MONISM IN ARITHMETIC. 23 



two old numbers whose difference may be made as small as we 

 please. Numbers possessing this property are called irrational num- 

 bers, in contradistinction to the numbers hitherto denned, which are 

 termed rational. The considerations which before led us to negative 

 rational numbers, now also lead us to negative irrational numbers. 

 The repeated application of addition and multiplication as of their 

 inverse processes to irrational numbers, (numbers which though not 

 exactly equal to previously defined rational numbers may yet be 

 brought as near to them as we please,) again simply leads to num- 

 bers of the same class. 



A totally new domain of numbers is reached, however, when we 

 attempt to impart meaning to the square roots of negative numbers. 

 The square root of minus 9 is neither equal to plus 3 nor to minus 

 3, since each multiplied by itself gives plus 9, nor is it equal to any 

 other number hitherto defined. Accordingly, the square root of minus 

 9 is a new number-form, to which, harmoniously with the principle 

 of no exception, we may give the definition that (I/ 9) 2 shall al- 

 ways be put equal to minus 9.* Keeping to this definition we see 

 at once that I/ a, where a is any positive rational or irrational 

 number, is a symbol which can be put equal to the product of l/-|- a 

 by i/ 7. In extending to these new numbers the rights of arith- 

 metical citizenship, in calling them also " numbers," and so shaping 

 their definition that we can reckon with them by the same rules as 

 with already defined numbers, we obtain a fourth extension of the 

 domain of numbers which has become of the greatest importance 

 for the progress of all branches of mathematics. The newly denned 

 numbers are called imaginary, in contradistinction to all heretofore 

 defined, which are called real. Since all imaginary numbers can be 

 represented as products of real numbers with the square root of 

 minus one, it is convenient to introduce for this one imaginary num- 

 ber some concise symbol. This symbol is the first letter of the word 

 imaginary, namely, /; so that we can always put for such an ex- 

 pression as I/ 9, 3 * 



If we combine real and imaginary numbers by operations of the 



* Henceforward we shall use the simpler sign 



