Analysis: Chapter 1: part 11: algebraic operations with real numbers: continued

(iii) Multiplication.

When we come to multiplication, it is most convenient to confine ourselves to positive numbers (among which we may include zero) in the first instance, and to go back for a moment to the sections of positive rational numbers only which we considered in articles 4-7. We may then follow practically the same road as in the case of addition, taking (c) to be (ab) and (O) to be (AB). The argument is the same, except when we are proving that all rational numbers with at most one exception must belong to (c) or (C). This depends, as in the case of addition, on showing that we can choose a, A, b, and B so that C-c is as small as we please. Here we use the identity

$C-c=AB-ab=(A-a)B+a(B-b)$.

Finally, we include negative numbers within the scope of our definition by agreeing that, if $\alpha$ and $\beta$ are positive, then

$(-\alpha)\beta=-\alpha\beta$, $\alpha(-\beta)=-\alpha\beta$, $(-\alpha)(-\beta)=\alpha\beta$.

(iv) Division.

In order to define division, we begin by defining the reciprocal $\frac{1}{\alpha}$ of a number $\alpha$ (other than zero). Confining ourselves in the first instance to positive numbers and sections of positive rational numbers, we define the reciprocal of a positive number $\alpha$ by means of the lower class $(1/A)$ and the upper class $(1/a)$. We then define the reciprocal of a negative number $-\alpha$ by the equation $1/(-\alpha)=-(1/\alpha)$. Finally, we define $\frac{\alpha}{\beta}$ by the equation

$\frac{\alpha}{\beta}=\alpha \times (1/\beta)$.

We are then in a position to apply to all real numbers, rational or  irrational the whole of the ideas and methods of elementary algebra. Naturally, we do not propose to carry out this task in detail. It will be more profitable and more interesting to turn our attention to some special, but particularly important, classes of irrational numbers.

More later,

Nalin Pithwa