MERITS AND DEFECTS 285 



results when a certain limited class of numbers is 

 involved, and applying them to numbers of a more 

 general class. Suppose a and b to be rational, 

 positive numbers, not zero ; we find, let us agree, 

 consistent results in the operation a-\-by a — b when 

 a>by and av^b, and a-^b. Let us now consider 

 the class composed of rational numbers, both 

 positive and negative ; suppose, moreover, that 

 we introduce o in order to give interpretation to 

 the operation a — a. If in this extended class of 

 numbers we admit the four operations a-\-by a — by 

 axb, a-^by trouble arises even after due considera- 

 tion has been given to the negative numbers. There 

 may arise the following well-known paradox. Let 

 a — b=\y then a^ — b'^ = a — b. Divide both sides of 

 the last equation by a — b^ and we have a-\-b=i, or 

 2=1. Where is the diffìculty ? The answer is 

 known to every schoolboy : We have used a — b^ or 

 o, as a divisor ; we have extended the operation of 

 division to the larger class of numbers, and to 

 zero, without first assuring ourselves that such an 

 extension is possible in every case ; division by zero 

 is inadmissible. 



243. A less familiar example is the following. 

 Let US suppose that, for real exponents, it is estab- 

 lished that ( A^)^ = A'-^. When we apply this process 

 to imaginary exponents, trouble arises. Take the 

 equation e""*'"" — e'^**'"\ where in and n are distinct 



integers, z'= 7- I, 7r=3'i4i59 • • ., and ^=2 7 18 

 . . . That this equation holds is evident, for ^''"" 



= cos 2?«7rH-/sin 2;;/7r = cos 2;/7r + /sin 2;/x = ^^''". If 



