CAN MINUS TIMES MINUS BE MINUS

As we have seen, the general application of ordinary algebraic rules entails various sorts of mathematical elements, such as imaginary numbers and quaternions. Accordingly, it also entails variations on the effects and validity of preestablished arithmetical operations and notions.

As suggested earlier, we might be able to dispose of all such complications if we could somehow change the rules that lead to the "existence" of imaginary expressions. And there is but one rule that leads to the introduction of imaginaries, namely, the extraction of roots of negative numbers. So let's just change that rule, to see what happens. We will then get some basic impressions of what an algebra based on unusual numerical rules would be like. Likewise, recall that for centuries mathematicians and teachers complained about double or multiple solutions of equations. Accordingly, let us experiment with rules designed to entail only a single solution for every equation. For the sake of the arguument, we will thus try to construct an artificial rudimentary arthmetic and a corresponding new algebra.

We should specify what we mean by the words "arithmetic" and "algebra." Owing to the old ambiguities relating to negative numbers and more, mathematicians traditionally restrict arithmetic as pertaining only to certain sorts of numbers, usually positive whole numbers and zero, and only to certain operations, such as addition, subtraction, multiplication, and division. As for algebra, mathematicians have defined it in very many distinct and discordant ways. To avoid such ambiguities and arbitrary restrictions, we may adopt more convenient and simple definitions. "Arithmetic" here means the study of the relations among numbers and operations where at least some numbers are not specified directly but are represented by letters.

The task of devising principles that are best suited for describing things of experience is postponed for the following chapter. For now, we will modify the traditional rules of signed numbers, that is, we will invent artificial rules, chiefly as a basic exercise demonstrating the flexibility or plasticity of the elements of mathematics. Hence we will, for now, view algebra essentially as a system of symbols. Nevertheless, we do not want these symbols to be completely devoid of meaning. Therefore, even though we will develop fanciful algebraic rules, we will aim to make those rules consonant with elementary arithmetical and geometrical notions.

To begin, instead of traditional rules such as

let us stipulate that now

Accordingly, let us change the traditional rule that

with its ambiguity about the double sign, that is, of the two possible solutions, to establish instead the simple rule that

If we can do this, without introducing inconsistencies and contradictions into our imaginary algebra, we will no longer need to conjure imaginary numbers for the solution of some problems. Thus we might dispense with complications embodied by theory of complex numbers. By formulating fictitious rules, and tracing some of their consequences, we will find whether they are indeed plausible, at least to some degree. Afterward, we will seek algebraic generalizations.

In accord with our makeshift rules on the extraction of roots, we can also stipulate the following rules:

and

where the parentheses have been included merely to indicate clearly that the operation of squaring is to be performed on these signed numbers. These operations are simply the reversse of the corresponding square root extrations. Note that here we need not assume that the exponent 2 is positive or negative. It is merely a symbol to indicate the operation of taking the second power of a number. Thus its effect is now symmetric on positive and negative numbers. To be sure, we might as well introduce positive and negative exponents also.

--negative math: HOW MATHEMATICAL RULES CAN BE POSITIVELY BENT, alberto a. martinez, Page 131-133