This is an expanded version of a book first published in 2000, which, to my knowledge, is unique in its declared aim of providing a ‘*robust bridge between high school and university mathematics’.* Moreover, it is one of the relatively few (contemporary) mathematics books declaring an emphasis on ‘pure’ mathematics. Several questions therefore spring to mind:

- What should be included within the guise of ‘pure mathematics’?
- Is there a gap between high school and university mathematics? If so, what is needed to bridge it? When should this bridging take place?

With respect to the first of the above questions, the author’s view can be gleaned by the following description and classification of the book’s twenty-two (shortish) chapters.

__Introduction to number systems and analysis:__

1. Sets and Proofs. 2. Number Systems. 3. Decimals 4. Inequalities 5. n^{th} roots and Rational Powers 8. Induction. 10. Introduction to Analysis.

__Theory of integers:__

11 The Integers. 12 Prime Factorisation. 14 Congruence of Integers. 15 More on Congruence.

__Introduction to discrete mathematics:__

17 Counting and Choosing. 18 More on Sets. 20 Functions. 21 Permutations.

__Functions, relations and countability:__

19 Equivalence Relations. 20 Functions. 22 Infinity.

Amongst many notable omissions are calculus, vectors, matrices, circular and hyperbolic functions and geometry (apart, that is, from some discussion of the Platonic solids).

As for the second question, the gap between high school and university mathematics is assumed to arise from what is supposed to be the handle-turning algorithmic approach that students are said to bring to their undergraduate studies, and the book aims to correct this imbalance by providing a ‘*fascinating introduction to the culture of mathematics’.* (Why, one wonders, should any mention of applied mathematics be excluded from this corrective process?)

However, experience of teaching undergraduates tells us that there isn’t a *single* definable gap in students’ mathematical knowledge, but a myriad of deficiencies — with wide variation from one student to another. Some such deficiencies may be addressed via much of the content covered in this book, but its scope is (inevitably) too narrow for it to cater for all needs. Whilst different tutors may form their own lists of gaps in student background knowledge, Liebeck has the tacit two-fold aim of ‘gap-filling’ and ‘attitude bridging’. But to what extent does he succeed?

For a start, the chapter on complex numbers covers most of the usual material in a familiar style, and it’s not too different to what may be covered at high school (or in English sixth forms), but it could serve as reasonable coverage for those who have missed out on this topic and the same could be said for the chapter on inequalities. On the other hand, chapter 10, introduces some new ideas from real analysis, yet in a no less a forbidding manner than many conventional approaches; so I’m not convinced that many of those ‘handle-turning’ high school maths students would find it inspirational. To my mind, motivation for the study of real analysis is maximised by integrating it into courses on calculus, where it can be seen to validate many of the algorithmic procedures with which students may be familiar.

For me, where this book succeeds is in its provision of insights into topics that students are likely to encounter as maths majors. Chapters 11 to 15 lay good foundations for later incursions into number theory. Then there is an all-pervading emphasis on methods of proof, which are discussed in a wide variety of contexts. Another of the book’s strengths lies in the author’s accessible style of presentation and the provision, at the end of each chapter, of interesting sets of exercises. It also commences a formalisation of many set-theoretic topics and makes some incursion into the world of discrete mathematics.

Overall, I would recommend this publication as a possible source of directed supplementary reading for first year maths majors. Also, for those undergraduates who wish to change to mathematics, and who have not done maths since high school, this book could form part of a necessary bridging course.

Peter Ruane is retired from university teaching and now dabbles in as many creative diversions as possible.