Authors: Gabriel Katz and Vladimir Nodelman
Publisher: World Scientific
Understanding that some students - particularly those from other disciplines - find it difficult to engage with the abstract concepts of modern mathematics, I was delighted to review this innovative book, which claims to present this material to those with "only a traditional exposure to high school algebra and some elements of calculus". However, I would say that readers will need to have a strong mathematical understanding, even if they are not accustomed to the rigour of a pure mathematics degree.
The book begins by demonstrating how rectangles can be graphically represented by a single point, and then examines the relationship between length and height, perimeter and area before moving on to quadratic equations and the relationship between the coefficients and roots. The authors provide an unusual graphical approach to this area of mathematics, weaving together aspects of abstract algebra, topology, complex analysis and number theory while keeping polynomial equations as the central theme.
An accompanying CD containing VisuMatica, a new graphical software package, is referred to throughout, and enables the reader to visually investigate the concepts discussed. Although VisuMatica is relatively straightforward - both for the reader's own use and to demonstrate to a class - students without a copy of the book will not have access to it. Nevertheless, many of the examples could be reproduced using similar readily available software. There is much here that will permit mathematics lecturers to enhance their traditional material, both to support students preferring a visual demonstration and also to stretch more confident learners.
Throughout the book there are exercises to complete and theorems to prove. No answers are included, which I fear will make the book less attractive to students, but the attentive reader soon realises that these are often provided in discussion further on. There are also "thought exercises": like the exercises that require pen and paper calculations or the use of software, they are designed to deepen the reader's understanding but they also foster good questioning habits designed to strengthen mathematical thinking.
Nowadays many maths lecturers have to prepare outreach material for taster days and masterclasses for school pupils of various ages and are often in need of new ideas. This book has plenty of material that could be adapted for such audiences with a minimum of extra work.
As treasurer of the British Society for the History of Mathematics, I particularly liked the sections of historical information that pepper the book. These passages can be easily skipped over, but in my opinion they add depth and context to the material rather than detracting from it, and provide the reader with a greater understanding of how mathematics has been developed over time.
Who is it for? University lecturers needing supplementary material, and a self-study volume for students wishing to visualise algebraic ideas.
Presentation: Written as a conversation discussing mathematical concepts, with copious essential diagrams.
Would you recommend it? It is an exciting addition to any mathematician's library.
Limits, Limits Everywhere: The Tools of Mathematical Analysis
Author: David Applebaum
Publisher: Oxford University Press