A Bridge to Advanced Mathematics

This book is intended as a reference manual for an introductory course in mathematical proofs. It aims to ease the transition from primarily calculus-based mathematics courses to more conceptually advanced proof-based courses. As such, it is intended for early undergraduate students who wish to become familiar with the language, fundamental knowledge and methods of abstract mathematics. Although this is a niche market with a lot of competition, the authors – Sebastian M. Cioaba and Werner Linde – have nevertheless come up with a relevant and unique proposal.

 

The originality of their book lies not in its content but in its structure. While most introductory proof manuals are organized around certain themes, (with, for example, a chapter on elementary logic, followed by another on naive set theory, then one on the rudiments of number theory …), the book A Bridge to Advanced Mathematics: From Natural to Complex Numbers, is based – as its subtitle indicates – around the concept of a number. Each of the six chapters deals with a number system and approaches it successively from different angles.

 

Proceeding from the most familiar to the least natural to the mind and moving in small steps from the concrete to the abstract, the authors deal in turn with natural, integer, rational, and real numbers. Before propelling their readers into a study of the field of complex numbers, Cioaba and Linde make a long digression to look at sequences of numbers and present some classical theorems in real analysis.

 

The chapters, all about 75 pages each, start with a section introducing, from a historical perspective, the number system to be discussed. All the content pertaining to logic (truth tables, types of proofs, etc.) and elementary set theory (set operators, functions, relations, cardinality, etc.) which hardly finds its place in the organization of the material adopted by the authors, is relegated to an appendix as long as the chapters themselves. This appendix also contains a brief discussion of Peano's axioms, the construction of integers and Dedekind's cuts.

 

In the short preface that precedes the main text, the authors outline how an instructor might use this book as a reference manual for a one-semester course totalling about 40 hours of instruction.  The scope of the book is broad enough to allow instructors some flexibility in the content that will be covered. One can choose to skip any of the subsections dealing with counting and binomial coefficients, graph theory, continued fractions or modular arithmetic without fear of running out of material to teach.

 

Around 150 tables and figures, a countless number of concise or detailed examples and more than 700 exercises are disseminated throughout this 525-page book. The ratio between routine exercises aimed at consolidating targeted knowledge and problems aimed at integrating and mobilizing a broader range of knowledge is reasonable.

 

At the very end of the book, there is an index of terms and notation elements as well as a bibliography of about thirty titles, including classic reference books, a few scholarly articles, and also a few books offering a historical or didactic perspective.

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Frederic Morneau-Guérin is a professor in the Department of Education at Université TELUQ. He holds a Ph.D. in abstract harmonic analysis.