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Essentials of mathematical thinking. (English) Zbl 1419.00007

Textbooks in Mathematics. Boca Raton, FL: CRC Press (ISBN 978-1-138-19770-1/pbk; 978-1-138-04257-5/hbk; 978-1-138-19771-8/ebook). xvi, 335 p. (2018).
The book under review is written by a renowned mathematician for non-mathematics students who might want to fulfill a mathematics requirement in their curriculum. However the book would also be excellent material for motivated high-school students who have an interest in mathematics and might wish to look up more on the subject. The author has done a remarkable job in catering to his audience and the book is full of exciting mathematics, written in a very friendly style which no doubt the students will like.
It would be impossible to describe all the topics touched upon in the book in a short review: it covers aspects from number theory, combinatorics, logic and even a little (disguised) differential geometry. The book is divided into 13 chapters, the first and the last are really an introduction and an epilogue to the whole book. The main material is covered in the 11 chapters in between. Among these, 8 (chapters 2 through 9) are quite easy and should be possible to be ‘taught’ to the intended audience without much of a prerequisite in mathematical training. The remaining 3 (chapters 10 through 12) are a bit more advanced containing topics like Turing machines, RSA encryption, non-Euclidean geometry, PageRank algorithm of Google and the Kakeya needle problem. If indeed used as a textbook (as the title suggests), the instructor might want to wade through this part of the book with some caution.
This is no doubt an interesting and very well written book, with many pros; but it would be remiss if we do not discuss some of the cons that the reviewer found in this book. First, the pros. The sheer breadth of topics introduced in the book is truly praise-worthy. Unlike other popular science books this book is not about topics in recreational mathematics, but about real mathematics. Serious discussion is given for most of the ideas presented and the non-mathematics students will surely learn a lot of mathematics. Exercises are sprinkled throughout (most of them are non-threatening) and could be solved with little effort by a motivated student. There is a certain inter-connectedness while introducing the different topics and this will be appreciated by both students and instructors.
Now to some cons. There are several missed opportunities in the book. While the reviewer understands that it is impossible to cover all ‘interesting’ things in a single book, still some of these ‘missed opportunities’ are glaring. For instance, a good description of the AKS primarily test could have been given (considering that the Kakeya needle problem is explained quite succinctly) since the concept of computational complexity and prime number testing was already introduced. The book is full of lots of pictures, some of them could be done without and some of them could be made much better. This aspect could have been avoided considering the excellence of the writing.
Overall, the book made a good impression on the reviewer both in terms of the writing and in terms of the material covered. The treatment of some of the topics is much better than can be found in other books. It would have been perhaps more helpful to have a longer bibliography (one that also avoids referring to advanced mathematics textbooks considering the intended audience of the book).

MSC:

00A09 Popularization of mathematics
00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.)
97A80 Popularization of mathematics (MSC2010)
97E40 Language of mathematics (educational aspects)
97E50 Reasoning and proving in the mathematics classroom
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