286 LIMITS AND FLUXIONS 



both sides of ^^""" = £* '"' are raised to the power i / 2, 

 vve obtain ^-""^ = ^-«\ Here ali the letters stand for 

 real numbers ; since ;;/ and n are not equal to each 

 other, this last equation is an absurdity. The 

 assumption that a rule of operation vab'd for real 

 exponents was valid also for imaginary exponents, 

 has led to papable error. 



Examples of this sort emphasise the need of 

 caution when operations, known to be valid for a 

 certain class of numbers, are applied to numbers 

 belonging to a larger class. Special examination 

 is necessary. These remarks are pertinent when 

 operations applicable to rational numbers are ex- 

 tended to a class which embraces both rational and 

 irrational numbers. What are the numbers called 

 irrational? It is hardly sufificient to say that an 

 irrational number is one which cannot be expressed 

 as the ratio of two rational numbers. A negative 

 definition of this sort does not even establish 

 the existence of irrational numbers. Considerale 

 attention has been paid to the definition of irrational 

 numbers as limits of certain sequences of rational 

 numbers. Thus, ^2 may be looked upon as the 

 limit of the sequence of rational fractions obtained 

 by the ordinary process of root-extraction, namely, 

 the sequence, i, 1*4, 1*41, I'4I4, I"4I42, . . . 

 This attempt to establish a logicai foundation 

 for irrational numbers was not successful. We 

 endeavour, in what foUows, to make this matter 

 plainer. 



244. Let US agree that in building up an aiith- 



