ADDITION. ADDITION, in Algebra, is the connecting the quantities together by their proper signs, and incorporating or uniting into one term or sum, such as are similar, and can be united. As 3a + 2b - 2a = a + 2b, the sum. The rule of addition, in algebra, may be divided into three cases: CASE I. When the quantities are like, and have like signs. ADD the co-efficients together, and set down the sum; after which set the common letter or letters of the like quantities, and prefix the common sign + * The reasons on which these operations are founded will readily appear, by 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 the 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, will make 8 times that thing. Thus, if a denote a shilling; then 3a is 3 shillings, and 5a is 5 shillings, and their sum 8 shillings. In like manner, 2ab and—7ab, or- 2 times any thing, and - 9 times that thing. 7 times the same thing, make 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, the one co-efficient must be subtracted from the other, to obtain the incorporated or united mass. As to the third case, where the quantities are unlike, it is plain that such quantities cannot be united into one, or otherwise added, than by means of their signs: thus, for example, if a be supposed to represent a crown, and b a shilling; then the sum of a and b can be neither 2a nor 26, 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 limited 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 the former is not possible. When we have several quantities, some affirmative and some negative; and the relation of these quantities can in the whole or in part be discovered; such incorporation of two or more quantities into one, is plainly effected by the foregoing rules. It may seem a paradox, that what is called addition in algebra should sometimes mean addition, sometimes subtraction, and sometimes both. But the paradox wholly arises from the scantiness of the name given to the algebraic process; from employing an old term in a new and more enlarged sense. Instead of addition, call it incorporation, or union, or striking a balance, or give it any name to which a more extensive idea may be annexed, than that which is usually implied by the word addition, and the paradox will vanish. When the quantities are like, but have unlike signs. ADD all the affirmative coefficients into one sum, and all the negative ones into another, when there are several of a kind: then subtract the less sum, or the less coefficient, from the greater, and to the remainder prefix the sign of the greater, and subjoin the common quantity or letters. Thus, + 5a and 3a, united, make + 2a. 5a + 4 a + 6a -3a + a + 3a VOL. I. OTHER EXAMPLES FOR PRACTICE. 2. +8x3 + 3y 5x3+4y 16x+5y + 3x3-7y + 2003 2y 5x 3/a+y—2x1√y+ √2 2 —8x (a+y)1—4.xy1 +3√/2 7x/a+y+3xy+2√2 2x(a+y)*+5x^/y+{×2a -9x3/a+y—8xy1 −8√√/2 10. -3 (ax+by+cz) —√x2+y2+a-b 24/ax+by+cz +(x2+y2)3—3 (a—b) 7 ax+by+cz+ -√/x2+y2+2 (a−b) 34/ (ax+by+cz) +(x2+y2)2 +a-b −51/ (ax+by+cz) + x2+y22. — 2(a—b) (ax+by+cz)*—√/x2+ y2 — 3(a—b) CASE III. When the quantities are unlike. HAVING collected together all the like quantities, as in the two foregoing cases, set down those that are unlike, one after another, with their proper signs. Note. It often happens that some one letter is considered the principal object in the calculation, and that the others are viewed as coefficients of this In this case their sums will assume a compound form : as in the following examples. one. (2a+cd-6b)x+(3b+8ad—c) y2 (a-3d) x+(3d-b) y2+(2b+4m) y In these cases, 2a, cd, 6b, &c. instead of being considered to form part of the components of the respective terms in which they appear, are collected under vincula, and the collection under each vinculum treated as a single quantity. Two other examples are added for the student's exercise. - a√x2-y2 + b √x2 + y2 - 5c √√√x2 + y2. 3d √x2 + y2 2 √x2 + y2+ 4a √x2 — y2 Sometimes these literal coefficients, are placed vertically under each other. tion and division. (a + b) √x-(2 + m) √y 4y2 + (a + c) x2 3x √y — (2d — e) xa (m + n) y2 + (b + 2c) √x 2x √x+12a√y instead of being collected horizontally, Specimens may be seen in multiplica SUBTRACTION. SET down in one line the first quantities from which the subtraction is to be made; and underneath them place all the other quantities composing the subtrahend; ranging the like quantities under each other, as in addition. Then change all the signs (+ and -) of the lower line, or conceive them to be changed; after which, collect all the terms together as in the cases of addition *. Note. When the sign is prefixed to a compound expression, it indicates that if the parenthesis is removed, the signs of all the terms must be changed. (ax bx+2cx2 3dx3) = 2cx2+3dx3. For - Thus, otherwise the sign equally. ax + bx would not affect all the terms within the parenthesis Note. If literal coefficients occur, they must be collected (the subtractive ones with their signs changed) as directed in note upon case iii. of addition. * This rule is founded on the consideration, that addition and subtraction are opposite to each other in their nature and operation, as are the signs and - by which they are expressed and represented hence, since to unite a negative quantity with a positive one of the same kind, has the effect of diminishing it, or subducting an equal positive one from it, therefore to subtract a positive (which is the opposite of uniting or adding) is to add the equal negative quantity. In like manner, to subtract a negative quantity, is the same in effect as to add or unite an equal positive one. So that, changing the sign of a quantity from + to - or from -to +, changes its nature from a subductive quantity to an additive one; and any quantity is in effect subtracted by barely changing its sign. |