PART VI. LXXX. ALGEBRA. ALGEBRA is a kind of specious arithmetic, or an arithmetic in letters; and is that science which teaches, in a general manner, the comparison of abstract quantities; by means whereof such questions are resolved whose solutions would be sought in vain from common arithmetic. Here every quantity, whether given or required, is commonly represented by some letter of the alphabet; the known or given quantities, for distinction's sake, being noted by the first letters, a, b, c, d, &c. and the unknown ones by the last letters, x, y, z, &c. There are, moreover, in algebra, certain signs or notes, made use of to show the relation and dependence of quantities one upon another, whose signification the learner ought first of all to be made acquainted with. (See the characters for abbreviation, next before page 1.) LXXXI. ADDITION. ADDITION in Algebra is performed by connecting the quantities by their proper signs, and joining in any sum such as can be united. For performing which, observe the following RULE. 1. If the quantities to be added are alike, and have the same sign, add the coefficients together, and to their sum prefix the common sign, and subjoin the common letter or letters. 2. If the quantities to be added are alike, but have unlike signs, add together the coefficients of the affirmative terms (if there be more than one), and do the same by the negative ones; and to their difference prefix the sign of the greater, adding the common letter or letters. 3. If the quantities to be added are unlike, write them down one after the other, with their proper signs and coefficients prefixed. Change the signs of the quantity to be subtracted into their contrary signs, and then add it, so changed, to the quantity from which it was subtracted (by the rule of addition): the sum arising will be the remainder. MULTIPLICATION of algebra is also performed by the following general RULE. Multiply the coefficients (if any) together, as in Sect. 4. and to their product join the letters, and prefix the proper sign before them, which, when the signs of the factors are alike, that is, both +, or both, the sign of the product is +; but when the signs of the factors are unlike, the sign of the product is-. (3) x+y+z a (5) 12x+6y 4a (6) -6d +76 (8) 2a-4b (9) aa+ab+b6 a-b DIVISION of algebraic quantities is the direct contrary to that of multiplication, and consequently performed by direct contrary operations. RULES. 1. When the quantities in the dividend have like signs to those in the divisor, and no coefficient in either, cast off all the quantities in the dividend that are like those in the divisor, and set down the other quantities with the sign + for the quotient. 2. When the quantities in the dividend have unlike signs to those in the divisor, then set down the quotient quantities, found as in the last rule, with the sign - before them. 3. If the quantities in the divisor cannot be exactly found in the dividend, then set them both down like a vulgar fraction, and find all the quantities of the same letters that are in the dividend and divisor, and proceed with the co-efficient as in Case 1, Sect. 38. 4. If the quantity to be divided is compound, range its parts according to the dimensions of some one of its letters, and proceed as in Sect. 5. 5. Different powers or roots of the same quantity are divided by subtracting the exponent of the divisor from that of the dividend, and placing the remainder as an exponent to the quantity given. EXAMPLES. Divisor. Dividend. d)ad+6d( (2) -d)-ad-bd( (3) a)aa+ab( -a)ab( (5) b)+ab-bd( (6) -bc) abc+bcd-bef( 7b) 42-db( (8) 2bx)8abx-18bxc( (9) 2b)ab-bb( 10) 20a)10ab- 15ac( 1 11) a-b)aaa-3aab+3abb-bbb( 12) a+b)aa+2ab+bb( (13) a+b)aa-bb( 14) 3-6)6-96( 15) 3x2-4x+5) 18x2-45x3+82x2-67x+40( 16) 4z-5a)48x376ax2-64a2x+105a2( (17) 3x+4a)81x-256a2( (18) 2x-3a)16+x-72a2x2 - 8104( (19) 2xyz)4xyxzz( (20) 20/2cy) 60ab10acxy( (21) x2) x2( (22) a+x)+( LXXXV. FRACTIONS. REDUCTION of algebraic fractions is of the same nature, and requires the same management, as that of numbers. A mixed quantity is reduced to an improper fraction by the rules in Sect. 38, Case 3. EXAMPLES. a2-ax (1) Reduce a-x+ to an improper fraction. An improper fraction is reduced to a mixed quantity, by the rule in Sect. 38, Case 4. |