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Elements of Mathematics

Gabor Toth
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Andrzej Sokolowski
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The book Elements of Mathematics by Gabor Toth is a part of the series of undergraduate texts in mathematics published by Springer and designed for students studying mathematics at North American universities. The book aims at presenting rigorous theories before calculus is introduced. Transparency of explanation and gradually built material are outstanding features of the textbook. In addition, solutions to some problems are designed using more than one approach, making it adaptable to various students' backgrounds.
The book consists of eleven chapters. The first five chapters present theories related, for example, to real analytic plane geometry or limits before advancing the analyses to various types of functions such as polynomial, rational, and transcendental. Trigonometry, along with Newton's cosine and sine series, is also included in that book.
Due to carefully arranged difficulty level, the prerequisite to understanding the book material does not pass high school algebra. The book makes itself accessible to a vast population of students. The book can enhance the undergraduate curriculum or serve as a reflective resource for graduate mathematics students. Embedded historical perspectives that correlate the mathematical theories help broaden the views on the language of mathematics that changes to make itself more understandable. 
While the theme of the textbook is to discuss mathematical theories, graphs and diagrams are very rare in the book, even when functions are concerned. Visualizing, for example, the concept of limits, function asymptotes, or general function behavior would make the book even more appealing to the readers. 
Andrzej Sokolowski, Ph.D., authored several books and research papers on developing students' mathematical reasoning and integrating this reasoning to improve understanding of science. He is also a mathematics and physics adjunct professor at Lone Star College, TX, USA.