Mathematical Fallacies, Flaws, and Flimflam

Mathematical Fallacies, Flaws, and Flimflam

Edward J. Barbeau
Series: Spectrum
Copyright Date: 2000
Edition: 1
Pages: 184
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  • Book Info
    Mathematical Fallacies, Flaws, and Flimflam
    Book Description:

    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.

    eISBN: 978-1-61444-518-0
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-viii)
    (pp. ix-x)

    Mathematics is a dangerous enterprise. 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.

    For the mathematics teacher, one source of such errors is the work of students. While students are responsible for a certain amount of...

  3. Table of Contents
    (pp. xi-xvi)
  4. Chapter 1 NUMBERS
    (pp. 1-14)

    Arithmetic is one of the first topics taught to elementary pupils. This is appropriate because of the prevalence of numbers in modern society and the need to have a numerate population. In particular, we often need to negotiate percentages and it is here that public understanding frequently falters. We begin with a few examples

    The columnMoney angles: where else to invest?by Andrew Tobias in the May 17, 1993, issue ofTimeoffers this advice for improving your financial worth:

    Buy staples in bulk when they’re on sale …. Consider a family that buys one bottle of wine each...

    (pp. 15-36)

    The April 3, 1994 issue of theWashington Postrecounted how a sports celebrity failed to answer the following questions on a high school equivalency test:

    1. If the equation for a circle isx2+y2= 34, what is the radius of the circle?

    2. If 6 − 50 =x+ 20, what isx?

    3. If 2xplus 3xplus 5x= 180, what isx?

    Bert Sugar, the publisher ofBoxing Illustrated, was not surprised at the failure. He opined that anyone who could answer the math questions “could probably qualify as a nuclear scientist.” The reporter’s reaction...

  6. Chapter 3 GEOMETRY
    (pp. 37-62)

    In a typical introductory course in abstract algebra, after you have proven the impossibility of trisecting an arbitrary angle using just straightedge and compasses, you sum up the argument as follows: “We have just shown that cos 20° is not constructible, and so we cannot construct a 20° angle either; thus we cannot trisect a 60° angle, and so we cannot trisect an arbitrary angle.”

    You can often create some consternation by continuing: “Now the fact that we cannot construct a 20° angle also shows that we cannotbisecta 40° angle and so you cannot bisect an arbitrary angle...

    (pp. 63-76)

    Mathematical reasoning has to be pursued with great care, as there are pits that beset the unwary. We will begin with proofs by induction before going to other aspects of finite mathematics. Effecting a proof by induction is a sophisticated procedure that many students find quite mysterious. As long as they harbour the suspicion that somehow they are assuming what they have to prove, they are likely to treat it as a rote process and fall into confusion.

    Perhaps the typographical error in the running head for page 589 of Michael Sullivan’sCollege Algebra(Prentice Hall, 1995) says it all....

  8. Chapter 5 PROBABILITY
    (pp. 77-90)

    Omicron and Upsilon are discussing Problem 297 fromFive Hundred Mathematical Challenges(MAA, 1996):

    A tennis club invites 32 players of equal ability to compete in an elimination tournament. (This proceeds in a number of rounds in which players compete in pairs; any losing player retires from the tournament.) What is the probability that two given players will compete against each other?

    Omicron. The solution seems straightforward enough. There have to be 31 games to knock out all but the ultimate winner. There are$\left( \begin{matrix} 32 \\ 2 \\ \end{matrix} \right)$possible pairs, so that the probability of a given pair being selected for a particular...

    (pp. 91-102)

    Proposition. Letnbe a nonnegative integer. The functionxnis constant.

    Proof. Observe that${{({{x}^{0}})}^{\prime }}=0$. Assume that the derivative ofxnis zero forn= 0, 1, 2, … ,k. Then

    \[({{x}^{k+1}})^{\prime } =(x\cdot {{x}^{k}})^{\prime } =x^{\prime } \cdot {{x}^{k}}+x\cdot ({{x}^{k}})^{\prime } \]

    is also zero sincex= (x1)= (xk)= 0. ♡

    Contributed by Alex Kuperman of the Israel Institute of Technology (Technion) in Haifa.

    Atx=c, the functiony= (xc)2=x2− 2cx+c2has a minimum, so that 0 =Dy=D(x2) − 2cD(x) =D(x2) − 2c. Butcis arbitrary andc=x....

    (pp. 103-116)

    Exercise. Integrate$\int{\frac{1}{x+1}dx}$.

    Solution(by a student on a quiz).

    \[\int{\frac{1}{x+1}=\int{\left( \frac{1}{x}+\frac{1}{1} \right)}}dx=\int{\frac{1}{x}}dx+\int{\frac{1}{1}}dx=\log x+\log 1=\log (x+1)+C.\diamond \]

    Contributed by Lewis Lum of the University of Portland in Oregon.

    A fair number of fallacies can be generated by simply neglecting the constant of integration when computing indefinite integrals. Here are a few examples.

    (a) In integrating$J=\int{dx/(x\log x)}$by parts withu= 1/logxanddv=dx/x, one readily obtainsJ= 1 +J. CancellingJyields 1 = 0.

    According to R. P. Boas of Seattle, WA, all swindles like this reduce to the form$\int{({{g}^{\prime }}/g)dx}$and he mentions the examples$\int{{{e}^{x}}{{e}^{-x}}dx}$and...

    (pp. 117-128)

    LetF(x,y) = (x+y)2. Setx=uvandy=u+v. Then

    \[\frac{\partial x}{\partial v}=-1,\frac{\partial y}{\partial v}=1,\frac{\partial F}{\partial x}=\frac{\partial F}{\partial y}=2(x+y).\]

    By the chain rule,

    \[\frac{\partial F}{\partial v}=\frac{\partial F}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial F}{\partial y}\frac{\partial y}{\partial v}=-2(x+y)+2(x+y)=0.\]

    But, from the definition,F(u,v) = (u+v)2whence ∂F/∂v= 2(u+v) = 2y. We seem to have found thaty= 0. ♣

    In the two computations of ∂F/∂v, the meaning of the variablesuandvis not consistent. We can clarify the first by writing the function ofuandvas the composite of two functions:




    so that$(F\circ g)(u,v)={{(2u)}^{2}}$. What we are really computing is the...

    (pp. 129-140)

    Choose a positive integernand letSbe the set of all real solutions (x1,x2, … ,xn,xn+1) of the equation

    \[{{x}_{1}}+{{x}_{2}}+\cdots +{{x}_{n}}+{{x}_{n+1}}=1.\] (*)

    Since any choice ofx1,x2, … ,xndetermines exactly one value of xn+1for which (*) holds, the mappingf: Rn+1→ Rndefined byf(x1,x2, … ,xn+1) = (x1,x2, … ,xn) mapsSone-to-one onto Rn. Moreoever,fpreserves addition and scalar multiplication (defined as usual). It follows thatSis isomorphic to Rnand is hence a vector space. In particular,Smust contain the zero...

    (pp. 141-150)

    Two separate footnotes appear on page 139 of the 1925 textbookA Practical Treatise on Fourier’s Theorem and Harmonic Analysis for Physicists and Engineersby Albert Eagle (Longman’s, Green & Co., London), where the author treats the proposition that a Fourier series converges to the average of the right and left limits of its parent function:

    A case in point is the series$\sin 1/t+\frac{1}{2}\sin 2/t+\frac{1}{3}\sin 3/t+\cdots $. Astincreases towards +∞, this series, as we have seen, approaches the limit of$+\frac{\pi }{2}$; while whentis negative and moves towards −∞ the series approaches the limit of$-\frac{\pi }{2}$while actuallyat...

  14. Chapter 11 PARTING SHOTS
    (pp. 151-160)

    Here is a multiple-choice question on the history of mathematics. The dates of birth and death of the Arab mathematician Thabit Ibn Qurra, who translated and commented on Greek higher mathematics, are

    (A) 826–901 (B) 833–902 (C) 836–901

    (D) 836–911 (E) All of the above.

    The correct answer is (E). See Al Abdullah Al-Daffa’,The Muslim contribution to mathematics(Humanities Press, 1977). The dates (A)–(D) are given, respectively, on pages 44, 13, 59, and 86.

    Sandra Z. Keith of St. Cloud State University in Minnesota writes:

    A student differentiatingf(x) = cos2xobtained the answer...

  15. References
    (pp. 161-162)
  16. Index of Topics
    (pp. 163-164)
  17. Index of Names
    (pp. 165-167)