Add-on to hit higher planes

This book is for anyone who enjoyed mathematics at a high level at school and intends to embark on further study. It is also for those who have studied the subject at an advanced level and now have time to pursue areas of interest.

Many people mastered trigonometry in solving right-angled triangles, using the definitions of sin( ? ) and cos( ? ) in terms of ratios. Later, several will have encountered Euler’s formula: e i? = cos( ? ) + i sin( ? ), but how many established the connection? This book helps in this task, and it is written in a way that is suitable not only for the student looking forwards and the graduate in reflective mode, but also for schoolteachers seeking material on which to base their accounts of mathematics.
The opening chapter on numbers goes quickly through the concepts from the integers to complex numbers and quaternions. Cryptography has become an important tool in e-commerce, and after a discussion of the primes and modular arithmetic, the RSA coding procedure is described.

The author opens the book by saying it is a well-known fact that university mathematics is hard - and, for many, analysis is the embodiment of this sentiment. This is the subject of the second chapter and, as well as pointing towards a unified view of trigonometry, it provides good motivation for students new to the subject. The chapter on algebra covers a large amount of material from linear equations to vector spaces to matrices, including eigenvalues/vectors. A practical dimension is given by a discussion of the simplex algorithm.

A chapter devoted to calculus and differential equations provides an overview of applications to classical applied mathematics. Probability is also covered well, with care taken to explain some rather counter-intuitive results.

The last chapter is on theoretical physics. As the author states, “part of the battle in becoming a theoretical physicist is to force oneself to really believe in the physical predictions of the underlying mathematics”. Stephen Hewson’s explanation of special relativity is good, and the argument leading to the Schršdinger equation is certainly persuasive.

The book concludes with a substantial body of exercises that aims to reinforce and extend the material.

Nigel Steele is emeritus professor of mathematics, Coventry University.