MATHEMATICAL INTRODUCTION. 

 CHAPTER I. 



NUMBER. 



1. Rationals and Irrationals. The primary objects of 

 study in Arithmetic are the natural numbers or integers, 1, 2, 3,... 

 forming an unlimited sequence. Of these any two a and b may 

 be added together, and we find the fundamental law that 



a + b = b + a. 



This is known as the commutative law. For more than two 

 numbers, we find that 



(a + b) + c = a + (b + c). 



This is known as the associative law. Any two numbers may be 

 multiplied together, and we find that multiplication is subject to 

 the commutative law, 



ab ba, 

 to the associative law, 



(ab) c = a (be), 



and in addition to the distributive law, 



a (b + c) = ab + ac. 



Defining the operation of subtraction as the inverse of addition, 

 so that c is defined as the result of subtracting b from a if b added 

 to c will give a, we find that the operation of subtraction is pos- 

 sible only if a is greater than 6. We are thus led to extend our 

 definition of numbers in such a way as to call that to which 6 

 must be added in order to give a, a number, in the case where a is 

 less than b. We are thus led to the conception of the negative 



^S W. E. 1 



