This book is based on a first year course for students of Economics and Finance at University College of Dublin, and it was part of a programme of Business Studies that required ‘sophisticated mathematics’ during each year of undergraduate study.

Because mathematics courses for engineers, accountants, economists etc, are all too often of the handle-turning, ‘ask-no-questions’ variety, it’s very refreshing to discover that Sean Dineen’s philosophy is the complete antithesis of such educational short-sightedness. In fact, Dineen’s approach is a true reflection of the attitude to mathematics shown by Laplace in his open lecture of January 20, 1795. For this book, it means that:

1. The vast majority of topics are set in an historical context.

2. Mathematics is seen as a way of knowing — not just as a body of knowledge.

3. Readers are led to explore the inter-connectedness of mathematical ideas.

4. The book will prepare students for ongoing mathematical study.

Basically, this book is suited to a course in one variable calculus and real analysis; but it would be wasteful to confine its use to students of business studies. For example, it is ideally suited for use on courses for future high school teachers, and it would lay good foundations for first year maths majors.

The first chapter centres upon an exploration of quadratic equations from the perspective of the formula that provides their roots. It examines the conditions under which this formula is applicable, and eventually considers analogous results for cubics. Because the approach is simultaneously heuristic and strongly historical, students will have begun to develop an expanded vision as to how, and why, mathematics is created.

This philosophy pervades the second chapter, which examines the role of diagrams and graphs in the mathematical thinking. Subsequent chapters then introduce the concepts of sets, functions and relations that are essential for the development of analytical ideas. Again, the presentation exemplifies a good combination of both horizontal and vertical thematic sequencing of ideas.

Real analysis begins (and continues) by avoiding the Weirerstass \(\varepsilon\)-\(\delta\) definitions. Convergence of real sequences is expressed with respect to lower and upper bounds, whilst continuity of a real valued function is defined in terms of its action upon a convergent sequence. Much of the material on differentiation and integration is standard, but the treatment is very much in the spirit of the aforementioned principles.

There’s no way round the fact that real analysis (\(\varepsilon\)-\(\delta\) or not) remains a significant hurdle for incipient mathematicians — and this book is no easy option. Moreover, there are three challenging chapters on constructional approaches to number systems that (although they may be omitted on a first reading) reinforce the theoretical underpinnings of real analysis. For example, the chapter that defines real numbers in terms of convergent sequences of dyadic numbers would challenge any honours mathematics student.

One other attractive feature of this book is the nature of the exercise sets. The problems are truly instructional and a sufficient number of them are supplied with solutions. Overall, this book is highly recommended as a refreshingly different introduction to undergraduate mathematics.

Peter Ruane’s career was centred upon primary and secondary mathematics education.