Books

Hardy: A Mathematician’s Apology

You rarely get an insight into the workings of a mathematician mind. However in, A Mathematician’s Apology G.H Hardy gives an excellent account of what mathematics is. Its usefulness (or lack of) and how it is more than just a science it is a creative art!

The apology was written as his powers as a mathematician were on the decline. This detail offers the reader an insight into how Hardy viewed his career as a mathematician and how he viewed some of his contemporaries. He mentions his work with Littlewood, his discovery of Ramanujam and his time at Oxford and Cambridge.

Hardy gives a brilliantly composed account of ‘real’ mathematics and this comes from his background in pure mathematics. He was known as ‘a real mathematician’ and this style of thinking comes across in the well-constructed, eloquent argument he proposes for his subject. His dislike of war and the lack of involvement that mathematics has had upon it is displayed in the book. He acknowledges the use of applied mathematics on ballistics however he hardly ranks them as ‘real’. He further states that the mathematics involved are “indeed repulsively ugly and intolerably dull.” You can’t help but like the language he uses.

This book is a must for anyone (not just mathematicians) as it is not only fantastically written but it offers an insight into one of the finest mathematical minds of the twentieth century.          

Elementary Number Theory (David M. Burton)

This text gives a good introduction to number theory and also some history behind the subject showing how it has evolved over time. The main use for the text book is exactly that. A text book by which a student (undergraduate) can use as reference during a course in number theory. If only I had discovered this book before I started a Cryptography course. It would have made the initial start easier as the book gives a nice introduction to the topic (Chapter 10).  Another use which this book has been designed for is for teachers. Not really as a teaching resource (unless you really want to stretch your students). More of a way for a teacher to gain better insight of the subject and so develop the level of their explanations in class.

Initially the book (chapter 1 – 9) lays the foundations of number theory ranging from divisibility, primes, congruence, Fermat’s theorem, functions, Euler’s theorem, Primitive roots and The Quadratic Reciprocity Law (note: the topics named before are not all chapters some are sections within a larger chapter). Each chapter naturally follows from the previous and within each chapter the “style” in which the mathematics is presented is pleasing. Each chapter usually starts with a fundamental theorem or “driving example” and these are expanded upon in an intuitive way through an appropriate lemma and corollary.

The remaining chapters of the book (chapters 10 -16) are more isolated chapters in which the reader can explore at a gentler pace. These chapters allow the reader to investigate modern development s within number theory (e.g. the Zeta Function), cryptography, Fermat’s Last Theorem, Fibonacci Numbers and Continued Fractions.

Throughout the text the author drops in historical background of the subject. The reader can choose whether or not to read these sections. I would recommend (if you have the time) just to have a quick read as these sections seem to bring the subject to life and giving it a purpose.

Overall this text is a good introduction to the area of number theory for someone who is wishing to venture into the area for the first time.