24 MONISM IN ARITHMETIC. 



first and second degree, always supposing that we follow in our 

 reckoning with imaginary numbers the same rules that we do in 

 reckoning with real numbers, we always arrive again at real or 

 imaginary numbers, excepting when we join together a real and an 

 imaginary number by addition or its inverse operations. In this 

 case we reach the symbol a-\- i . b, where a and b stand for real num- 

 bers. Agreeably to the principle of no exception we are permitted 

 to reckon with a -f ib according to the same rules of computation as 

 with symbols previously defined, if for the second power of i we 

 always substitute minus i. 



In the numerical combination a-\- ib, which we also call num- 

 ber, we have found the most general numerical form to which the 

 laws of arithmetic can lead, even though we wished to extend the 

 limits of arithmetic still further. Of course, we must represent to 

 ourselves here by a and b either zero or positive or negative rational 

 or irrational numbers. If b is zero, a -f- ib represents all real num- 

 bers ; if a is zero, it stands for all purely imaginary numbers. This 

 general number a -f ib is called a complex number, so that the com- 

 plex number includes in itself as special cases all numbers hereto- 

 fore defined. By the introduction of irrational, purely imaginary, 

 and the still more general complex numbers, all combinations be- 

 come invested with meaning which the operations of the third de- 

 gree can produce. For example, the fifth root of 5 is an irrational 

 number, the logarithm of 2 to the base 10 is an irrational number. 

 The logarithm of minus i to the base 2 is a purely imaginary num- 

 ber ; the fourth root of minus i is a complex number. Indeed, we 

 may recognise, proceeding still further, that every combination of two 

 complex numbers, by means of any of the operations of the first, second, 

 or third degree will lead in turn to a complex number, that is to say, 

 never furnishes occasion, by application of the principle of no ex- 

 ception, for inventing new forms of numbers. 



A certain limit is thus reached in the construction of arithmetic. 

 But such a limit was also twice previously reached. After the in- 

 vestigation of addition and its inverse operations, we reached no 

 other numbers except zero and positive and negative whole num- 

 bers, and every combination of such numbers by operations of the 



