l8 MONISM IN ARITHMETIC. 



together now both these extensions of the domain of numbers, we 

 arrive at negative fractional numbers. 



We pass over the easily deduced rules of computation for frac- 

 tions and shall only direct the reader's attention to the connexion 

 which exists between fractional and non-fractional or, as we usually 

 say, whole numbers. Since the number 12 lies between the num- 

 bers 10 and 15, or, what is the same thing, io<i2<i5, and since 

 10:5 = 2, 15:5 = 3, we say also that 12:5 lies between 2 and 3, or 

 that 



In itself, the notion of '-iess than " has significance only for results 

 of counting. Consequently, it must first be stated what is meant 

 when it is said that 2 is less than -^-. Plainly, nothing is meant by 

 this except that 2 times 5 is less than 12. We thus see that every 

 broken number can be so interpolated between two whole numbers 

 differing from each other only by i that the one shall be smaller 

 and the other greater, where smaller and greater have the meaning 

 above given. 



From the above definitions and the laws of commutation and 

 association all possible rules of computation follow, which in virtue 

 ol tne principle of no exception now hold indiscriminately for all 

 numbers hitherto defined. It is a consequence of these rules, again, 

 that the combination of two such numbers by means of any of the 

 operations defined must in every case lead to a number which has 

 been already defined, that is, to a positive or negative whole or frac- 

 tional number, or to zero. The sole exception is the case where 

 such a number is to be divided by zero. If the dividend also is 

 zero, that is, if we have the combination , the expression is one 

 which stands for any number whatsoever, because any number what- 

 soever, no matter what it is, if multiplied by zero gives zero. But 

 if the dividend is not zero but some other number a, be it what it 

 will, we get a quotient form to which no number hitherto defined 

 can be equated. But we discover that if we apply the ordinary arith- 

 metical rules to 0-r- all such' forms may be equated to one another 

 both ~hc;n a is positive and also when a is negative. We may there- 

 fore invent two new signs for such quotient forms, namely -j- oo and 



