Barbeau E.J. Mathematical Fallacies, Flaws and Flimflam

The Mathematical Association of America, 2000. — 167 p. — ISBN 0883855291.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.Numbers
How to get drunk and rich at the same time
Fifty per cent more for fifty per cent less
Whose real world?
United in purpose
A case of black and white—but not so much black
Effects of changing temperature
To those that have shall be given
Distributing addition over multiplication
Distributing exponents over sums
An exponential mess
A product of logarithms
A divisibility property
All perfect numbers are even
Why Wiles' proof of the Fermat Conjecture is false
A quick (?) proof of irrationality
A rational combination of two transcendentals
How the factorial works
Dollars and sense
Algebra and Trignometry
Do you know how to split the atom?
The number of tickets
A superficial volume problem
The end justifies the means
How to solve a quadratic equation
A new method for solving a cubic
An old method for solving a cubic
An exponential equation
Logarithms distribute over sums
The multiplication rules for logarithms
A lack of technical unanimity
A straightforward cancellation
An application of the Cauchy-Schwarz Inequality
Surprising symmetry
Factoring homogeneous polynomials
Polynomial detection
The remainder theorem
The zero polynomial
An inductive fallacy
On not identifying equations and identities
A surd equation
The disappearing solution
Solving an inequality
An appearance of finite geometric sequences
Glide-reflecting the sine curve
A trigonometric identity
Floored by an Olympiad problem
A New Identity for the Ceiling Function
Geometry
The impossibility of angle bisection
Trisecting an angle with ruler and compasses
A luney way to square a circle
The Steiner-Lehmus Theorem
A geometry problem
A case of irregularity
A counterexample to Morley's Theorem
Going for the stars
Identifying the angle
The speeder's delight
A solution to problem 480
Tangency by double roots
A puzzling graph
The wilting lines
The height of a trapezoid
Forces with a given resultant
A linear pythagorean theorem
The surface area of a sphere
Drenching a sphere
Volume of a tin can
The Puptent Problem ,
The spirit is willing but the ham is rotten '
Finite Mathematics
Rabbits reproduce; integers don't
All positive integers are equal
Every second square is the same
Four weighings suffice
Perron's paradox
There is a unique positive integer
A criterion for a cyclic graph
Doggedly bisexual
Equal unions
Surjective functions
Hockey ranking
Spoiled for choice
Arranging a collection
A full house
Which balls are actually there?
Red and blue hats
An invalid argument
A logical paradox
Probability
Meeting in a knockout tournament
Where the grass is greener
How to make a million
A problem of Lewis Carroll
Nontransitive dice
Three coins in the fountain
Getting black balls
An encounter in the cafeteria
The car and goats and other problems
Your lucky number is in Pi
Calculus: Limits and Derivatives
All powers of .r are constant
Differentiating the square function
3 equals 2
The shortest distance from a point to a parabola
A foot by any other name
A degree of differentiation
The derivative of the sum is the sum of the derivatives
Differentiating xx
Double exponential
Iterated exponential
Calculation of a limit
Which is the correct asymptote?
Every derivative is continuous
Telescoping series
Calculus: Integration and Differential Equations
A new way to obtain the logarithm
The constant of integration
The integral of log sin x
Evaluation of a sum
Integrals of products
L'Hopital's Rule under the integral sign
A power series representation
More fun with series
Why integrate?
The disappearing factor
Cauchy's negative definite integral
A positive vanishing integral
Blowing up the integrand
Average chord length
Area of an ellipse
Infinite area but a finite volume
An Euler equation
Solving a second-order differential equation
Power series thinning
Calculus: Multivariate and Applications
Variable results with partial differentiation
Polar paradox?
Polar increment of area
Evaluating double integrals
One-step double integration
The converse to Euler's theorem on homogeneous functions
The wrong logarithm
The conservation of energy according to Escher
Calculating the average speed
Maximizing a subtended angle
Hanging oneself with a minimum of rope
Throwing another fallacy out the window
Generalizing an approach to the radius of curvature
The lopsided uniform rod
Linear and Modern Algebra
A proof that 0=1
Matrices and the TI-81 graphics calculator
The Schwarz-Cauchy Inequality
An entrance examination question
Nonsquare invertible matrices
An inversion conundrum
The Cayley-Hamilton Theorem
All groups are simple
Groups with separate identities
The least common multiple order
The number of conjugates of a group element
Even and odd permutations
How large is the set of degenerate real symmetric matrices?
Advanced Undergraduate Mathematics
Troublemakers
The countability of the reals
The plane constitutes an uncountable set
A consequence of the nearness of rationals to reals
A universal property of real subsets
A topological spoof
Is there a nonmeasurable set?
Is there a nonmeasurable set?
Is there a function continuous only on the rationals?
The continuum hypothesis
A heavy-duty proof that 1 = 0
All complex numbers are real
Opening the floodgates
Parting Shots
Ibn Qurra
Reading a calculator display
Infallibility of a symbolic manipulation program
A perpetual motion machine
The bouncing ball
Watch your ears!
Positive series with a negative sum
A Centennial tribute to Sam LoydIndex of Topics
Index of Names