Algebraic operations with real numbers.
We now proceed to meaning of the elementary algebraic operations such as addition, as applied to real numbers in general.
(i), Addition. In order to define the sum of two numbers and , we consider the following two classes: (i) the class (c) formed by all sums , (ii) the class (C) formed by all sums . Clearly, in all cases.
Again, there cannot be more than one rational number which does not belong either to (c) or to (C). For suppose there were two, say r and s, and let s be the greater. Then, both r and s must be greater than every c and less than every C; and so cannot be less than . But,
and we can choose a, b, A, B so that both and are as small as we like; and this plainly contradicts our hypothesis.
If every rational number belongs to (c) or to (C), the classes (c), (C) form a section of the rational numbers, that is to say, a number . If there is one which does not, we add it to (C). We have now a section or real number , which must clearly be rational, since it corresponds to the least member of (C). In any case we call the sum of and , and write
If both and are rational, they are the least members of the upper classes (A) and (B). In this case it is clear that is the least member of (C), so that our definition agrees with our previous ideas of addition.
We define by the equation .
The idea of subtraction accordingly presents no fresh difficulties.