ABSTRACT
This chapter is an essential prerequisite for Section II of the book. Algebra deals with the manipulation of symbols. In high school, algebra is based on the properties of the real numbers. This means that when you write x you mean a real number. The properties you use in solving equations involving x are therefore also the properties of real numbers. The problem is that if those properties are not spelt out, then doing algebra becomes instinctive: you simply apply a method without thinking. That is fine if the only algebra you meet is the algebra of the real numbers, but will cause you serious problems as soon as you have to do algebra requiring the application of different properties. For example, in Chapter 8 matrices are introduced where (A+B)2 6= A2 + 2AB+B2, and in Chapter 9 vectors are introduced where a× (b× c) 6= (a×b)× c. The first job, therefore, is to spell out in detail what those properties might be and to make explicit the implicit rules that you know. This is done in Section 4.1. The terminology introduced is essential in appreciating the rest of the book, and although there is a lot to take in and it is abstract, the more examples of algebraic systems you meet the more the terminology will reveal its worth. In Section 4.2, we use this terminology to describe precisely the algebraic properties of the real numbers, the very properties which underlie high-school algebra. Sections 4.3 and 4.4, on the solution of quadratic equations and the binomial theorem respectively, are likely to be familiar, but the material is presented in a possibly unfamiliar way. With that, the main business of this chapter is concluded. There are then two optional sections included for background but not needed in Section II. In Section 4.5, Boolean algebras are introduced. They have properties that contrast radically with those of the real numbers. For example, a+a= a always holds as does the unnerving a+(b · c) = (a+b) · (a+ c). Historically, Boolean algebras were another milestone in the development of algebra in the nineteenth century. They are also an important tool in circuit design and can be regarded as an algebraic premonition of the computer age. In Section 4.6, we return to the real numbers but this time it is their order-theoretic properties that come under the spotlight. This enables us to connect the material in this book with that in any standard introduction to analysis.
