and and and and nx, n b. x yaxay+axy, 5. Add together the following quantities. XII. The subtraction of simple quantities, as has already been observed, is performed by giving the sign to the quantity to be subtracted, and writing it before or after the quantity, from which it is to be taken. If it is required to subtract c+d from a + b it is plain that the result will be a + b —c d, for the compound quantity c+d is made up of the simple quantities c and d, which being subtracted separately would give the above result. 22 From 22 subtract 13- -7. and 13—7—6. 22 - 6 = 16. The result then must be 16. But to perform the operation on the numbers as they stand, first subtract 13, which gives 139. This is too small by 7 because the number 13 is larger by 7 than the number to be subtracted, therefore in order to obtain a correct result the 7 must be added; thus 22 13716, as required. This quantity is too small by c because 6 is larger than b—e by the quantity c. Hence to obtain a correct result c must be added, thus ab+c. This reasoning will apply to all cases, for the terms affected with the sign in the quantity to be subtracted diminish that quantity; hence if all the terms affected with + be subtracted, the result will be too small by the quantities affected with -, these quantities must therefore be added. The reductions may be made in the result, in the same manner as in addition. Hence the general RULE. Change all the signs in the number to be subtracted, the signs to, and the signs to +, and then proceed as in ad+ dition. 1. From Examples in Subtraction. a2x+3by-5 a c3-16 Subtract 3 ax+by-2 ac3-22 Operation. a2x+3by-5 a c3 — 16 -3a2 x-by+2 ac3 +22 -2a2x+2by-3ac3 +6 α 3b x2-7 ax3 + 13 13bc-3 ax3-8. Ans. 3b x2-13bc-4ax+ 21. 17 a❜y + 13 ay—a—3 42 a xy-4 a x 143-17 y 6. From Subtract 7. From Subtract a+3abc-1 1+3abc-a. 3abz+2ab7z 2 ab-7z2 abz. Multiplication of Compound Quantities. XIII. Multiplication of compound quantities is sometimes expressed without being performed. To express that a+b is to be multiplied by cd, it may be written a+b xc-d with a vinculum over each quantity, and the sign of multiplication between them; or they may be each enclosed in a parenthesis and written together, with or without the sign of multiplication; thus (a + b) × (c — d) or (a + b) (c or (a+b) (cd). In the expression a+b (c-d), b only is to be multiplied by c Multiply a+bby c. d. It is evident that the whole product must consist of the product of each of the parts by c. a+b When some of the terms of the multiplicand have the sign - they must retain the same sign in the product. ax +3 abx by 13 ab* x3 7. 8. Multiply ab by c, also 23-5 by 4. Since the quantity a-b is smaller than a by the quantity b, the product ac will be too large by the quantity bc. This quantity must therefore be subtracted from a c. When both multiplicand and multiplier consist of several terms, each term of the multiplicand must be multiplied by each term of the multiplier. It is evident that if a + be taken c times and then d times, and the products added together, the result will be c + d times a+b. difference between + 8 and 12 is 20, or between +b and cis b+c. Hence it follows, that to subtract a quantity which has the sign -, we must give it the opposite sign, that is, it must be added. X. The learner, by this time, must have some idea of the use of letters, or general symbols, in algebraic reasoning. It has been already observed that, strictly speaking, we cannot actually perform the four fundamental operations on these quantities, as we do in arithmetic; yet in expressing these operations, it is frequently necessary to perform operations so analogous to them, that they may with propriety be called by the same -names. Most of these have already been explained; but in order to impress them more firmly on the mind of the learner, they will be briefly recapitulated, and some others explained which could not be introduced before. Note. Algebraic quantities, which consist of only one term, are called simple quantities, as + 2 a, -3 a b, &c.; quantities which consist of two terms are called binomials, as a +-b, a — 3b+2c, &c.; those which consist of three terms are called trinomials; and in general those which consist of many terms are called polynomials. Simple Quantities. ia To ex The addition of simple quantities is performed by writing them after each other with the sign+between them. press that a is added to b, we write a + b. To express that a, b, c, d, and e are added together, we write a+b+c+d+e. It is evidently unimportant which term is written first, for 358 is the same as 5 + 3 + 8, or as 8 + 5 + 3. So a+b+c has the same value as +a+c. It has been remarked (Art. I.) that x + x + x may be written 3 x. This is multiplication; and it arises, as was observed in Arithmetic, Art. III., from the successive addition of the same quantity. 3x, it appears, signifies 3 times the quantity , that is, a multiplied by 3. So b+b+b+b+b may be written 56. In the same manner, if x is to be repeated, any number of times, for instance as many times as there are units in a, we write a x, which signifies a times x, or a multiplied by a. |