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 \(\sqrt{3}\), \(-\sqrt{3}\), \(\sqrt[3]{7}, \dots\).
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