It is plain that, now that we have extended our conception of number, we are bound to make corresponding extensions of our conceptions of equality, inequality, addition, multiplication, and so on. We have to show that these ideas can be applied to the new numbers, and that, when this extension of them is made, all the ordinary laws of algebra retain their validity, so that we can operate with real numbers in general in exactly the same way as with the rational numbers of § 1. To do all this systematically would occupy a considerable space, and we shall be content to indicate summarily how a more systematic discussion would proceed.

We denote a real number by a Greek letter such as $$\alpha$$, $$\beta$$, $$\gamma, \dots$$; the rational numbers of its lower and upper classes by the corresponding English letters $$a$$$$A$$; $$b$$$$B$$; $$c$$$$C$$; …. The classes themselves we denote by $$(a)$$$$(A), \dots$$.

If $$\alpha$$ and $$\beta$$ are two real numbers, there are three possibilities:

$$(i)$$ every $$a$$ is a $$b$$ and every $$A$$$$B$$; in this case $$(a)$$ is identical with $$(b)$$ and $$(A)$$ with $$(B)$$;

$$(ii)$$ every $$a$$ is a $$b$$, but not all $$A$$’s are $$B$$’s; in this case $$(a)$$ is a proper part of $$(b)$$,1 and $$(B)$$ a proper part of $$(A)$$;

$$(iii)$$ every $$A$$ is a $$B$$, but not all $$a$$’s are $$b$$’s.

These three cases may be indicated graphically as in Fig. 4.

In case (i) we write $$\alpha = \beta$$, in case (ii) $$\alpha < \beta$$, and in case (iii) $$\alpha > \beta$$. It is clear that, when $$\alpha$$ and $$\beta$$ are both rational, these definitions agree with the ideas of equality and inequality between rational numbers which we began by taking for granted; and that any positive number is greater than any negative number.

It will be convenient to define at this stage the negative $$-\alpha$$ of a positive number $$\alpha$$. If $$(a)$$$$(A)$$ are the classes which constitute $$\alpha$$, we can define another section of the rational numbers by putting all numbers $$-A$$ in the lower class and all numbers $$-a$$ in the upper. The real number thus defined, which is clearly negative, we denote by $$-\alpha$$. Similarly we can define $$-\alpha$$ when $$\alpha$$ is negative or zero; if $$\alpha$$ is negative, $$-\alpha$$ is positive. It is plain also that $$-(-\alpha) = \alpha$$. Of the two numbers $$\alpha$$ and $$-\alpha$$ one is always positive (unless $$\alpha = 0$$). The one which is positive we denote by $$|\alpha|$$ and call the modulus of $$\alpha$$.

Example IV

1. Prove that $$0 = -0$$.

2. Prove that $$\beta = \alpha$$, $$\beta < \alpha$$, or $$\beta > \alpha$$ according as $$\alpha = \beta$$, $$\alpha > \beta$$, or $$\alpha < \beta$$.

3. If $$\alpha = \beta$$ and $$\beta = \gamma$$, then $$\alpha = \gamma$$.

4. If $$\alpha \leq \beta$$, $$\beta < \gamma$$, or $$\alpha < \beta$$, $$\beta \leq \gamma$$, then $$\alpha < \gamma$$.

5. Prove that $$-\beta = -\alpha$$, $$-\beta < -\alpha$$, or $$-\beta > -\alpha$$, according as $$\alpha = \beta$$, $$\alpha < \beta$$, or $$\alpha > \beta$$.

6. Prove that $$\alpha > 0$$ if $$\alpha$$ is positive, and $$\alpha < 0$$ if $$\alpha$$ is negative.

7. Prove that $$\alpha \leq |\alpha|$$.

8. Prove that $$1 < \sqrt{2} < \sqrt{3} < 2$$.

9. Prove that, if $$\alpha$$ and $$\beta$$ are two different real numbers, we can always find an infinity of rational numbers lying between $$\alpha$$ and $$\beta$$.

[All these results are immediate consequences of our definitions.]

1.  I.e. is included in but not identical with $$(b)$$.↩︎