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Beyond the Quadratic Formula

Ron Irving
Mathematical Association of America
Publication Date: 
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MAA Classroom Resource Materials
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Mark Hunacek
, on

Once, a long time ago, one could often find in college catalogs a course called Theory of Equations. These courses must have been fairly common, because for awhile there were quite a few books with this phrase (or some mild variation thereof) as the title. Indeed, a professor at my own undergraduate alma mater, Sam Borofsky, wrote one, which competed with similar books by (among others) Uspensky, MacDuffee, Cajori, and L. E. Dickson.

These books all had similar content. They generally discussed polynomials, complex numbers, the formulas for finding roots of cubic and quartic polynomials, theorems (like Sturm’s theorem) for getting information about roots even if they couldn’t be found explicitly, theorems for approximating roots, and (perhaps) the Fundamental Theorem of Algebra.

However, courses based on these books started (probably in the late 1960s or early 1970s) getting phased out of the undergraduate curriculum, presumably because they were being supplanted by an increased emphasis on abstract algebra. The texts listed above began to be considered hopelessly old-fashioned, and were consigned to dusty shelves in libraries or used book stores (or, on one occasion, a shelf at the semi-annual fundraising book sale for the public library of the city of Ames, where I once found and purchased Uspensky’s book for the princely sum of 25 cents.)

In the process of shifting emphasis from the old theory of equations courses to more modern courses in abstract algebra, some mathematical content necessarily got lost. While it is not uncommon to teach polynomials and their arithmetic in introductory algebra courses, few of them, for example, do much with the solutions of cubics and quartics.

Some upper-level algebra texts touch on these topics in connection with Galois theory; see, e.g., Cox’s Galois Theory, Newman’s A Classical Introduction to Galois Theory, Artin’s Algebra or Dummit and Foote’s Abstract Algebra. These, however, are not elementary texts, and the material on cubics and quartics does not appear until the student has already learned quite a bit of algebra. Saul Stahl’s Introductory Modern Algebra: A Historical Approach, the second edition of which is imminent, is one of the few truly introductory texts which address these topics at an early stage in the student’s development. If an undergraduate student these days is exposed to the subject at all, it is likely to be in a course on the history of mathematics, assuming that student’s university has such a course. (I suspect that these courses as well may not be as commonly offered as they once were.) It seems fair to say, therefore, that most undergraduate mathematics majors graduate from college without ever seeing the details of these formulas, or, for that matter, perhaps even not knowing that such formulas even exist.

And this brings me to the book under review, which attempts to remedy this situation. Combining mathematics and history, this text tells, in a way accessible to beginning students, the interesting story of how formulas came to be discovered for the roots of third and fourth degree polynomials, and why nobody will discover corresponding formulas for fifth (or higher) degree polynomials.

Specifically, the book begins with an introductory chapter on polynomials (treated informally as formal expressions rather than rigorously defined), and is followed by a chapter on quadratic equations, in which the familiar quadratic formula is derived from several different points of view. The study of cubic equations begins in the next chapter, which discusses Cardano’s formula. One of the more amusing aspects of Cardano’s formula is that even nice, simple numbers can wind up being represented by horrendous sums of cube roots of expressions involving square roots; lots of examples are provided in the book. See, for example, page 54 of the text, where the number 2 appears as

\[ \sqrt[3]{10\vphantom{\hat{E}}+6\sqrt{3}} + \sqrt[3]{10\vphantom{\hat{E}}-6\sqrt{3}}.\]

Even more interesting, when the cubic polynomial has three distinct real roots Cardano’s formula calls for square roots of negative numbers. Thus, finding real roots requires us to use complex numbers. Chapter 3 proceeds up to the point where this becomes apparent, and then, because complex numbers intrude into the search for roots, the author then pauses in the development to discuss, in the next chapter, complex numbers, and with this knowledge in hand, proceeds in chapter 5 to finish off the discussion of cubic polynomials.

The information gleaned in chapters 3 and 5 is then applied to the study of quartic polynomials in chapter 6, where, again, several different methods (those of Ferrari, Descartes and Euler) are looked at. The final chapter of the text then looks to higher-degree polynomial equations: it begins with an exposition (without proofs) of the insolvability of the quintic and mentions more recent work connecting quintic polynomial equations with complex dynamics. The book then proceeds to a discussion of the Fundamental Theorem of Algebra (first a historical overview is given, and then, in the last section of the text, the result is proved, using the Fundamental Theorem of Symmetric Polynomials, which in turn is discussed, but not completely proved, earlier in the chapter).

Any reader who dislikes “hands-dirtying” calculations will probably be horrified by a lot of the material found here. But these calculations are all part of what makes this material so interesting; it is just fascinating to see how the relatively simple quadratic formula extends to much more complicated formulas for polynomials of degree three and four. This extension has considerable historical significance: while many people think it was the possible appearance of the square root of a negative number in the quadratic formula that led to the creation of complex numbers, in fact, it was not. Cardano was quite content to label non-real solutions as useless, but it was the appearance of these square roots of negative numbers when the roots of the polynomial were known to be real that forced consideration of them, notably by Raffael Bombelli.

Because the bulk of this book is about formulas for cubic and quartic equations, its focus is fairly narrow, and the amount of material covered is somewhat less than one finds in the old theory of equations books of yore. Indeed, the author notes in the preface that the material covered in the text “parallels that found in a few chapters of” L.E. Dickson’s book.

Notwithstanding this narrow focus, though, the book is almost 250 pages long. Several factors account for this. For one thing, the exposition is quite leisurely, with very little background assumed on the part of the reader. (Witness the fact, for example, that complex numbers are discussed pretty much from scratch.) In fact, it would probably not be an overstatement to say that much of this book should be comprehensible to bright high school students with sufficient discipline to work through details on their own.

Such work is definitely required of the reader. A good deal of the book is written in what might be called the “inquiry based learning” approach, in which the author teases information out of the student by the use of carefully selected problems, generally broken up into manageable chunks and with adequate hints provided when necessary.

Another reason for the length of the book is the unusually detailed focus on history. Each chapter, except the first and last, ends with a fairly detailed historical section. The section in chapter 2, for example, is about fifteen pages long, and covers not only the history of the quadratic formula through several cultures but also touches on related developments in algebra.

I did notice, however, that only a handful of the historical sources cited in these chapters were published in the last ten years. I mention this because the August/September 2013 issue of the MAA Focus contains an article (“The New Historian’s Bookshelf”) by Fernando Gouvêa which points out that the last decade has resulted in a “remarkable outpouring” of new writings, which “have opened up whole new vistas” in mathematical history. With specific reference to Indian mathematics (which the author discusses in this book), for example, Gouvêa points out that Mathematics in India by Plofker “should be everyone’s introduction to Indian mathematics.” Plofker’s book does not appear in the 70-item bibliography of this text. Now, I freely admit that I am in no sense a mathematical historian; I have little personal familiarity with any of the recent scholarship Gouvêa talks about in his article, and cannot say whether any of it would have made much difference in Irving’s historical discussions, but Gouvêa’s article did give me food for thought, and as a result of it I have taken to looking more carefully at the dates of sources in history discussions than I did previously. (Another of the books cited by Gouvêa in his article, Robson’s Mathematics in Ancient Iraq, does appear in the book’s bibliography.)

This little (potential) quibble aside, there is a great deal to like about this book. It is clearly written and will teach the reader a lot of mathematics that current undergraduates may rarely see. Future teachers, in particular, may find quite a lot of value here, since it clearly conveys the idea that the standard quadratic formula, which most students find boring, is really the tip of a very interesting iceberg.

Mark Hunacek ( teaches mathematics at Iowa State University.

1. Polynomials
2. Quadratic Polynomials
3. Cubic Polynomials
4. Complex Numbers
5. Cubic Polynomials, II
6. Quartic Polynomials
7. Higher-Degree Polynomials
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