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In classical text books on the study of Algebra, as well as in all standard dictionaries, the subject is usually described as a generalization and extension of Arithmetic. For the student whose introduction to arithmetic began with a consideration of operations on physical things, such as apples and oranges, this all too brief attempt at a definition oftentimes proves to be an impediment to grasping the true nature of algebra.More accurately, Algebra is a way of reasoning about the nature of metaphors and their underlying lattice structures. This is to be taken in much the same spirit as the definition of Geometry as a way of reasoning about space.
It is imperative to make this leap from the profane to the abstract world of symbols as early on as possible in order to facilitate an easy entry into this further dimension. Moreover, the very concept of Number itself can be, for all intents and purposes, apprehended on this purely symbolic level, relieved of the weight of substantiality. With this cutting of the tether holding it to the ground the inherent self-consistency of algebra makes its presence known, even though algebra's utility as a practical analytic tool is beyond question.
Basically what we need in order to have an algebra is a set of symbols closed under at least one operation. These symbols are usually numbers although this restriction is not necessary as long as each element of the set in question maintains a formally defined internal relationship (that is, the set is well-ordered). And by closed we mean simply that our operation does not produce any symbol-forms that cannot be found within the set.
What we will be concerned with here, however, is an algebraic structure not quite so primitive. We will want to be able to solve equations and manipulate functions for which we need the structure most high school and first-year college students come to know and love: the field of real numbers [with extension to the two-dimensional complex]. Without bothering to go into its formal axiomatic definition, which can be found in a variety of sources, we will plunge right into its most brutal and cryptic blood and guts.
Threaded through the following journey I will expositorily point out to the reader, from time to time, the particular significance of material that will come into play later in development, and also those peculiar ideas that I have found in my years of teaching to produce either consternation or genuine impasses. It seems that most students, for whatever reasons, trip over the same ideas, ideas that, if not grasped at the time of their introduction, tend to create a lasting confusion and sense of anxiety. This in turn may and often does produce the danger of a lowering of self-esteem, an increase in one's mistaken belief as to the difficulty of the material, and/or a completely false and negative assessment of the individual's ability to 'do math.' We will attempt to eradicate these potentialities.
The rudiments of a mathematical sense are: number, size, order, and form. Mathematicians are concerned with form and relationship, and differences in patterns. As mathematics has developed through the eons it has realized greater generalization, broader definitions relegating once all-encompassing spaces and structures to the status of special cases. Algebraic curriculums, by and large, tend to evolve from single ideas to the more general cases inductively, introducing definitions and theorems as they come up, sequentially. What we are going to attempt here, for the most part, is the opposite approach. It seems to me that once the high-ground has been taken, details of the terrain can more readily be seen and understood.