12. The number $\sqrt{2}$

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) $(a{a}^{\prime })$, where $a$ and $a^{\prime }$ are positive rational numbers whose squares are less than $2$, (ii) $(A{A}^{\prime })$, where $A$ and $A^{\prime }$ 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 $\sqrt{3}$, $-\sqrt{3}$, $\sqrt[3]{7},\dots$.