Let us now return for a moment to the particular irrational number which we discussed in §§ 4-5. We there constructed a section by means of the inequalities $$x^{2} < 2$$, $$x^{2} > 2$$. This was a section of the positive rational numbers only; but we replace it (as was explained in § 8) by a section of all the rational numbers. We denote the section or number thus defined by the symbol $$\sqrt{2}$$.

The classes by means of which the product of $$\sqrt{2}$$ by itself is defined are (i) $$(aa’)$$, where $$a$$ and $$a’$$ are positive rational numbers whose squares are less than $$2$$, (ii) $$(AA’)$$, where $$A$$ and $$A’$$ are positive rational numbers whose squares are greater than $$2$$. These classes exhaust all positive rational numbers save one, which can only be $$2$$ itself. Thus $(\sqrt{2})^{2} = \sqrt{2}\sqrt{2} = 2.$

Again $(-\sqrt{2})^{2} = (-\sqrt{2})(-\sqrt{2}) = \sqrt{2}\sqrt{2} = (\sqrt{2})^{2} = 2.$ Thus the equation $$x^{2} = 2$$ has the two roots $$\sqrt{2}$$ and $$-\sqrt{2}$$. Similarly we could discuss the equations $$x^{2} = 3$$, $$x^{3} = 7, \dots$$ and the corresponding irrational numbers $$\sqrt{3}$$$$-\sqrt{3}$$, $$\sqrt{7}, \dots$$.