)
Effects of changing temperature	5	(1)
To those that have shall be given	5	(1)
Distributing addition over multiplication	6	(1)
Distributing exponents over sums	6	(1)
An exponential mess	7	(1)
A product of logarithms	7	(1)
A divisibility property	8	(1)
All perfect numbers are even	8	(1)
Why Wiles' proof of the Fermat Conjecture is false	9	(1)
A quick (?) proof of irrationality	10	(1)
A rational combination of two transcendentals	10	(1)
How the factorial works	11	(4)
Dollars and sense	11	(4)
Algebra and Trignometry	15	(22)
Do you know how to split the atom?	15	(1)
The number of tickets	15	(1)
A superficial volume problem	16	(1)
The end justifies the means	16	(1)
How to solve a quadratic equation	17	(1)
A new method for solving a cubic	17	(1)
An old method for solving a cubic	18	(1)
An exponential equation	19	(1)
Logarithms distribute over sums	19	(1)
The multiplication rules for logarithms	20	(1)
A lack of technical unanimity	20	(1)
A straightforward cancellation	20	(1)
An application of the Cauchy-Schwarz Inequality	21	(1)
Surprising symmetry	21	(1)
Factoring homogeneous polynomials	22	(1)
Polynomial detection	22	(1)
The remainder theorem	22	(1)
The zero polynomial	23	(1)
An inductive fallacy	24	(1)
On not identifying equations and identities	25	(2)
A surd equation	27	(1)
The disappearing solution	28	(3)
Solving an inequality	31	(1)
An appearance of finite geometric sequences	32	(1)
Glide-reflecting the sine curve	32	(1)
A trigonometric identity	32	(1)
Floored by an Olympiad problem	33	(1)
A New Identity for the Ceiling Function	34	(3)
Geometry	37	(26)
The impossibility of angle bisection	37	(1)
Trisecting an angle with ruler and compasses	37	(2)
A luney way to square a circle	39	(2)
The Steiner-Lehmus Theorem	41	(1)
A geometry problem	42	(1)
A case of irregularity	43	(2)
A counterexample to Morley's Theorem	45	(1)
Going for the stars	46	(1)
Identifying the angle	46	(2)
The speeder's delight	48	(2)
A solution to problem 480	50	(1)
Tangency by double roots	51	(1)
A puzzling graph	52	(2)
The wilting lines	54	(1)
The height of a trapezoid	55	(1)
Forces with a given resultant	56	(1)
A linear pythagorean theorem	57	(2)
The surface area of a sphere	59	(1)
Drenching a sphere	59	(1)
Volume of a tin can	60	(1)
The Puptent Problem	61	(1)
The spirit is willing but the ham is rotten	62	(1)
Finite Mathematics	63	(14)
Rabbits reproduce; integers don't	63	(1)
All positive integers are equal	64	(1)
Every second square is the same	64	(1)
Four weighings suffice	65	(1)
Perron's paradox	65	(1)
There is a unique positive integer	66	(1)
A criterion for a cyclic graph	66	(1)
Doggedly bisexual	67	(1)
Equal unions	68	(1)
Surjective functions	69	(1)
Hockey ranking	69	(1)
Spoiled for choice	70	(1)
Arranging a collection	70	(1)
A full house	71	(1)
Which balls are actually there?	72	(1)
Red and blue hats	73	(1)
An invalid argument	74	(1)
A logical paradox	75	(2)
Probability	77	(14)
Meeting in a knockout tournament	77	(1)
Where the grass is greener	78	(2)
How to make a million	80	(1)
A problem of Lewis Carroll	81	(1)
Nontransitive dice	82	(1)
Three coins in the fountain	83	(1)
Getting black balls	84	(1)
An encounter in the cafeteria	84	(2)
The car and goats and other problems	86	(4)
Your lucky number is in Pi	90	(1)
Calculus: Limits and Derivatives	91	(12)
All powers of x are constant	91	(1)
Differentiating the square function	91	(1)
3 equals 2	91	(1)
The shortest distance from a point to a parabola	92	(1)
A foot by any other name	93	(1)
A degree of differentiation	94	(1)
The derivative of the sum is the sum of the derivatives	95	(1)
Differentiating xx	95	(1)
Double exponential	96	(1)
Iterated exponential	96	(2)
Calculation of a limit	98	(1)
Which is the correct asymptote?	99	(2)
Every derivative is continuous	101	(1)
Telescoping series	102	(1)
Calculus: Integration and Differential Equations	103	(14)
A new way to obtain the logarithm	103	(1)
The constant of integration	103	(1)
The integral of log sin x	104	(1)
Evaluation of a sum	105	(1)
Integrals of products	106	(1)
L'Hopital's Rule under the integral sign	107	(1)
A power series representation	107	(1)
More fun with series	107	(1)
Why integrate?	108	(1)
The disappearing factor	109	(1)
Cauchy's negative definite integral	109	(1)
A positive vanishing integral	110	(1)
Blowing up the integrand	110	(1)
Average chord length	111	(1)
Area of an ellipse	112	(1)
Infinite area but a finite volume	112	(1)
An Euler equation	113	(1)
Solving a second-order differential equation	114	(1)
Power series thinning	115	(2)
Calculus: Multivariate and Applications	117	(12)
Variable results with partial differentiation	117	(1)
Polar paradox?	118	(1)
Polar increment of area	118	(1)
Evaluating double integrals	119	(1)
One-step double integration	119	(1)
The converse to Euler's theorem on homogeneous functions	120	(1)
The wrong logarithm	121	(1)
The conservation of energy according to Escher	122	(1)
Calculating the average speed	123	(1)
Maximizing a subtended angle	123	(1)
Hanging oneself with a minimum of rope	124	(1)
Throwing another fallacy out the window	124	(2)
Generalizing an approach to the radius of curvature	126	(1)
The lopsided uniform rod	126	(3)
Linear and Modern Algebra	129	(12)
A proof that 0 = 1	129	(1)
Matrices and the TI-81 graphics calculator	129	(1)
The Schwarz-Cauchy Inequality	130	(1)
An entrance examination question	131	(1)
Nonsquare invertible matrices	131	(1)
An inversion conundrum	132	(1)
The Cayley-Hamilton Theorem	133	(1)
All groups are simple	134	(1)
Groups with separate identities	135	(1)
The least common multiple order	135	(1)
The number of conjugates of a group element	136	(1)
Even and odd permutations	137	(4)
How large is the set of degenerate real symmetric matrices?	139	(2)
Advanced Undergraduate Mathematics	141	(10)
Troublemakers	141	(1)
The countability of the reals	142	(1)
The plane constitutes an uncountable set	142	(1)
A consequence of the nearness of rationals to reals	143	(1)
A universal property of real subsets	144	(1)
A topological spoof	145	(1)
Is there a nonmeasurable set?	145	(1)
Is there a nonmeasurable set?	146	(1)
Is there a function continuous only on the rationals?	146	(1)
The continuum hypothesis	147	(1)
A heavy-duty proof that 1 = 0	148	(1)
All complex numbers are real	148	(1)
Opening the floodgates	148	(3)
Parting Shots	151	(10)
Ibn Qurra	151	(1)
Reading a calculator display	151	(1)
Infallibility of a symbolic manipulation program	151	(2)
A perpetual motion machine	153	(1)
The bouncing ball	153	(3)
Watch your ears!	156	(1)
Positive series with a negative sum	156	(5)
A centennial tribute to Sam Loyd	157	(4)
References	161	(2)
Index of Topics	163	(2)
Index of Names	165