There are four parts to be noticed in Division, viz. The Quotient, or answer to the question, shows how many times the divisor is contained in the dividend. If there be any thing left after the operation is performed, it is called the Remainder; sometimes there is a remainder and sometimes there is not. The remainder, when there is any, is of the same denomination as the dividend. Division is both Simple and Compound. PROOF. Multiply the divisor and quotient together, and add the remainder, if there be any, to the product; if the work be right, that sum will be equal to the dividend. SIMPLE DIVISION Is the dividing of one number by another, without regard to their values: As 56, divided by 8, produces 7 in the quotient : That is, 8 is contained 7 times in 56.* RULE. Having drawn a curve line on each side of the dividend and placed the divisor on the left hand of it, Seek how many times the divisor is contained in the least number of the figures of the dividend on the left hand that do contain According to the rule, we refolve the dividend into parts, and find, by trial, the number of times the divisor is contained in each of thofe parts; and the only thing which remains to be proved,is, that the feveral figures of the quotient, taken as one number, according to the order, in which they are placed, are the true quotient of the whole dividend by the divifor; which may be thus demonstrated. Dem. The complete value of the first part of the dividend, is, by the nature of notation, 10, 100, 1000, &c. times the fimple value of what is taken in the operation; accordingly, as there are 1, 2, or 3, &c. figures ftanding before it ; and, confequently, the true value of the quotient figure, belonging to that part of the dividend, is alfo 10, 100, 1000, &c. times its fimple value; but the true value of the quotient figure, belonging to that part of the dividend, found by the rule, is alfo 10, 100, 1000, &c. times its fimple value; for there are as many figures fet before it, as the number of remaining figures in the dividend: therefore the first quotient figure, taken in its complete value from the place it stands in, is the true quotient of the divifor in the complete value of the first part of the dividend. For the fame reafon, all the rest of the figures of the quotient, taken according to their places, are, each, the true quotient of the divifor, in the complete value of the feveral parts of the dividend belonging to each; because, as the first figure, on the right hand of each succeeding part of the dividend, has a lefa number of figures standing before it, so ought their quotients to have; and fo they are actually ordered; confequently, taking all the quotient figures in order, as they are placed by the rule, they make one number, which is equal to the fum of the true quotients of all the feveral parts of the dividend; and is, therefore, the true quotient of the whole dividend by the divifor. That no obfcurity may remain, in the demonstration, it is illustrated by the following example." E it, and place the answer on the right of the dividend for the quo tient; multiply the divisor by this quotient figure, place the product under those left hand figures of the dividend; then subtract it therefrom, and bring down the next figure of the dividend to the right hand of the remainder: If, when you have brought down a figure to the remainder, it is still less than the divisor, a cypher must be placed in the quotient, and another figure be brought down; after which, you must seek, multiply and subtract, till you have brought down every figure of the dividend: tients, or, the Answer. Explan. It is evident the dividend is refolved into thefe parts, 74000+500-+004-3; for the first part of the dividend is confidered only as 74; but yet it is, truly, 74000; and therefore its quotient, instead of 2, is 2000, and the remainder 24000; and fo of the reft; as may be seen in the operation. When there is no remainder to a divifion, the quotient is the abfolute and perfect answer to the question; but where there is a remainder, it may be obferved, that it goes fo much towards another time as it approaches the divifor; thus, if the remainder be half the divifor, it will go half of a time more, and fo on; in order, therefore, to complete the quotient, put the laft remainder to the end of it, above a line, and the divifor below it. Hence the origin of vulgar fractions, which are treated of hereafter. It is fometimes difficult to find how often the divifor may be had in the numbers of the several steps of the operation: The best way will be to find how often the first figure of the divifor may be had in the first, or two first figures of the dividend, and the answer, made lefs by one or two, is, generally, the figure wanted: but, if, after subtracting the product of the divifor and quotient from the dividend, the remainder be equal to, or exceed the divifor, the quotient figure must be increased accordingly; or, if the product of the divifor and quo tient figure exceed the dividend, then the quotient figure must be proportiona bly leffened. The reafon of the method of proof is plain; for, fince the quotient is the number of times the dividend contains the divifor, the product of the quotient and divisor, must, evidently, be equal to the dividend. 1. Pivifor. Dividend. Quotient. 3)175817(58605 15 25 24 Proof. 18 18 T 17 15 2 Rem. 58605 Quotient. X3 Divisor +2 175817 EXAMPLES. In this example, I find that 3, the divisor, cannot be contained in the first figure of the dividend; therefore, I take two figures, viz. 17, and inquire how often 3 is contained therein, which finding to be 5 times, I place the 5 in the quotient, and multiply the divisor by it, setting the first figure of the multiplication under the 7 in the dividend, &c. I then subtract 15 from 17, and find a remainder of 2, to the right hand of which I bring down the next figure of the dividend, viz, 5; then I inquire how often the divisor 3, is contain. ed in 25, and, finding it to be 8 times, I multiply by 8, and proceed as before, till I bring down the 1, when, fiuding 1 cannot have the divisor in 1, I place 0 in the quotient, and bring down 7 to the 1, and proceed as at the first. Observe, that, in multiplying by 3, I add in the 2. There are several other methods made use of to prove division; as follow, viz. RULE I. Subtract the remainder from the dividend; divide this number by the quotient, and the quotient, found by this divifion, will be equal to the former divifor, when the work is right. RULE II. Add the remainder and all the products of the feveral quotient figures multiplied by the divifor together, according to the order in which they ftand in the work, and the fum, when the work is right, will be equal to the dividend. Here the numbers to be added are the products of the divifor by every figure of the quotient, feparately; and each by its place, poffeffes its complete value; therefore, the fum of the parts together with the remainder, must be equal to the whole. I will illuftrate the whole by an example proved according to the feveral different methods. 123456789)121932631112635269( Some operations are more readily performed by the following particular rules. CASE I. When there is one cypher, or more, at the right hand of the divisor: It or they must be cut off; also cut off the same number of figures from the dividend, and then proceed as in Case first: But the figures which were cut off from the dividend must be placed at the right hand of the remainder.f 9 8 7 6 5 4 3 2 1 Proof by Addition. We need only to refer to the example, except for the proof by addition; where it may be remarked, that the Afterifms fhew the numbers to be added, and the dotted lines their order. + The reafon of this contraction it is eafy to conceive; for cutting off the fame figures from each, is the fame as dividing each of them by 10, 100, 1000, &c. and it is evident, that as often as the whole divifor is contained in the whole dividend, so often must any part of the divisor be contained in the like part of the dividend; this method is only to avoid a needless repetition of cyphers, which would happen in the common way, as may be feen by working one of the examples of this cafe in the common way without cutting off the cyphers. Note. In dividing by 10, 100, 1000, &c. when you cut off as many figures from the dividend, as there are cyphers in the divisor, your work is done; those figures, cut off at the right hand, are the remainder, and those on the left, the quotient, as above. CASE II Short Division may be used when the divisor does not exceed 12. it is performed by the following RULE. First, seek how often the divisor can be had in the first figure, or figures of the dividend; which, when found, place in the quotient; then, mentally, multiply your divisor by the figure placed in the quotient, and subtract the product from the like number of the left hand figures of your dividend, and the remaining units, if any, must be accounted so many tens, which you must suppose to stand at the left hand of the next figure in the dividend, and to be reckoned with it; then, seek how often you can have your divisor in those two figures; but, if nothing remain, you must then seek how often your divisor is contained in the next figure, or figures, and thus proceed till you have done. |