Basic Mathematics - Serge Lang

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In this part we develop systematically the rules for operations with numbers, relations among numbers, and properties of these operations and relations: addition, multiplication, inequalities, positivity, square roots, n-th roots. We find many of them, like commutativity and associativity, whichrecur frequently in mathematics and apply to other objects. They apply to complex numbers, but also to functions or mappings (in this case, commutativity does not hold in general and it is always an interesting problem to determine when it does hold).

Even when we study geometry afterwards, the rules of algebra are still used, say to compute areas, lengths, etc., which associate numbers with geometric objects. Thus does algebra mix with geometry. The main point of this chapter is to condition you to have efficient reflexes in handling addition, multiplication, and division of numbers. There are many rules for these operations, and the extent to which we choose to assume some, and prove others from the assumed ones, is determined by several factors.

We wish to assume those rules which are most basic, and assume enough of them so that the proofs of the others are simple. It also turns out that those which we do assume occur in many contexts in mathematics, so that whenever we meet a situation where they arise, then we already have the training to apply them and use them. Both historical experience and personal experience have gone into the selection of these rules and the order of the list in which they are given. To some extent, you must trust that it is valuable to have fast reflexes when dealing with associativity, commutativity, distributivity, cross-multiplication, and the like, if you do not have the intuition yourself which makes such trust unnecessary. Furthermore, the long list of the rules governing the above operations should be takenin the spirit of a description of how numbers behave.

It may be that you are already reasonably familiar with the operations between numbers. In that case, omit the first chapter entirely, and go right ahead to Chapter 2, or start with the geometry or with the study of coordinates in Chapter 7. The whole first part on algebra is much more dry than the rest of the book, and it is good to motivate this algebra through geometry. On the other hand, your brain should also have quick reflexes when faced with a simple problem involving two linear equations or a quadratic equation.

Hence it is a good idea to have isolated these topics in special sections in the book for easy reference. In organizing the properties of numbers, I have found it best to look successively at the integers, rational numbers, and real numbers, at the cost of slight repetitions. There are several reasons for this. First, it is a good way of learning certain rules and their consequences in a special context (e.g. associativity and commutativity in the context of integers), and then observing that they hold in more general contexts. This sort of thing happens very frequently in mathematics. Second, the rational numbers provide a wide class of numbers which are used in computations, and the manipulation of fractions thus deserves special emphasis. Third, to follow the sequence integers rational numbers-real numbers already plants in your mind a pattern which you will encounter again in mathematics. 

This pattern is related to the extension of one system of objects to a larger system, in which more equations can be solved than in the smaller system. For instance, the equation 2x = 3 can be solved in the rational numbers, but not in the integers. The equations x2 = 2 or 10x = 2 can be solved in the real numbers but not in the rational numbers. Similarly, the equations x2 = — 1, or x2 = —2, or 10x = —3 can be solved in the complex numbers but not in the real numbers. It will be useful to you to have met the idea of extending mathematical systems at this very basic stage because it exhibits features in common with those in more advanced contexts.