Finally it is evident that the classes l and r both

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Finally, it is evident that the classes L and R both exist; and they form a section of the positive rational numbers or positive real number z which satisfies the equation z 3 = 3 z + 8. The reader who knows how to solve cubic equations by Cardan’s method will be able to obtain the explicit expression of z directly from the equation. (ii) The direct argument applied above to the equation x 3 = 3 x + 8 could be applied (though the application would be a little more difficult) to the equation x 5 = x + 16 , and would lead us to the conclusion that a unique positive real number exists which satisfies this equation. In this case, however, it is not possible to obtain a simple explicit expression for x composed of any combination of surds. It can in fact be proved (though the proof is difficult) that it is generally impossible to find such an expression for the root of an equation of higher degree than 4. Thus, besides irrational numbers which can be expressed as pure or mixed quadratic or other surds, or combinations of such surds, there are others which are roots of algebraical equations but cannot be so expressed. It is only in very special cases that such expressions can be found. (iii) But even when we have added to our list of irrational numbers roots of equations (such as x 5 = x +16) which cannot be explicitly expressed as surds, we have not exhausted the different kinds of irrational numbers contained in the continuum. Let us draw a circle whose diameter is equal to A 0 A 1 , i.e. to unity. It is natural to suppose * that the circumference of such a circle has a length capable of numerical measurement. This length * A proof will be found in Ch. VII .
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[I : 16] REAL VARIABLES 29 is usually denoted by π . And it has been shown * (though the proof is unfortunately long and difficult) that this number π is not the root of any algebraical equation with integral coefficients, such, for example, as π 2 = n, π 3 = n, π 5 = π + n, where n is an integer. In this way it is possible to define a number which is not rational nor yet belongs to any of the classes of irrational numbers which we have so far considered. And this number π is no isolated or ex- ceptional case. Any number of other examples can be constructed. In fact it is only special classes of irrational numbers which are roots of equations of this kind, just as it is only a still smaller class which can be expressed by means of surds. 16. The continuous real variable. The ‘real numbers’ may be re- garded from two points of view. We may think of them as an aggregate , the ‘arithmetical continuum’ defined in the preceding section, or individ- ually . And when we think of them individually, we may think either of a particular specified number (such as 1, - 1 2 , 2, or π ) or we may think of any number, an unspecified number, the number x . This last is our point of view when we make such assertions as ‘ x is a number’, ‘ x is the mea- sure of a length’, ‘ x may be rational or irrational’. The x which occurs in propositions such as these is called the continuous real variable
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