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Essays on Numbers and Figures

V. V. Prasolov
American Mathematical Society
Publication Date: 
Number of Pages: 
Mathematical World 16
[Reviewed by
Steve Morics
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I cook most of the meals in our household, but when I get a night off, my wife makes something she calls hot dish, a favorite meal from her childhood. It's made of spaghetti noodles, browned ground beef, a can of tomatoes, and... Well, there is no "and", that's it. There are no seasonings, no cheese, no sauce (other than the liquid the tomatoes were packed in). I suppose it's not bad as it is, but with a little work, it could be much more satisfying.

The same goes for V.V. Prasolov's Essays on Numbers and Figures, an interesting little book published as part of the American Mathematical Society's Mathematical World series. Aimed at advanced high school students, undergraduates, mathematics teachers, and others with an avocational interest in mathematics, the series "...features well-written, challenging expository works that capture the fascination and usefulness of mathematics. " There's no attempt to capture mathematics' usefulness here, though. The twenty different subjects in this collection are here simply because Prasolov has been fascinated with them at one time or another.

The topics covered by the essays range over geometry, topology, number theory and polynomials. Titles range from the broad (Conjugate Numbers) to the specific (The Van der Waerden Theorem on Arithmetical Progression) to the unique (One Butterfly and Two Butterflies Theorems). Prasolov has lectured on all of these topics to high school students in Russia and Israel. The essays are short. All twenty essays fit into 76 pages, with some essays taking as few as two. The geometry chapters are a bit longer to make space for helpful diagrams.

By the author's own admission, the essays have little to do with each other; each is supposed to be a self-contained snapshot of the subject at hand. However, there is a bit more overlap than the author admits to. Brocard points, the topic of Chapter 14, show up again in the last two chapters. Notation defined in Chapter 14 is used, but not defined, in Chapter 12. The chapters on conjugate numbers and sums of squares of polynomials share results concerning products of powers of conjugates.

At their best, the essays pose a question or state an interesting result, and then proceed to explore in an intuitive way some of the consequences leading from the initial topic. The first essay, Conjugate Numbers, begins by noticing that 2 plus the square root of 3, raised to increasing powers, gets closer and closer to an integer value. This leads to a discussion of the roots of certain types of polynomials through a natural sequence of questions and explanations. Another essay begins by mentioning the four-color theorem, and then asks, "What happens if we start coloring in vertices?" What happens a few pages later is Whitney's theorem on chromatic polynomials.

At their weakest, the essays read like very condensed textbook chapters. The essay on isogonal conjugate points covers seven theorems in the space of five pages. Brocard points are introduced in an engaging manner, but their properties are covered by running through a laundry list of theorems with no evident motivation. Theorems are good, of course, but this book's stronger essays motivate or explore one result at a time, rather than attempt to provide comprehensive coverage of a topic.

The Mathematical World web page describes this book as being appropriate for independent study, but some words of caution are called for before handing one of these essays over to an eager but inexperienced student. Knowledge of calculus is required to understand some of the essays; proof by induction is used in some of the others. Typographical errors, sometimes in formulas, occur about once per essay, which seems like a lot considering their length. There is some unfamiliar notation ("ctg" for cotangent, for example), and, as mentioned above, notation used in one essay may be defined in another. Finally, these things are dense. Many steps are left out and a younger student, especially one weaned on today's less rigorous calculus texts, may have a tough time following the thread of an argument. Of course, the compactness of the essays provides the students with some practice in reading journal articles in small, accessible pieces. Still, if an inexperienced student is handed a two-page essay to work on, he or she isn't likely to anticipate the amount of time needed to work through it without some words of guidance.

A little extra work could have taken this book from the ranks of the interesting to the really exceptional. The overall impression one takes is that the author had a bunch of outlines for talks lying around, decided there were enough for a book, and just threw them together without much thought or effort. Simply reorganizing the essays, with an eye towards combining similar topics and notation, would go a long way toward improving the utility of the book. A careful reading to make sure unfamiliar notation is defined before it is used would prevent some frustration on the part of a student user. A few references would also be appreciated. In keeping with the spirit of the book, only one or two sources for each essay would be needed, ideally consisting of the sources which prompted Prasolov to look at the topic in the first place, or perhaps the first sources to which he turned when he began his explorations. The "well-known" theorems aren't well-known by the intended audience, and a couple of references can give first time mathematical researchers a push to their school's library.

As it stands, though, the book is still a worthwhile resource for mathematics departments to have on hand. While resources for interesting projects and explorations for calculus classes as a whole are easy to come by, the supply of potential independent study topics for the motivated first or second year student is much smaller. High schools with committed mathematics teachers who get students looking for more to do also may want to give this book a look. The essays are too dense for Prasolov to fully express his enthusiasm and love for his subject, but this collection serves as a nice resource for accessible topics beyond the standard curriculum around the level of calculus.

Steve Morics is assistant professor of mathematics at the University of Redlands. Having never before lived west of the Mississippi, he's trying to adjust to life in southern California, with only two seasons (dry and wet) instead of four, freeway interchanges that look like modern sculptures, and summer temperatures which can range from the 110's during the day to the low 60's at night. His first mathematical love was combinatorics, and he's now working on research in mathematics education and the connections between mathematics and politics.

  • Conjugate numbers
  • Rational parametrizations of the circle
  • Sums of squares of polynomials
  • Representing numbers as the sum of two squares
  • Can any knot be unraveled?
  • Construction of a regular 17-gon
  • The Markov equation
  • Integer-valued polynomials
  • Chebyshev polynomials
  • Vectors in geometry
  • The averaging method and geometric inequalities
  • Intersection points of the diagonals of regular polygons
  • The chromatic polynomial of a graph
  • Brocard points
  • Diophantine equations for polynomials
  • The Pascal lines
  • One butterfly and two butterflies theorems
  • The Van der Waerden theorem on arithmetical progressions
  • Isogonal conjugate points
  • Cubic curves related to the triangle