and the product of a, b, and c, by either abc, or acb, or bac, or bca, or cab, or cba; as it matters not which quantities are placed or multiplied first. But it will be sometimes found convenient in long operations, to place the several letters according to their order in the alphabet, as abc, which order also occurs most easily or naturally to the mind. 22. Likewise, the several members, or terms, of which a compound quantity is composed, may be disposed in any order at pleasure, without altering the value of the significa. tion of the whole. Thus, 3a-2ab+4abc may also be writ. ten 3a+4abc-2ab, or 4abc+3a-2ab, or -2ab+3a+4abc, &c.; for all these represent the same thing, namely, the quantity which remains, when the quantity or term 2ab is subtracted from the sum of the terms or quantities 3a and 4abc. But it is most usual and natural, to begin with a positive term, and with the first letters of the alphabet. SOME EXAMPLES FOR PRACTICE. In finding the numeral values of various expressions, or combinations, of quantities. Supposing a 6, and b=5, and c=4, and d=1, and e=0. Then 1. Will a2+3ab-c2= 36 +90-16110. 2. And 2a3-3a2b+c3= 432-540+ 64 = 44. 3. And a2x (a+b)-2abc = 36× 11-240 = 156. 4. And a3 + c2 = 216 +16=12+16= 28. 5. And Zac+c2 or (2ac + c2)=/64 = 8. 6. And c+ 7. And = =7. a2-√(b3-ac) 36-1 35 = 2a-√(b2+ac) 12-75 8. And √(b3-ac)+ √(2αc+c2)=1+8=9. 9. And b3-ac+ √(2ac+c2) = √(25—24+8) = 3. 10. And ab+c-d = 183. 11. And 9ab-1062 + c = 24. 25. And 3ac2 + 3⁄4/a3 — b3 = 292.497942. 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: one, when the quantities are like, and their signs like also; a second, when the quantities are like, but their signs unlike; and the third, when the quantities are unlike. Which was performed as follows*. * 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, 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 8 shillings. In like manner, 2ab and -7ah, or-2 times any thing, and -7 times the same thing, make 9 times that thing. ADDITION. 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 + or Thus, 3a added to 5a, makes 8a. And -2ab added to -7ab, makes —9ab. And 5a+76 added to 7a + 3b, makes 12a + 10b. 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 cau 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, and sometimes subtraction. 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 vanishes. When the Quantities are alike, but have Unlike Signs. Add all the affirmative co-efficients 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 co-efficient, from the greater, and to the remainder prefix the sign of the greater, and subjoin the common quantity or letters. So + 5a and - 3a, united, make + 2a. 3a, united, make — 2a. |