Elementary algebra

About this book

The purpose of this text in Elementary Algebra is not only to help the reader develop the traditional skills associated with this subject, but also to imbue him with a correct understanding and intuition of those ideas and practices that make mathematics meaningful. There is continuous emphasis on the fact that we are dealing with numbers rather than ink marks or, as they are sometimes called, "expressions." For example, instead of being asked to "factor x2 — 4x + 3," the reader is directed to "find all real numbers x for which x2 — 4x + 3 = 0." Thus, various skills, including "factoring," are developed through activities in which explicit use is made of the properties of numbers. This approach makes it unnecessary to use such nearly undefinable (and, on the level of elementary algebra, conceptually empty) terms as, for example, "variable." in line with this approach, the term "equation" always refers to a relationship between the coordinates of points on a graph. Throughout, numerical and geometric intuitions interlace and bolster each other: real numbers are to points on the number line what functions and equations are to their graphs. In each case, when one is studied, so |l the other; and the reader is led, through exposition and examples, to "ice" both whenever he considers either one. The modern spirit of this text is to be found not in adherence to passing fads, such as cumbersome "modern" notation or undue emphasis on "axiom-atics" or a lengthy discussion of set-theoretic subtleties, but rather in the consistent correctness of the mathematics involved combined with a sympathetic recognition of the readers' inexperience. Sets, of course, are encountered at every turn, and are referred to as such (for, after all, how else can one speak of the domain of a function, a graph, or the real line itself?). However, not until Chapter 11 (which is entirely devoted to sets) is any fuss made about them. Throughout, "concepts" and definitions are saved for when they are needed.