1: Mental Skills
Even a superficial glance at history shows ... great innovators ...did vast amounts of computation and gained much of their insight in this way. I deplore the fact that contemporary mathematical education tends to give students the idea that computation is demeaning drudgery to be avoided at all costs.
Harold M. Edwards (1936–)
Fermat’s Last Theorem
We start our journey in a way that should be accessible to everyone – with a quick romp through important ideas from secondary school mathematics. The content is at times very elementary; but the problems often hint at something more challenging. The items included in this first chapter also highlight selected facts, techniques and ideas. Some of this early material is included to introduce certain ideas and techniques that later chapters will assume to be “known”. A few problems appeal to more advanced ideas (such as complex numbers ), and are included here to indicate that “mental skills” are not restricted to elementary material.
Pencil and paper will be needed, but the items tend to focus on things which a student of mathematics should know by heart, or should learn to see at a glance, or should be able to calculate inside the head. In later problems (e.g. from Problem 18 onwards) the emphasis on mental skills should be interpreted as “ways of thinking”, rather than being taken to mean that everything should be done in your head. This is especially true where extended calculations or proofs are required.
Some of the items in this chapter (such as Problems 1 and 2) should be thoroughly familiar, and are included to underline this fact, rather than because we anticipate that they will need much active attention. Most of the early items in this first chapter are either core or auxiliary . However, there are also some real gems , which may even warrant a place in the the standard core .
The chapter is largely devoted to underlining the need for mastery of a repertoire of instantly available techniques, that can be used mentally, quickly, and flexibly to analyse less familiar problems at sight. But it also seeks to emphasise connections . Hence readers should be prepared to challenge their previous experience, in case it may have led to methods and results being perceived too narrowly.
We repeat the comment made in the section About this book . The “essence of mathematics”, which is referred to in the title, is largely implicit in the problems, and is there for the reader to extract. There is some discussion of this essence in the text interspersed between the problems. But, to avoid spilling the beans prematurely, and hence spoiling the problems, many important observations are buried away in the solutions, or in the Noteswhich follow many of the solutions.