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Paradoxes and Sophisms in Calculus

Sergiy Klymchuk and Susan Staples
Mathematical Association of America
Publication Date: 
Number of Pages: 
Classroom Resources Materials
Problem Book
[Reviewed by
Mark Hunacek
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This short book, which is apparently a reprint of a book published in New Zealand about seven years ago, is a compilation of about 60 examples in calculus. The examples are of two basic kinds: true but counter-intuitive results (these are the “paradoxes” of the title, though I am not sure that something can be called a paradox just because it’s surprising) and false results with purported proofs that “look formally correct” but contain a “subtle mistake or flaw” (these are the “sophisms”). There are approximately twice as many sophisms as there are paradoxes.

Roughly the first half of the book consists of statements of the various results, and the second half consists of explanations and solutions, along with occasional references for further reading. The preface states that the book is intended to enhance the teaching and learning of a first-year calculus course, though some of the discussions, I thought, would not be out of place in a beginning real analysis course.

The preface also states that the book explores topics usually covered in first-year calculus: “functions, limits, derivatives and integrals”. This is mostly correct, though I would explicitly add to this list infinite sequences and series. Also, some of the examples also involved some understanding of ideas from physics (such as center of mass or angular velocity), and a few involved concepts, such as volume of solids of revolution, usually taught in multi-variable calculus courses. On the other hand, some of the examples really involved nothing more than high-school algebra.

The examples are of variable quality. Some, for example, are virtually obvious. For example, one sophism purports to prove that 1 = 2 by calculating the limit of \(x \sin(1/x)\) as x approaches 0, in two different ways: first one correctly calculates the limit to be 0, and secondly one incorrectly calculates it to be 1 by replacing \(x\) by \(t = 1/x\) and using the fact that \(\sin t / t\) approaches 1 as \(t\) approaches 0. Of course the problem here is that when we replace \(x\) by \(t= 1/x\), the variable t does not approach 0 as x does. I don’t see anything “subtle” or “formally correct” about this; if a student were to write this kind of thing on an exam, that response would likely elicit an exasperated sigh.

A fair number of the other sophisms are equally obvious, such as the one where the “subtle mistake” is evaluating the limit of \(n^{1/n}\) (as n approaches infinity) by saying it is the n-th root of infinity, or infinity.

As another example, another sophism gives the following argument that purports to show that 1 = 2: start with the equation x2= x + x + …. + x (x summands); differentiate to obtain 2x = 1 + 1 + … + 1 = x, and divide by x. Here even the authors don’t give what seems to me the easiest reason why this argument makes no sense: they say (correctly) that the sum rule for differentiation does not apply in cases where there is a variable sum, but an even more obvious reason, I think, is that the initial equation does not even make sense unless x is a positive integer.

As obvious as these are, the award for the most groan-worthy example in the book must surely go to example 20 from chapter 3, which purports to “prove” that 2 = –2 because they are both square roots of 4. One would think that any student who really needs to think about this does not belong in a calculus class.

On the other end of the scale, some of the problems are simply too difficult to expect most students to solve. One asks for an example of a function that is differentiable at a point but not continuous in any neighborhood of that point. The authors give a reasonably simple example (define f(x) to be x2 if x is rational and 0 if x is irrational) and prove that it works, but this is hardly the kind of problem that one sees in a first-year calculus course; even if the authors gave the example up front and asked for a verification, it would still be a more-than-substantial exercise for such a course.

Likewise, the problem that follows this states as a fact that continuous, non-differentiable functions exist, and asks the reader: “Can you envision this improbable phenomenon?” For most beginning calculus students, the correct (i.e., truthful) answer to this question is a simple “no”. In the solutions part of the book, the authors give a brief discussion of this idea, but I don’t see much value in asking this particular question of the students in the first place.

There are, however, a number of results that fall between these two extremes and are both fairly interesting and accessible. Most struck me as being fairly well known, but there is some value in having them assembled in one easy-to-find place. The very first problem in the book, for example, involves repeatedly placing one brick on top of another but a little to the right (as you look at them from the side); the question is what is the maximum overhang? The answer, a nice application of the divergence of the harmonic series, is that there is no maximum (at least in theory, if not actual practice). This problem, according to the book, was first popularized by Martin Gardner in 1964, and the authors also provide references to a pair of articles from the American Mathematical Monthly in 2009 that discuss it.

Another nice paradox discussed in the book, which I first learned about in Nahin’s Mrs. Perkins’s Electric Quilt, concerns a ladder leaning against a wall, which begins to slide down as the base moves further and further away from the wall; a bit of calculus manipulation leads to the conclusion that by the time the ladder hits the ground it is moving with infinite velocity. (The explanation here is that the top of the ladder does not always maintain contact with the wall.)

Other examples expose the reader to fractals (the Sierpinski carpet and Koch snowflake), the Cantor set, the cycloid, Gabriel’s Horn (a figure with infinite surface area but finite volume) and others. As noted before, I suspect most of these would be best done in lecture rather than having the student try and solve problems on them.

This book is a companion of sorts to Klymchuk’s Counterexamples in Calculus, unseen by me but reviewed a few years back in this column. The reviewer there, Allen Stenger, thought that that book would be most useful if “read by the teacher and not by the student”. I think this is a correct statement for this book as well. Its principal value, it seems to me, is as a reference for instructors of calculus courses, who can incorporate some of the more difficult questions into their lectures and, if they think it worthwhile, assign some of the easier ones as homework questions. I’ll be teaching an introductory calculus course next fall and though I won’t be assigning this book as a supplemental text, I’ll probably flip through it from time to time when preparing lectures.


Mark Hunacek ( teaches mathematics at Iowa State University. 

I. Paradoxes
II. Sophisms
III. Solutions to Paradoxes
IV. Solutions to Sophisms
About the Authors