a be taken must be prefixed; thus 5 a denotes that the quantity a is to be taken 5 times, and 3 be represents three times be, and 7 √u2 + b denotes that √u2 + is to be taken 7 times, &c. The numbers thus prefixed are called 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 that are expressed by the same letters under the same powers, or which differ only in their co-efficients; thus, 3bc, 5bc, and 8bc, are like quantities, and so are the radicals 2 √ and 7 But unlike quantities are those which are expressed by different letters, or by the same letters with different powers, as 2 a b, 5 a b2, and Sab. When a quantity is expressed by a single letter, or by several single letters multiplied together, without any intervening sign, as a, or 2 ab, it is called a simple quantity. But the quantity which consists of two or more such simple quantities, connected by the signs + or, is called a compound quantity; thus, a2 a b + 5 a b c is a compound quantity; and the simple quantities a, 2 ab, 5 ab c, are called its terms or members. If a compound quantity consist of two terms, it is called a binomial; of three terms, a trinomial; of four terms, a quadrinomial, &c. of many terms, a multinomial. If one of the terms of a binomial be negative, the quantity is called a residual quantity. The reciprocal of any quantity is that quantity inverted, or unity divided by it; thus is the reciprocal of a and is the reciprocal of a' The letters by which any simple quantity is expressed may be ranged at plea. sure, and yet retain the same signification; thus ab and ba are the same quantity, the product of a and b being the same with that of b by 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, a2ab+5a2b may be written a +5 a2 b — 2ab, or -2ab+a+5 a2 b, for all these represent the same thing or the quantity which remains, when from the sum of a and 5 a2 b, the quantity 2 a b is deducted. AXIOMS. 1. If equal quantities be added to equal quantities, the sums will be equal. 2. If equal quantities be taken from equal quantities, the remainders will be equal, together with 6 b and 5 a to be subtracted, is sign of the quantity to be subtracted, and then 14 b to be subtracted. 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. 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 to add, and 9 times b to take away; i. e. we have, upon the whole, 6 times b to take away: and thus the sum of all the quantities is 2a - 6 b. If several similar quantities are to be added together, some with positive and some with negative signs, take the difference between the sum of the positive and the sum of the negative co-efficients, prefix the sign of the greater sum, and annex the common letters. adding it to the other by the rules laid down in the last article. Ex. 1. From 2 bx take cy, and the difference is properly represented by 2 bx-cy; because the prefixed to cy shews that it is to be subtracted from the other; and 2 b xcy is the sum of 2 bx and -cy. In this example the co-efficients are united; a- -p. x3 is equal to a x3 — px3; —— q. x2 is equal to bx2+9x2; and 1—1.x=x a x3-b x2 -x If the quantities to be multiplied have coefficients, these must be multiplied together as in common arithmetic; the sign and the literal product being determined by the preceding rules. Thus, 3 a X 5b15 ab; because 3 X a x 5 x 6 = 3 x 5 x ax b = 15 ab; 4 x x -11 y=- · 44 x y ; — 9 b x −5c=+ 45 bc; -6dx 4m-9 -24 m d. The powers of the same quantity are multiplied together by adding the indices; thus, a2 Xa3 = a; for aa X aaa aaaaa. In the same manner, a′′ × a′′ =a”+"; and -3 a2 x3 × 5axy2 = 15 a3 x1 y2. If the multiplier or multiplicand consist of several terms, each term of the latter must be multiplied by every term of the former, and the sum of all the products taken, for the whole product of the two quantities. Ex. 1. Mult. a+b+x by c+d Ans. ac+be+xc+ad+bd+x d Here a+b+x is to be added to itself e+d times, i. e. c times and d times. Ex. 2. Mult. a + b - x by C- - d Ans, ac+bcxc-ad-bd +x d Ex. 5. Mult. by-5a2+4bd ·15 a 37 abd — 20 b2 d2 Ex. 6. Mult. a2+2ab+b2 by a2-2ab+b2 a*+2ab+a+b2 Ans. a1 -2a3b-4a2 b2 — 2a b3 +a2b2+2ab+b Ex. 7. Mult. 1 — x + x2 — x3 by 1+x 1-x+x2. Ans. 1 * * *-x^ Ex. 8. Mult. x2-p x + q by x+a x3-px2+9x Ans. x3 — p—a. x2+q—ap.x+aq Here the co-efficients of x2 and x are collected; -p-a. x2 = − p x2+a x2; and q-ap.x=qx-ap x DIVISION. To divide one quantity by another, is to determine how often the latter is contained in the former, or what quantity multiplied by the latter will produce the former. Thus, to divide a b by a is to determine how often a must be taken to make up a b; that is, what quantity multiplied by a will In the division of simple quantities, if the co-efficient and literal product of the divisor be found in the dividend, the other part of the dividend, with the sign determined by the last rule, is the quotient. The reason of this, and the foregoing rule, is, that as the whole dividend is made up of all its parts, the divisor is contained in the Thus, =c; because ab multiplied by whole, as often as it is contained in all the abc ab • gives a b c. parts. In the preceding operation we inquire first, how often a is contained in a2, which gives a for the first term of the quotient, then multiplying the whole divisor by it, we have a2-ab to be subtracted from the dividend, and the remainder is ab+b2, with which we are to proceed as before. The whole quantity a22ab+b2 is in reality divided into two parts by the process, each of which is divided by a— therefore the true quotient is obtained. -b; Ex. 5. =manam n. a; i. e. a is contain ——a).x3—px2+qx—r(x2+a—p.x+a2—pa+q ed in xy, mn times, or it measures ay by the units in mn. Now it appears from what has been said, that a-pbc, and b—qc = d; every quantity therefore which measures a and b, measures pb, and a-p b, or c; hence also it measures qc, and b - qc, or d; that is, every common measure of a and b measures d. The quantity 2 x2, found in every term of one of the divisors, 2 a2x2 - 2 xa, but not in every term of the dividend, a3 — a2 x — a x2 +3, must be left out; otherwise the quotient will be fractional, which is contrary to the supposition made in the proof of the rule; and by omitting this part, 22, no common measure of the divisor and dividend is left out; because, by the supposition, no part of 2 x is found in all the terms of the dividend. To find the greatest common measure of three quantities abc; take d the greatest common measure of a and b, and the greatest measure of d and c is the greatest common measure required. In the same manner, the greatest common measure of four or more quantities may be found. If one number be divided by another, and, the preceding divisor by the remainder, according to what has been said, the remainder will at length be less than any quantity that can be assigned. |