ADDITION. ADDITION, in Algebra, is connecting the quantities together by their proper signs, and uniting in simple terms such as are similar. In addition there are three cases. CASE I. When like quantities have like signs. RULE.* Add the coefficients together, to their sum join the common letters, and prefix the common sign when neces sary. EXAMPLES. * The reasons, on which these operations are founded, will readily appear from a little reflection on the nature of the quantities to be added, or collected together. For with regard to the first example, where the quantities are 3a and 5a, whatever a represents in one term, it will represent the same thing in the other; so that 3 times any thing, and 5 times the same thing, collected together, must needs make 8 times that thing. As, if a denote a shilling, then 3a is 3 shillings, and 5a is 5 shillings, and their sum is 8 shillings. In like manner -2ab and —7ab, or 2 times any thing and -7 times the same thing, make times that thing. -9 As to the second case, in which the quantities are like, but the signs unlike; the reason of its operation will easily appear by reflecting, that addition means only the uniting of quantities together by means of the arithmetical operations denoted by their signs + and —, or of addition and subtraction; which being of contrary or opposite natures, one coefficient must be subtracted from the other, to obtain the incorporated or united mass. As united into one, or otherwise Thus, for example, if a be As to the third case, where the quantities are unlike, it is plain, that such quantities cannot be added than by means of their signs. supposed to represent a crown, and b a shilling; then the sum of a and b can be neither za nor 2b, that is, neither 2 crowns nor 2 shillings, but only 1 crown plus 1 shilling, that is, a+b. In this rule, the word addition is not very properly used, being much too scanty to express the operation here performed. The business of this operation is to incorporate into one mass, or algebraic expression, different algebraic quantities, as far as an actual incorporation or union is possible; and to retain the algebraic marks for doing it in cases, where an union is not possible. When we have several quantities, some affirmative and others negative, and the relation of these quantities can be discovered, ip whole or in part; such incorporation of two or more quantities into one is plainly effected by the foregoing rules. It máy seem a paradox, that what is called addition in algebra should sometimes mean addition, and sometimes subtraction. But the paradox wholly arises from the scantiness of the name, given to the algebraic process, or from employing an old term in a new and more enlarged sense. Instead of addition, call it incorporation, union, or striking a balance, and the paradox vanishes. Subtract the less coefficient from the greater, to the re mainder prefix the sign of the greater, and annex their common letters or quantities. * In example 3, the coefficients of the two quantities, viz. +2c and -26, are equal to each other, therefore they destroy one another, and so their sum makes o, or *, which is frequently used, in algebra, to signify a vacant place. NOTE. When many like quantities are to be added together, whereof some are affirmative and others negative; reduce them first to two terms, by adding all the affirmative quantities together, and all the negative ones; and then add the two terms according to the rule. Thus, 11. Add 4a2+7a*—3a2+12a2—8a2+a2—5a2 together. First, 4a+ya+12a2+a2=24a2, the sum of the affirmative quantities, And3a8a5a2-16a2, the sum of the nega tive. Then 24a-16a8a", the sum of the whole. 12. Add 5ax2-4ax2+10ax2-8ax2--6ax2 together. First, 5ax+10ax2=15ax2, And-4ax-8ax2-6ax2-18ax2; Therefore the sum of these quantities is 15ax-18a x2=—zax2. CASE III. When the quantities are unlike. RULE. Set them down in a line, with their signs and coefficients prefixed. Sum 15a2-763+3✔✅✅ab+6c2—5d+xy—2y3-+-4* 2. * Here the first column is composed of like quantities, which are added together by case 1. The terms 963 and +963 destroy one another; and the sum of 1263 and +563 is —763, by case 2. The sum of +7 ab and -4ab is +3√ab. In like manner, +10 and -6 together make +4; and the rest of the terms being unlike, they are set down with their respective signs and coefficients prefixed, conformably to case 3. |