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The Equations World
Equations are the lifeblood of mathematics, science, and technology, and this book examines equations of all kinds. With his masterful ability to convey the excitement and elegance of mathematics, author Boris Pritsker explores equations from the simplest to the most complex—their history, their charm, and their usefulness in solving problems.
The Equations World bridges the fields of algebra, geometry, number theory, and trigonometry, solving more than 280 problems by employing a wide spectrum of techniques. The author demystifies the subject with efficient hints, tricks, and methods that reveal the fun and satisfaction of problem solving. He also demonstrates how equations can serve as important tools for expressing a problem's data, showing the ways in which they assist in fitting parts together to solve the whole puzzle. In addition, brief historical tours reveal the foundations of mathematical thought by tracing the ideas and approaches developed by mathematicians over the centuries. Both recreational mathematicians and ambitious students will find this book an ample source of enlightenment and enjoyment.
The Equations World bridges the fields of algebra, geometry, number theory, and trigonometry, solving more than 280 problems by employing a wide spectrum of techniques. The author demystifies the subject with efficient hints, tricks, and methods that reveal the fun and satisfaction of problem solving. He also demonstrates how equations can serve as important tools for expressing a problem's data, showing the ways in which they assist in fitting parts together to solve the whole puzzle. In addition, brief historical tours reveal the foundations of mathematical thought by tracing the ideas and approaches developed by mathematicians over the centuries. Both recreational mathematicians and ambitious students will find this book an ample source of enlightenment and enjoyment.
336 pages, Paperback
Published August 14, 2019
4 people are currently reading
16 people want to read
About the author
Boris Pritsker
9 books2 followersBoris Pritsker was born in Kiev, Ukraine, and studied mathematics at Kiev State Pedagogical University, where he graduated summa cum laude with a degree in Mathematics and Education (the equivalent of a U.S. Bachelor of Science). Pritsker served as a mathematics teacher in two high schools, including a specialized school for gifted and talented students featuring advanced curricula in algebra, geometry, trigonometry, and calculus. During this time, he developed a series of educational seminars exploring classic mathematical problems and advanced topics that extend beyond the standard secondary school curriculum. In algebra and trigonometry, Pritsker examined strategies for solving various types of equations and inequalities, and systems of equations and inequalities. Additionally, he applied non-traditional techniques to an in-depth analysis of Fibonacci numbers. The seminars highlighted several sophisticated topics rarely addressed in mathematical literature. These included utilizing Diophantine equations to identify common or extraneous solutions within trigonometric equations and applying non-traditional techniques to solve multivariable equations. Through his work with the Fibonacci sequence, Pritsker established the necessary and sufficient conditions for a natural number to be identified as a Fibonacci number.
In the realm of geometry, the seminars covered non-trivial construction problems involving inaccessible vertices, as well as Steiner’s and Mascheroni’s challenges—constructions using a straightedge only or a compass only. Along with traditional synthetic techniques, the coursework integrated isometric and homothetic transformation properties. The seminars prompted students to derive alternative proofs for fundamental theorems, such as Fagnano’s Problem, Apollonius’s Theorem, and the Nine-Point Circle. Beyond pure geometry, the coursework examined the relationship between the inversion in a circle and mechanical engineering applications. Additionally, students investigated the intersection of geometry and physics, utilizing Archimedes’ Law of the Lever to discover new geometric properties.
Pritsker also trained school math teams for participation in math competitions; several students from these math teams won the prestigious local and regional math Olympiad contests.
After immigrating to the United States, Pritsker earned an MBA magna cum laude from the Zicklin School of Business at Baruch College, City University of New York. A licensed CPA in New York State, he has spent the past twenty-seven years at CBIZ Inc., a national accounting, insurance, and financial consulting firm, where he rose to the position of Director.
Driven by a lifelong devotion to mathematical research and education, Pritsker has spent years exploring challenging topics in Euclidean geometry, algebra, trigonometry, and pre-calculus. His findings and original problems have been featured in educational journals across the Soviet Union, the United States, Singapore, and Australia.
Pritsker authored internationally acclaimed mathematics books “Geometrical Kaleidoscope”, Dover Publications, 2017, “The Equations World”, Dover Publications, 2019, “Mathematical Labyrinths. Pathfinding”, World Scientific, 2021, "Expanding Mathematical Toolbox: Interweaving Topics, Problems, and Solutions", CRC Press, Taylor & Francis Group, 2023, “Geometrical Kaleidoscope” Second Edition, World Scientific, 2024, and "Learning Geometry Through Problem Solving", World Scientific, 2026. The books have been featured with detailed positive reviews in the Mathematical Association of America and ZBmath web-sites. “Geometrical Kaleidoscope” was assigned a BLL rating and the MAA Basic Library List Committee suggested that undergraduate mathematics libraries consider the book for acquisition. “Mathematical Labyrinths. Pathfinding” was translated into Turkish language and published in 2025 year, “Matematikte Problem Çözme” - Zihin Fırtınaları Ve Sıra Dışı Yolculukl
In the realm of geometry, the seminars covered non-trivial construction problems involving inaccessible vertices, as well as Steiner’s and Mascheroni’s challenges—constructions using a straightedge only or a compass only. Along with traditional synthetic techniques, the coursework integrated isometric and homothetic transformation properties. The seminars prompted students to derive alternative proofs for fundamental theorems, such as Fagnano’s Problem, Apollonius’s Theorem, and the Nine-Point Circle. Beyond pure geometry, the coursework examined the relationship between the inversion in a circle and mechanical engineering applications. Additionally, students investigated the intersection of geometry and physics, utilizing Archimedes’ Law of the Lever to discover new geometric properties.
Pritsker also trained school math teams for participation in math competitions; several students from these math teams won the prestigious local and regional math Olympiad contests.
After immigrating to the United States, Pritsker earned an MBA magna cum laude from the Zicklin School of Business at Baruch College, City University of New York. A licensed CPA in New York State, he has spent the past twenty-seven years at CBIZ Inc., a national accounting, insurance, and financial consulting firm, where he rose to the position of Director.
Driven by a lifelong devotion to mathematical research and education, Pritsker has spent years exploring challenging topics in Euclidean geometry, algebra, trigonometry, and pre-calculus. His findings and original problems have been featured in educational journals across the Soviet Union, the United States, Singapore, and Australia.
Pritsker authored internationally acclaimed mathematics books “Geometrical Kaleidoscope”, Dover Publications, 2017, “The Equations World”, Dover Publications, 2019, “Mathematical Labyrinths. Pathfinding”, World Scientific, 2021, "Expanding Mathematical Toolbox: Interweaving Topics, Problems, and Solutions", CRC Press, Taylor & Francis Group, 2023, “Geometrical Kaleidoscope” Second Edition, World Scientific, 2024, and "Learning Geometry Through Problem Solving", World Scientific, 2026. The books have been featured with detailed positive reviews in the Mathematical Association of America and ZBmath web-sites. “Geometrical Kaleidoscope” was assigned a BLL rating and the MAA Basic Library List Committee suggested that undergraduate mathematics libraries consider the book for acquisition. “Mathematical Labyrinths. Pathfinding” was translated into Turkish language and published in 2025 year, “Matematikte Problem Çözme” - Zihin Fırtınaları Ve Sıra Dışı Yolculukl
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Displaying 1 - 5 of 5 reviews
April 14, 2020
...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 server as thorough explanation on 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 new topics of the type seen in contests matched with highly developed problem-solving techniques and a near absence of application and theory. ...
[Look for my entire review at MAA Reviews]
[Look for my entire review at MAA Reviews]
January 23, 2020
In my pre-calculus class in college, I really struggled to understand several topics including exponential and logarithmic functions, division and factoring of polynomials, and trigonometric equations. Moreover, I did not see the point in solving these types of equations in the first place. This book helped me, and its creative questions and explanations taught me to be able to both think through and break down problems like a true mathematician. I greatly enjoyed this reading, and this book should be helpful to anyone who does not see the point in solving algebraic equations or struggles to grasp this kind of material in the first place. This book really saved me big time. It also... it made me feel genuinely happy again when I solve math problems, just like when I was a kid. There are many difficult problems covered in this book, which I did not go over in the details yet, but it will be challenging to try them, especially the Diophantine equations with multiple variables which I had never faced before.
December 30, 2019
Being a computer science major, I was looking for a math problem-solving book to practice before the exam. The goal was to find the book covering trigonometric equations, exponential and logarithmic equations. The book has what I was looking for and much more.
Very enjoyable and challenging. Problems range from easy to very difficult. The author provides excellent explanations and introduces multiple techniques for problems solving. What I found the most valuable is that you get the explanations of reasons for applying a specific technique to a specific problem, the book opens your eyes how to approach the unusual problems and make the solution manageable. Lots of exercises for your own practice with the detailed solutions and answers provided at the end of the book. One of the best books of this type I have ever found.
Very enjoyable and challenging. Problems range from easy to very difficult. The author provides excellent explanations and introduces multiple techniques for problems solving. What I found the most valuable is that you get the explanations of reasons for applying a specific technique to a specific problem, the book opens your eyes how to approach the unusual problems and make the solution manageable. Lots of exercises for your own practice with the detailed solutions and answers provided at the end of the book. One of the best books of this type I have ever found.
May 30, 2023
As a professional mathematician I have to admit that I was very impressed with the variety of techniques and many interesting suggestions for solving different types of equations presented in this book. The book provides a unique view on math problem solving, and I believe it should be extremely helpful to high school and college students interested in math. The book has several nice reviews on Amazon, but it is surprising it did not entice real interest in math community. In my mind this one deserves to be a bestseller.
June 13, 2020
Great problem-solving book which covers challenging topics through studying various types of equations in algebra, trigonometry, number theory, and geometry. It is not a routine set of problems but rather very well-written and organized treatise of various types of equations and multiple techniques for their solutions.
I found many interesting topics not seen anywhere before. I highly recommend it
I found many interesting topics not seen anywhere before. I highly recommend it
Displaying 1 - 5 of 5 reviews