The book “Everything in mathematics makes sense” is intended for school students in grades 8-12 who study mathematics and also for their teachers. The main point of this book is that mathematics is not about mechanical usage of formulas but about understanding. The author devotes much attention in his book to concepts such as definitions, proofs, logic. The author emphasizes that traditional textbooks do not give the understanding and therefore can’t teach someone to apply mathematics to problems of real life. This book draws attention to the beauty of mathematics and to the fact that if mathematics seems complicated to someone then he/she just does not realize how complicated real life is. In contrast to this, a traditional approach makes the perception that mathematics is cumbersome and useless. The author emphasizes that understanding of mathematics extends the abilities of a human applicable to real life and that real life still requires qualities beyond computer button pressing… This textbook provides an evidence that a usage of calculators is harmful for learning mathematics as well as mechanical memorising of formulas. On the other hand, the knowledge of the multiplication table and the ability to perform simple arithmetical operations without calculators is necessary for everyone. There are many examples of incorrect formulations and reasonings from other sources collected in the book.

The author has many decades of mathematics teaching experience. The given book is a product and generalization of this experience. The textbook contains materials of different topics. The explanation of each topic is provided with a recommendation in which grade the learning of this topic is optimal. There are many tasks in the book. Some tasks are provided with a solution explained in detail, some tasks contain hints, but some of them are intended for autonomous solution. Also, the book contains many questions for understanding checking. In contrast to traditional textbooks of mathematics, this book contains many proof tasks. There are many
illustrative pictures following the material in this book. The book is written in a “cheery” style, contains epigraphs with citations of famous scientists and humorous examples of mathematical fallacies from Internet.

Emanouil’s book underlines that everything in mathematics is based on rigorous definitions and proofs. The accurate definitions of many fundamental notions are given in the book as well as the accurate proofs of statements that are usually avoided in school-books. Also, this book explains why these definitions have a sense and why we use precisely these definitions but not other ones, tells about these definitions as a part of a whole harmonic picture. The book emphasizes the difference between a definition and a provable statement. It provides us with examples when the human intuition leads to a wrong answer.

Much attention is given to problems with parameters. The reason of this increased attention is that problems with parameters are the kind of problems that can’t be solved mechanically without thinking. The author gives a lot of different variations of equations and inequalities with parameters requiring radically different approaches for their solution. In addition to the traditional topics that are explained in school-books of mathematics, this book covers several interesting nonstandard topics that are very helpful for the development of intellectual and mathematical facilities of school students, including diophantine equations, functional equations, combinatorics and combinatorial geometry, the inclusion-exclusion method, comparison of infinite quantities, equations and inequalities with integer and fractional parts. The book analyzes and disassembles interesting questions such as, for example, proofs of irrationality for concrete numbers, the existence and uniqueness of the decomposition into prime numbers, the sums and products of divisors, counting relatively prime numbers, the fixed-point theorem. Some of interesting questions being considered are too difficult for complete explanation in a school-book; the textbook provides the references to external sources where the reader can find the detailed explanations.

This textbook can be used for preparing school students to mathematical competitions. When i was 10-11 years old, i attended Emanouil’s special course of mathematics for school students that he led in Murmansk in 1995-1997. That course gave a significant contribution to my results in mathematical competitions (olympiads). It was the first lecture course where i was acquainted with basic notions of logic, set theory and mathematical analysis, the Dedekind cuts, the CantorBernstein-Schroeder theorem, the difference between countable and continuum sets and many other concepts.

In addition to the words above, this textbook pays attention to the fact that there are many unsolved problems in mathematics having a simple formulation that can be understood by school students. Some examples of these problems are provided in the book. Also, Emanouil’s book draws attention to some paradoxical situations in mathematics. For example, it is known that almost all real numbers are transcendent, but it is very difficult to prove the transcendence for a specific number. The book mentions interesting things such as undecidable statements, the Godel incompleteness theorem, the continuum hypothesis, raises philosophical questions. This book is recommended for everyone who wants to understand mathematics and its beauty.

Sergey Volkov, senior scientist

Skobeltsyn Institute of Nuclear Physics (Moscow, Russia) http://sinp.msu.ru/en
Joint Institute of Nuclear Research (Dubna, Russia) http://www.jinr.ru/main-en/
Address:
Russia, 141800, Moscow region, Dmitrov city, Oboronnaya 10-122
Phone:
+79166693672
E-mail: volkoff sergey@mail.ru, sergey.volkov.1811@gmail.com