This work is ideal for undergraduate students with little prerequisite knowledge required. It can support a typical pre-calculus college course as a reference and additional text. Advanced secondary students will find much of the content relevant, especially early sections on “story problems”. On the surface, the flow of chapters and exercises make this appear to be a textbook. More accurately, the deep and narrow scope of each section serves as a thorough explanation of specific sub-topics, perhaps as preparation for Mathematical Olympiads and similar contests. Fittingly for Olympiad-level study, this work expects familiarity with intermediate topics. These are taken to a higher level, with a few topics of the type seen in contests matched with highly developed problem-solving techniques and a near absence of application and theory. This is a reference work richly augmented with detailed examples and solved exercises.

One pillar technique exploited throughout is Vieta's formulas applied to the quadratic polynomial. Vieta's formulas relating the coefficients of a polynomial to sums and products of its roots are, in my opinion, too little explored in textbooks in this area. Here they are referred to as “Viète's formulas”, avoiding the common Latinized form of the name. Regardless of the appellation, there is much detailed, quality content around these formulas to be used in study or perhaps as classroom capsules to add to a lecture.

Other topics covered well and in detail here include:

- Cardano's formula
- Integer solutions to nonlinear systems
- Substitution techniques for complex expressions
- Cyclotomic polynomials
- Diophantine equations including non-standard ones
- Fibonacci sequence beyond the basics

One weakness is an avoidance, largely, of graphical or geometric illustration, even where it could be most helpful. For instance, while one rare illustration shows graphs of root functions for describing domain, there is no such display to illustrate intersections as solutions to systems. I appreciate the in-depth exploration of domain which is better than in most comparable textbooks that I have seen. The result is an algebra-centric approach. For instance, two dozen pages on trigonometric topics have only four illustrations.

This book is well referenced and adequately indexed.

Tom Schulte, web coordinator for the Louisiana-Mississippi MAA Section, teaches mathematics online for multiple institutions from the shores of Castine Bayou in Louisiana.

## Comments

## The Equations World

“The Equations World” is a very well thought out and written problem-solving book. It contains not just lots of fresh and bright mathematical ideas, but interesting historical tours into foundations of equations’ solutions and their development through the centuries. The review reads as positive, and points out special features of the book that the reviewer appreciated. However, in my opinion the final negative comments un-fairly overshadow this mostly positive picture and provide the wrong impression on the true narrative of the book. It is one of the best problem-solving books on equations analyses I’ve seen, and for this reason, I want to share my thoughts about some of the missing points in this review.

What the reviewer mentioned as a drawback “One weakness is an avoidance, largely, of graphical or geometric illustration, even where it could be most helpful” perhaps reflects his view of applying illustrations wherever possible. On occasion function graphs could even be misleading. For example, there are polynomials, whose graphs are indistinguishable from zero on an interval. On such occasions the algebra and its graphical illustration may seem even to contradict each other. I agree though that graphs and illustrations should be used when they help to simplify solutions. The reviewer stated that “while one rare illustration shows graphs of root functions for describing domain, there is no such display to illustrate intersections as solutions to systems”. It is not clear what problem he was referring to here; he did not mention the chapter and problem number. Yet, there are several occasions when the author nicely utilizes graphs in his solutions. If you look, for instance, at problem 12 from chapter 13, it provides great illustration of using the Cartesian coordinate system method for solving problems; a very useful tool for demonstrating links between algebra and geometry.

Speaking about graphs and geometrical illustrations, “The Equations World” has many such illustrations where they appear to be applicable and useful; for instance, in the first two chapters. Moreover, at the end of chapter 12, the author emphasizes that “Since many mathematical ideas have both an algebraic and geometrical aspect, always keep your eyes open for possible links between algebraic and geometric interpretations of the items involved”. This statement is well supported by illustrating close connections between algebra and geometry in problems 19 and 20 from this chapter. While solving the algebraic problem 19, the author applied pure geometrical techniques. On the other hand, while solving the geometrical problem 20, its solution was simplified by applying algebraic methods for the system of equations solutions. Both problems are supported with geometrical illustrations. There are other interesting instances of links between math disciplines throughout the book. In chapter 6, for example, the famous construction problem of the impossibility of trisecting the angle with the use of a compass and straightedge is clarified with a trigonometry use through cubic equation analyses; and the cyclotomic equations reveal the connections to the constructability of regular polygons.

As the reviewer mentioned, the author took an algebra-centric approach. It is understandable; the book is devoted to the theory of equations. However, one of its undeniable strengths that I see is in its practical illustrations of multiple connections among math disciplines through applying equations solutions.

Another good pedagogical approach is incorporating the elements of mathematical research throughout the book. This was best demonstrated in chapter 11. While discussing solutions of three variations of un-orthodox Diophantine equation and posing a few questions unexpectedly emerging from the obtained results, the author arrives at very interesting conclusions leading to formulating the necessary and sufficient conditions for a natural number to be a Fibonacci number. It is indeed a fascinating example that illustrates the relevance of problems solving in new discoveries.

I agree with the reviewer that this is the ideal book for undergraduate students and secondary students who take advance algebra classes. The book should be a valuable addition to any university’s math library.

## The comment on 6/20/2020 was

The comment on 6/20/2020 was posted by Yury

Grabovsky, Professor of Mathematics at Temple University, PA.

## Hello from the Reviewer

Salutations, YGRABOVSKY

It is good to hear from another reader that like myself sees value in “The Equations World”.

While agreeing with most of your points, even that “On occasion function graphs could even be misleading”, I remain of the opinion that there’s a paucity of graphical and geometric illustration compared to what if present would elevate the presentation for the target audience. I agree that “there are several occasions when the author nicely utilizes graphs in his solutions.”

Indeed, there are several portions “demonstrating links between algebra and geometry.” Personally, I feel geometry itself let alone its connections to algebra are poorly represented in American secondary and college education and anyone looking to fill that gap will find this text a useful tool. As a teacher in mathematics, I find it a useful resource in preparing material for my students.

That you for sharing your considered thoughts here.

Tom Schulte