Third example. 2789 9639475(3456,621. 12724 15687 17425 691 92. In the preceding operations of arithmetic it has been shown that the calculations may be made either from left to right, or from right to left; but that the latter course is the more convenient. The calculation in division may, in like manner, be carried from right to left; for the divisor may be subtracted from the dividend and remainders time after time till the latter are exhausted, and the times of subtraction assembled to form the quotient; or the divisor, and its multiples by 10, 100, 1000, &c. &c. may be subtracted as often as possible, and the ones, tens, hundreds, &c. &c. combined into a quotient number. In either case the calculation may proceed from right to left. But the calculation by the rule of division, properly so called, is carried from the left to the right of the dividend. The reason for so doing is this: the dividend is the sum of the partial products of the divisor by the units, the tens, the hundreds, &c. &c. of the quotient. In the addition of these partial products, tens arising from the product of units of the first order are combined as ones with units of the second order, tens of the second with units of the third, &c. Now, in the process of decomposition the tens carried must be restored to the units of that order from which they came; and it is only by commencing the decomposition with the highest figures of the dividend that these tens can be disengaged in such a manner as to admit at once of combination with the units from which they are derived, and to which they belong. 93. To divide the greater of two whole numbers by the less. General Rule. Arrange the numbers thus, Or thus, Divisor) Dividend (Quotient. Divisor. Dividend (Quotient. Take from the left of the dividend as many figures as are contained in the divisor, or (when the absolute number expressed by these figures is less than the divisor) one figure more. These form the first partial dividend, of which, also, the last figure has the same relative value as the first figure of the quotient. Find by trials (or by a table as in Article 88) how often this partial dividend contains the divisor: the figure expressing the number of times is the first partial quotient. Write this figure in its proper place; multiply the divisor by it, and subtract the product from the first partial dividend. Annex the next figure of the dividend to the remainder, to form a second partial dividend. If the second partial dividend is greater than the divisor, find, as before, how often it contains the divisor. Annex to the first partial quotient the figure expressing the number of times; multiply the divisor by it, subtract the product from the second partial dividend, and to the remainder annex the next figure of the dividend. The third partial dividend is thus obtained. But if the second partial dividend is less than the divisor, annex to the first partial quotient a zero, to preserve the place of that order of units which is not expressed by a significant figure; and bring down the next figure of the dividend, as if the second partial dividend were a remainder. In this manner, whether the second partial division gives to the quotient a significant figure or a zero, the third partial dividend is attained. The process with the third partial dividend, the fourth. . . . the last, is the same as with the second. When the last remainder is zero the division is exact; when it is composed of one or more significant figures, these are written over the divisor, and annexed to the quotient, in order to complete it, as in the example of Article 81. 94. Exercises in the division of whole numbers: 40th. 41st. 38th. 274616097456÷ 579486= 473896. 39th. 38110890724211194÷594786043= 64074958. 29857547202÷ 498673= 59874. 962674279624÷ 24373969= 39496. 95. If two factors are multiplied together, and the product is divided by one of the factors, the result is the other factor. Whence, if any number of factors are employed to form a product, and this product is divided by these factors one after another in a reverse order, it is evident that the products in a reverse order, and the original multiplicand, must be in succession recovered. Now, to form the product of a multiplicand by 2, 3, 4, . . . . factors, either the multiplicand is multiplied by one factor, the product by a second, this by a third, &c. &c., in any order; or the product of all the factors is formed, and the multiplicand multiplied by this product (Art. 71). Consequently, if the product of any multiplicand by several factors is divided by these factors one after another, the last quotient is the multiplicand; or, if the same product is at once divided by the product of all the factors, the quotient is also the multiplicand. Whence, it is indifferent to divide a number by several divisors in succession, or at once by the product of these divisors. 96. It is sometimes convenient to break down a large divisor into factors each less than 10, and to divide by these factors one after another, instead of by the whole divisor. It follows from the last article that the results given by the two methods are the same. But when the calculation by the unbroken divisor ends in a remainder, that by the factors of the divisor must exhibit remainders terminating some or perhaps all the divisions. From these remainders it may be necessary to find the whole remainder left when the division is performed at once. The remainder from the division of the dividend by the first factor of the divisor is evidently composed of units of the dividend; and the remainder from the division of the first quotient by the second factor of the divisor is, in like manner, composed of units of the first quotient. Similarly the remainder from the third division is composed of units of the second quotient, &c. &c. The manner, therefore, of reducing the remainder of the second division into units of the first dividend being discovered, the other reductions must be made in the same way. Whence it is sufficient to investigate the means of reducing this remainder. Then, since D is composed of d repeated as often as there are units in q, it follows that to 1 unit in q correspond 1 xd units in D; consequently, from any number in q to recover the corresponding number in D, it is necessary to multiply the former by d. 97. As the whole calculation may be best explained through the medium of an example, let it be required to divide 359482 by 105, which is equal to the continual product of the factors 3, 5, and 7. Calculation: 3)359482 5)119827+1 first remainder. 7)23965+2 second remainder. 3423+4 third remainder. Explanation. The quotient resulting from the division of 359482 by 3 is 119827, and the remainder 1. This remainder (Art. 96) is 1 unit of the dividend. The quotient resulting from the division of 119827 by 5 is 23965, and the remainder 2. This remainder is composed of units of the first quotient; and I unit of the first quotient corresponding to 3 units of the dividend, the second remainder, 2, corresponds to 2x3=6 units of the dividend. The quotient resulting from the division of 23965 by 7 is 3423, and the remainder 4. This remainder is composed of units of the second quotient. Now 1 unit of the second quotient corresponds to 5 units of the first : Therefore 4 units of the second quotient correspond to 4×5= 20 units of the first. Again, 1 unit of the first quotient corresponds to 3 units of the dividend : Therefore 20 units of the first quotient correspond to 20×3=60 units of the dividend. Whence 4 units of the second quotient correspond to 60 units of the dividend. The 1st remainder is equal to 2d 3d The sum of the three remainders is 67 The whole remainder which would be obtained if the division were performed at once is therefore 67 : And 359482+105=3423-67 The several remainders may be reduced into one number briefly, as follows: 3d rem. 2d div. 2d rem. 5 20, and 20+ 2 =22. 98. If any of the divisions are exact, let 0 be written as the remainder. This done, the following practical rule for the reduction of all the remainders into one is perfectly general: Multiply the last remainder by the last divisor but one, and to the product add the preceding remainder; multiply this sum by the next preceding divisor, and to the product add the next preceding remainder; repeat this process until all the divisors and remainders are included. The last result is the number sought. 99. It is only when a number can be broken into factors, each not greater than the greatest number contained in the first horizontal or first vertical column of a multiplication table, that the employment of this method can bring any practical advantage. Whether a given number can be decomposed into such factors may be ascertained by trials, thus: it may be divided by 2 as often as the division can be made exactly; then by 3, 5, 7, &c. in the same manner. If a last remainder equal to zero is found, the product of the divisors is equal to the given number. Some numbers cannot be broken into factors, being divisible by themselves only, and by unity, which is a divisor of every whole number." 8 100. The division of one general expression of quantity, a, by another, b, is indicated thus, Supposing the division exact, and the quotient c, the preceding expressions The division being inexact, the quotient c, and the remainder d, they When the quantities composing the dividend and divisor are unlike, the division and its result must be represented in one or other of the preceding forms; but when they are composed of similar factors, the quotient is expressible in terms of the common quantity. When the dividend, therefore, and divisor are powers of the same root, the quotient is obtained by giving that root an exponent equal to the difference between the exponents of the dividend and divisor. a. If the dividend is composed of the product of two or more factors, and the divisor involves one or more factors which are either similar to the factors of the dividend or are powers of the same root, the fractional expression of the quotient admits of the same kind of simplification: a2b2 a2 Thus, a2b2÷a2=" = a2 a2 ·xb2=1xb2=b2. For the continuation of this subject see Article 154, on the reduction of fractions to the lowest terms. b. The product of a multiplicand composed of several terms by a multiplier consisting of one term is obtained by multiplying each term of the multiplicand by the multiplier (Art. 76 c). Consequently, in division (Art. 79), the dividend being expressed by several terms, and the divisor by one term, the quotient is a quantity consisting of as many terms as the dividend; and the product of the first term of the quotient by the divisor is the first term of the dividend; the product of the second term of the quotient by the divisor is the second term of the dividend, &c. &c. Whence, the divisor and dividend being given, the quotient in this case is obtained by dividing each term of the dividend by the divisor, and taking the sum of the partial quotients for the result required: a5 a2b2 ab a+ (a3+a2b2+ab)+ab=ab + ab +ab=b+ab+1. For the division of one polynomial expression by another see Part II. |