OF SUBTRACTION. S taken from 6 times 8 leaves 5 times 8; twice 8 taken from 9 times 8 leaves 7 times 8; and, whatever a may represent, 3 times a taken from 13 times a leaves 10 times a; 7 times the sum of a and b taken from 9 times the sum of a and b leaves twice the sum of a and b. Hence 12 a a 11 a, 15 ab-7 a b = 8 ab, &c. Again, if the difference between 4 and 7 were required to be taken from 12; if 7 were taken away, the remainder would be 5; but in deducting 7, too much would be taken away by 4; if, therefore, to the remainder 5, the number 4 were added, the sum 9 would be the required result, or 12 — 7 7+ 4 = 9. If from a + b it were required to deduct d 4 12 c; a + b d would be less than the true remainder by c; for, in subtracting d from a + b, too much by c had been taken away. If therefore c were added to a + b d, the sum a + b d + c would be the required difference of the two given quantities. Hence, the subtraction of algebraical quantities is performed by changing the signs of all the terms in the quantity to be subtracted; and, adding the terms with the signs so changed, to the quantity from which the subtraction is to be made. EXAMPLES IN SUBTRACTION. 1. From 17 a take 3 a? Answer, 14 a. 2. What is the difference between 8 a b and 3 ab? Answer, 5 a b. 3. What is the difference between 15 b + c and 6b-3 c? Answer, 9 b + 4 c. 4. What is the difference between 4 a b + c and a b 3ab8c. -7 c? Answer, 6. From a2 + 2 a b + b2 take a2 2 ab + b2? Answer, 4 a b. OF MULTIPLICATION. 4 multiplied by 6 denotes that 4 is to be taken 6 times, or that 6 fours are to be added together, and the sum of these 6 fours, or the product of 4 by 6 is represented by 6 × 4 or 24. 7 a multiplied by 3 denotes that 7a is to be taken 3 times, and the product is represented by 7a x 3, or 21 a. a multiplied by 5 denotes that a is to be taken 5 times; and the product is represented by 5 a. a multiplied by b, denotes that a is to be taken as often as there are units in b, and the product is represented by ab; and, in like manner, if C were multiplied by b, the product would be represented by bc; and the product of a and b, added to the product of c and b, would therefore be represented by a b + cb, or by a + c . b. Now, if the sum of a and c, or a + c, were to be multiplied by b, the product would contain as many times a and as many times cas there are units in b; or a + c multiplied by b would be represented as above, by a b + c b, or by a + c. b Again, suppose a to be the greater, and c the less of two quantities, and let the difference of a and c be multiplied by b; for every time that a is taken, c must be subtracted; and therefore the product will be represented by as many times a as there are units in b, diminished by as many times c as there are units in b, or a b cb will represent the required product. Let now a + c be multiplied by b d. If a + c were multiplied by b alone, the product, as we have already seen, would be abcd. But in multiplying by b, we have multiplied by a quantity which is too great by d, for the multiplier b d is only the excess of b above d; therefore, from the product of a + c by b, we have to deduct as many times a + c as there are units in d, or we have to subtract a d + c d from ab + cb, and the remainder ab+cbad cd, is the required product. Finally, let a c be multiplied by bd. This signifies that α c is to be taken as often as there are units in the difference of b and c; or, if a - c be multiplied by b, as many times ac must be deducted from the product as there are units in d; or, from the product of a - c by b, the product of a Now, the product of a cd. c by bisa b c by d must be subtracted. cb, and the product of a And a b c b ad cd ab by d is a d с c b We hence deduce the following rule for the multiplication of alge braical quantities. Multiply every term of the quantity to be multiplied by each term separately, of that by which the multiplication is to be made; observing when the signs of the terms multiplied together are alike, t make the sign of the product +; but when the signs of the terms are unlike; and the sum of the several products will be the total product required. Note.-As 52 signifies 5 x 5, and 54 signifies 5 × 5 × 5 × 5; 52 x 54 will be the same as 5 x 5 x 5 x 5 × 5 × 5, or 56; and, in the same way, a3 × a1 = a. a. a × a. a. a. a. = a7 ; therefore different powers of the same quantity are multiplied by the addition of their indices. EXAMPLES IN MULTIPLICATION. 1. Multiply 3 a by 11? Answer, 33 a. 2. Multiply a by a? Answer, a2. 3.Multiply a + b by a-b? Answer, a2 — b2. 4. Multiply a - b by a a2 2ab+b2. b, or find the square of ab? Answer, 5. Multiply a + b by a + b, or find the square of a + b? Answer, a2 + ab + b2. 6. Multiply a3 by a2? Answer, a5. 7. Multiply 5 ab by 7 a? Answer, 35 ao b. a? Answer, 35 a2 + 7 ab. 8. Multiply 5 a + b by 7 9. Multiply a + b by a? Answer, a2 + a b. OF DIVISION. 3 times 9 divided by 3 gives 9 for the quotient; and 3 times 9 divided by 9 gives 3 for the quotient. In like manner, a times b, or a b, divided by b, gives a for the quotient; and ab divided by a gives b for the quotient, Again, the quotient of 3 divided by 7 is represented fractionally by 3 12 7 ; 12 divided by 16 is represented fractionally by cr as 12 and 16' 16 have a common divisor 4, the quotient in its lowest terms 12 In like manner, ma divided by mb is represented by mb; or by the equivalent fraction— Further, 128 divided by 16, is expressed by Again, for the signs, + a x + b is + a b, therefore + a - ab is + a, and + a Therefore in dividing, it is to be observed, that like signs in the factors give in the quotient, and unlike signs in the factors give in the quotient. We have hence the following general rule for division :-Place the dividend over the divisor in the form of a fraction, and this fraction will be the value of the quotient. If either the whole numerator and denominator, or each of their terms, have a common divisor, divide by that divisor, and the result will be the quotient in simpler terms. 7 × 7 × 7 7 or to a2; the division of different powers of the same quantity, is effected by subtracting the index of the divisor from that of the dividend. any fractional expression whose numerator and denominator are equal, is equal to 1. Fractions having a common denominator, are therefore added, by placing the sum of their numerators; and subtracted, by placing the difference of their numerators, over the common denominator. Hence, when the fractions to be added, or subtracted, are of different denominators, they are reduced, as in common vulgar fractions, to equivalent fractions, having a common denominator; and then the sum or the difference of their numerators so reduced, is placed over the common denominator, for the required sum or difference of the fractions. Algebraic fractions are therefore reduced to a common denominator, added, and subtracted, exactly in the same manner as common vulgar fractions are. EXAMPLES IN ADDITION AND SUBTRACTION OF FRACTIONS. The multiplication and division of algebraic fractions are therefore performed in the same manner as in vulgar fractions. Multiplication being effected by taking the product of the numerators and denominators respectively, for the numerators and denominators of the fractional product; and division, by taking, in the same manner, the product of the dividend and the reciprocal of the divisor. EXAMPLES IN MULTIPLICATION AND DIVISION OF FRACTIONS. |