ALGEBRA 



CHAPTER I. 



INTRODUCTION. 



1, Definitions. When we wish to use a general term which 

 shall inclnde in its meaning any intelligible combination of alge- 

 braic symbols and quantities, the word Expression will be adopted. 

 Thus 



^2 j\ r 2 I i I \ c^ -\- d^ -\- abed 



ix^—d) {ax^-\-bx-\-e)\ ■ — — ^Ai,_l / — 



may be called expressioris. It includes the words polynomial, 

 fraetio7i, and radieal and more besides. 



When we wish to call attention to the fact that certain specified 

 quantities appear in an expression it may be called a Funetion of 

 those quantities. Thus if we desire to point out that x appears in 

 the first expression above, it would be called a fnnetion of x. If 

 we wish to state that a, b, e, and d occur in the second expression, 

 we would call it o. fu7ietion of a, b, e, and d. If we wivSh to say 

 that y occurs in the last expression, it may be called Sifunetioti of 

 y, or if we wish to say that a, b, and y occur in it, we would 

 speak of it as 3.fu?iefio?i of a, b, a?idy. A formal definition of the 

 word function would be : 



A Function of a quantity is a name applied to any mathemati- 

 cal expression in which the quantity appears. 



2. Definition. An expression is Integral with respect to any 

 quantity or quantities, that is, is an integral function of those 

 quantities, when the quantities named do not appear in any man- 

 ner as divisors. Thus 5,r^-f|jf— >/2 is integral with respect to 

 x ; that is, is an integral function of x. 



a—b-\-a-b ab 



x^-^xy X 



is integral with respect to a and b, but fractional with respect to 



