Subtraction of algebraic quantities is performed by changing the sign of the quantity to be subtracted, and then adding the two quantities together, agreeably to the rules of addition. In the first example, where 3 am is to be taken from 8a+ 3 m, if we change the sign of 3 a, which is + into -, and then add these two quantities by the second rule of addition, the result will be + 5 a; and in subtracting -m from +3 m, we change the into +; and then adding their two quantities together, the result is + 4 m. If the question had been proposed, to take away only 3 a from 8a+ 3 m, the remainder would evidently have been 5 a 3 m; but as the sum to be subtracted is less than 3 a by once the value of m, the remainder must be greater than it would have been, on the first supposition, by an additional value of m; that is, it must be 5 a + 4 m, as above shown. Multiplication of algebraic quantities is performed according to the following rules. When the quantities to be multiplied have like signs, the sign of the product will be +; and when they have unlike signs, it will be -. When the quantities given are simple, find the sign of the product by the above rule; to which annex the product of the co-efficients, if any, and then all the letters, which will give the product required; thus, in the first example following, where 8am is to be multiplied by 3x, we mul tiply the co-efficients 8 and 3, giving 24 for the co-efficient of the product, and write down the letters of the multiplicand and multiplicator, prefixing the sign +, as the factors have like signs. Again, when the factors are compound quantities, each term of the multiplicand is to be multiplied by each term of the multiplicator; and the sum of these several products, collected according to the rules for addition, will give the product required. Prod. an+mn - nx - a c − cm + cx + az + mz−x Z In the first of these examples, 5 a multiplied by 2 a, give 10 aa; and both having the like sign+, the product has also that sign; in the same way m multiplied by 2a, gives 2 am, also a positive quantity; but the third term, — 2c, multiplied by + 2a, gives 4 a c, with the sign, because the signs of the factors are unlike. In the second example, where the multiplicator consists of two terms, a multiplied by x, gives xx; then z multiplied byx, gives xx; again working with z, the second term of the multiplicator, we have zx, or more properly xz, (in order to place the letters as they stand in the alphabet, for the value is the same,) which is written under the second term, and % by z equal to zz: then summing up these two lines of product, beginning either at the right or the left hand, we have x x + 2 x z + zz for the product required. ༤ In the third example, the characters of the multiplicand and multiplicator being all different, none of the products obtained by working with the several terms of the multiplicator can be combined together; consequently the three lines of product must be written down successively in one line, as was directed to be done in performing addition. These operations may be illustrated by using arithmetical numbers; take, for instance, the second, where + is to be multiplied into itself or by x+x. Let x stand for 4, and z for 6; then the operation would appear in the following shape: Here the component parts of the given quantity are multiplied separately together, and the result is the sum of the product of 4 by 4, of twice 4 by 6, and of 6 by 6, making 100, which is equal to the product of the whole of these component parts, or 10 multiplied into itself. In this example, in summing up the separate products, we find+am and—a m, which destroy each other; that is, if the one be subtracted from the other, as must be done, since they have unlike signs, there will be no remainder; a point, therefore, or asterisk, is placed in the general product: and in the same manner the two quantities-ax and +ax destroy each other, on which account another asterisk is placed in the product, and the following terms are successively brought down. To illustrate this example by common arithmetic, let a represent 8, 6, and x 4, then the question will be thus performed. 8+6-4 64 -36 The products arising from the repeated multiplication of a quantity into itself, and its several products, are called its powers; and the quantity itself is the root; thus a is the root, and if multiplied into itself, the product, or second power, will be aa; which again multiplied by a, will give the third power, or aaa; the 4th power will be aaaa, and so on; but as this manner of expression would soon become inconvenient, and liable to error, it has been the practice to express the different powers by a small figure placed over and towards the right hand of the root; thus the 2d power, aa, may be expressed by a2, the 3d power aaa by a3, the 4th power by at, &c. This small figure is called the exponent, or index; and, by adding these exponents, the same expression is obtained as if the root had been repeatedly multiplied; thus a + a* = a3, and a2 + a+ = a, &c. Division of algebraic quantities is performed agreeably to the following rules. When the signs of the divisor and the dividend are like, the sign of the quotient is +; but if they be unlike, the sign is -. 1st. When the divisor is simple, and a part, of or found in each term of the dividend, you must divide the co-efficient of each term of the dividend by the co-efficient of the divisor, and expunge, or withdraw from each term, the letter or letters of the divisor, and the result will be the the quotient. Thus, if it be required to divide 18mx by 3m, dividing the co-efficient 18 by 3, we have 6 for the co-efficient of the quotient; and mx being a product of which m is one of the factors, this symbol being taken away or expunged, the remaining symbol x will belong to the quotient, which will then be 6x. Again divide 5 a3m + 25 alm 5 am by 5am, and the quotient will be a +5b-m, thus: Sam |