ALGEBRA. 



quantity ; and the simple quantities, a. 2 

 a b, 5 ab c, are called its terms or mem- 

 bers. If a compound quantity consist of 

 two terms, it is called a binomial ; of 

 three terms, a trinomial ; of four terms, 

 a quadiinomial ; and of many terms, a 

 multinomial. If one of the terms of a 

 binomial be negative, the quantity is call- 

 ed a residual quantity. The reciprocal of 

 any quantity is that quantity inverted, or 



unity divided by it; thus is the reci- 



procal of - and - is the reciprocal of 



a. The letters by which any simple 

 quantity is expressed may be ranged 

 at pleasure, and yet retain the same 

 signii. cation ; thus <i b and b a are the 

 same quantity, the product of a and b 

 being the same with that or' 6 and a. The 

 several terms of which any compound 

 quantity consists may be disposed in Any 

 order at pleasure, provided they retain 

 their proper signs. Thus a 2 a b -J- 

 5 a 1 b may be written a-\- 5 a 1 b 2 a 6, 

 or 2 a b-\-a-\-5 a* b, for all these repre- 

 sent the same thing, or the quantity which 

 remains, when from the sum of a and 5 

 a- b, the quantity 2 a b is deducted. 



AXIOMS. 1. If equal quantities be add- 

 ed to equal quantities, the sums will be 

 equal. 



2. If equal quantities be taken from 

 equal quantities, the remainders will be 

 equal. 



3. If equal quantities be multiplied by 

 the same, or equal quantities, the pro- 

 ducts will be equal. 



4. If equal quantities be divided by the 

 same, or equal quantities, the quotients 

 will be equal. 



5. If the same quantity be added to und 

 subtracted from another, the value of'the 

 latter will not be altered. 



6. If a quantity be both multiplied and 

 divided by another, its value will not be 

 altered. 



ADDITION OF ALGEBHAICAL Q.UA3T- 

 TITIES. 



The addition of algebraical quantities is 

 performed by connecting those that are un- 

 like with their proper signs, and collecting 

 those that are similar into one sum. 



Add together the following unlike 

 quantities : 



Ex. 1. ax 



bu 



Ex. 2. < 



-4-3 z x 



Ans. a x b u + 3 z 2y 



Ans. Q+6-fSc x 4 y+3 z 



It is immaterial in what order the quan- 

 tities are set down, if we take care to 

 prefix to each its proper sign. 



When any terms are similar, they may 

 be incorporated, and the general expres- 

 sion for the sum shortened. 



1. When similar quantities have the 

 same sign, their sum is found by taking 

 the sum of the co-efficients with that 

 sign, and annexing the common letters. 



Ex. 3. 4 a 5 b 

 2 a 66 

 9 3 6 

 Ans. 15 a 146 



Ex. 4. 4 a j c 10 6 d e 



6a-c 9 6 d c 



11 a- c 3 b d e 



Ans. 21 a* c ^226 de 



The reason is evident ; 4 a to be add- 

 ed, together with 2 a and 9 a to be add- 

 ed, makes 15 a to be added ; and 5 b to 

 be subtracted, together with 6 b and 3 a 

 to be subtracted, is 14 b to be subtract- 

 ed. 



2. If similar quantities have different 

 signs, their sum is found by taking the 

 difference of the co-efficients with the 

 sign of the greater, and annexing the 

 common letters as before. 



Ex. 5. 7 a+3 b 



5 a 9 b 

 Ans. 2 a 66 



Ex. 6. 6 a+4 6-f 9 c 



9a+36-fl6c 



-j- 12 a 7 620 c 



Ans. 9 a * + 5 c 



In the first part of the operation we 

 have 7 times a to add, and 5 times a to 

 take away; therefore, upon the whole, 

 we have 2 a to add. In the latter part, 

 we have 3 times b to add, and 9 times b 

 to take away; *. e. we have upon the 

 whole, 6 times b to take away: and thus 

 the sum of all the quantities is 2 a 66. 



If several similar quantities are to be 

 added together, some with positive and 

 some with negative signs, take the differ- 

 ence between, the sum of the positive 



