Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/485

 A R. I T H M Approximates. If the decimals to be added run on to a great, many places, it will be fufficient in moft cafes to ufe only four or five places, and obferve to increafe the figure at which you break off by an unit, if the rejefted figure on the right exceed 5. And in adding fuch approximates, omit the right-hand figure of the fum, as uncertain, but take in the carriage. Follows an example at large, and the fame contradled. contratted. Ex. at large. 12.23529 + 12.2352946 8.15789 + 8.15789325 7.08696— 7.086968435 6.32143 + 6.32143482 4-75 4-75 38.551591105 38.5515 certain. Rule II. When all or any of the given decimals are repeaters, give every repeater the fame number of places, and one place more than the longeft finite; and for every nine in the right-hand column carry 1. or to its fum add I for every nine, and then carry at ten. Examp. 7484= 748-3'3 6534= 653.06 844— 84./i 254= 25.8* aT.e’e 15S7t=i5S7.-^ In this example the fum of the right-hand column is 24, which contains 9 twice, and 6 over ; fo fet down 6 and carry 2 : Or to the fum 24 add 2, for the two nines, which makes 26; fo fet down 6 and carry 2. Proceed with the reft as in integers^ The fums, differences, and produ&s, of interminate decimals, are always interminate, unlefs they end in a cipher. A repeating digit is the numerator of a vulgar fraction, whofe denominator is 9; and hence, in adding a column of repeating digits, every 9 of the:fum is f, or an unit, to be carried ; and what is over a juft number of nines is fo many ninth-parts. Or, if to the fum of a column of repeating digits, 1 for every 9 contained in it be added, we then carry 1 for every ten; but what is over a juft number of tens will ftill continue to be ninth-parts. If in any. example the repeating figures happen all to be reiterated, the carriage from the right-hand column adjufts the column on the left, or makes every ten of them equal to an unit of the next fuperior column, tbr. Thus, if we imagine a column of the repeating figures reiterated on the right of any example, the carriage from it would adjuft the right-hand column of the example. Rule III. When all or any of the given decimals are circulates, make all the circles conterminous, find the number of tens to be carried from the left-hand column

E T I C K. 403 of the circles, add this carriage to the right hand column, and proceed as in addition of integers. If repeaters be mixed with the circulates, give the repeaters the form of circulates, by extending the repeating figures till they become conterminous with the other circles. If finite decimals are joined with the circulates, extend the finite parts of all the circulates to as many places as there are figures in==the longeft finite. Examp. 4 .428571, = .428571, 4 =.857142,= .857142, tt — .4S> — •454S4J» tt=-37°> = *370370. 2.110630 In order to find the carriage from the left-hand column of the circles, add the column next to it on the right, faying, 7 + 5 + 5+ 2 = 19; from which carry 1, and fay, 1+3+4 + 8 + 4 = 20; from which carry 2, and go on to add the right-hand column of the circle, faying, the carriage 2 + 5 + 2 + 1=10; fofet down o, and carry 1, and proceed with the reft as in integers. The adding the carriage from the left-hand column of the circles to the column on the right hand, arifes from the flux of numbers; for as the circles repeat infinitely, if we fuppofe a new fett of the fame circles to be repeated upon the right of our examples, it is plain, that in adding them the carriage from the left-hand column of the new fett would naturally fall into the right-hand column of our example. The operation here is the fame as in addition of vulgar fra&ions ; for every circle is the numerator of a vulgar fradlion, whofe common denominator is 999999 ; and if the circles or numerators be added, without minding any carriage from the left-hand column, the fum will be 2110628. And 999999)2ii°628(244444f 1999998 110630 But, by pointing off from the fum of the circles fix figures towards the right, we divide.by 1006000, inftead of dividing by 999999 ; which gives indeed the fame quot, but makes the remainder too fmall. Now, that the carriageTfigure from the left-hand column of the circles, is the integral part of the quot, and at the fame time the difference between the true and falfe remainder, is evident; for the quotient-figure 2, muHiplied into the two. divifors 1000000.and 999999, gives two produdlsj whofe difference is 2; and confequently, if the greateft product, viz. 2.X 1000000= 2000000, be fubtradted from the dividend, the refult. will want 2 of the true remainder. To prevent fuch errors, and to put the work on a fure. footing, find the. carriage from, the left-hand column: of the circles, add this carriage to the right-hand column, divide the fum. by 1000000, and you will have a true quot,' and a true, remainder. The learner may look back to divifion <oP integers,.