ALGERBA. 



and a + b into a -f- c, signify the same 

 thing as (a + A) X (a -f c), or the pro- 

 duct of -f A by a + c . The sign -=- 

 is the sign of division, as it denotes that 

 the quantity preceding it is to be divided 

 by the succeeding quantity. Thus, c-r-A 

 signifies that r is to be divided by A ; and 

 (a + 6) -T- (a -f- r), that a -f- A'is to be 

 divided by a + c. The mark ) is some- 

 used as a note of division; thus 

 -f- A) a A denotes that a A is to be divi- 

 ded by ti -f A. But the division of alge- 

 braic quantities is most commonly ex- 

 it by placing the divisor under the 

 dividend, with a line between them, like 



a vulgar fraction. Thus, - represents the 

 quantity arising by dividing c by A, or 

 the quotient, and - represents the 



quotient of n+A divided by o-f c. Quan- 

 tities thus expressed are called algebraic 

 fractions. 



The sign ^/ expresses the sqviare root 

 of any quantity to which it is prefixed; 

 thus x/ 25 signifies the square root of 25, 

 or 5, because 5x5 is 25 ; and ^/ (u A) 

 denotes the square root of n A; and 



denotes thesquare root of 



, or of the quantity arising from 

 d 

 the division of a A -f A c by d; but 



, which has the separating 



u 



line drawn under v/, signifies that the 

 square root of a A+A c is to be first ta- 

 ken, and afterwards divided by </;*so 

 that if a were 2, A 6, c 4, and d 9, 



vAHl) would be * 

 d 



orjj; but 



| would be 



d '. 

 ^/ 4, which is 2. The sign ^/ with a 



figure over it is used to e\ press the cubic 

 er biquadratic root, &c. of any quantity ; 

 thus x / 64 represents the cube root of 

 64, or 4. because 4x4x4 is 64; and x / 

 (a A-}-rf/) the cube root of a A-f-c </. In 

 like nunne r */ 16 de notes the biquadratic 

 root of I6,oi 2x2x2x2 is 16, 



an ' *f (a A+c d, denotes the bi quadra 

 tic root of a A -j- c </; and so of others. 

 Quantities thus expressed are called r.i- 

 dical quantities, or surds; of which those, 

 ing of one term only, as N / a and 

 v/ (' ;ied simple sum 



tl\o>e consisting of several terms or num- 

 bers i asv' (a j A J ) and -^ (a 4 A+Ar) 



are denominated compound surds. 

 ther commodious method of c-xpr> 

 radical quantities is that which d 

 the root by a vulgar fraction, placed at 

 the end of a line drawn over the quantity 

 gi\en In this notation, the square root 

 is expressed by $, the cube root by i 

 the biquadratic root by i, &c. TI 

 expresses the same quantity with ^/ a, 

 i. e. the square root of a, and ( J +a A) JL 

 the same as -^ (a-+a A), i. e. the eubt: 

 root of n'+rt A ; and denotes the 

 cube root of the square of a, or the 

 square of the cube root of a ; and (a-j-z)i 

 the seventh power of the biquadratic root 

 of a+z; and so of others; (a 1 ) ^ is a, 

 a?) i is o, &c. Quantities that have no ra- 

 dical sign (v/) or index annexed to them, 

 are called rational quantities. The sign 

 =, called the sign of equality, signifies 

 that the quantities between which it oc- 

 curs are equal. Thus 2+3 = 5, shews 

 that '2 plus 3 is equal to 5; and .r= A 

 shews that x is equal to the difference of 

 n and A. The mark : : signifies that the 

 quantities between which it stands arc 

 proportional. As a : A :: c : <f denote* thai 

 u is in the same proportion to A as e is to 

 d ; or that if a be twice, thrice, or four 

 times, &c. as great as A, c will be twice, 

 thrice, or four times, &c. as great as d. 

 When anyquantityis to be taken more than 

 once, the number, which shows how many 

 times it is to be taken, must be prefixed; 

 thus 5 a denotes that the quantity a is to 

 be taken 5 times, and 3 A c represents 

 three times A c,and 7 ^/ (a'xA 1 ) denotes 

 that v/ (a'+A 1 ) is to be taken 7 times 

 &.c. The numbers thus prefixed . i 

 ed co-efficients ; and if a quantity have 

 no co-efficient, unit is understood, and it 

 is to be taken only once. Similar or like 

 quantities are those tliut are e\] 

 by the same letters under the same pow- 

 ers, or which differ only in their co-effici- 

 ents; thus, 3 A r, 5 A c, and 8 A c, are like 

 quantities, and so are the radicals 



like quantities are those which are ex- 

 1 by different letters, or hy the 

 same letters \\ ith different powers, as 2 a 

 A, 5 a A 1 , and ,1 a- h. \\hen a quantity is 

 expressed by a single letter, < 

 single letters multiplied together, without 

 any intervening > >< 2 <i A, it is 



called a simple (pia-it <;\ I tut the quan- 

 tity which consists (,f two or more such 

 simple quantities, connected by the signs 

 -f- or , is called a compound quantity ; 

 thus, 2o-(-5a6cis a compound 



