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 ARITHMETIC K. 405 General Role. the'circles', 8 cannot be fubtrafted from' 2 without borrowing ; therefore add i to the right-hand figure of the Give fo many decimal places to the produtf, on the minor, andfay, i+2=3, and 3 from 4, and 1 remains. righr, as are in both factors; and if the product ha-s The reafon is obvious : for^ fuppofing the circles reitera- not fo many figures, fupply that defed by prefixing cited on the right of the example, it would be, 8 from 2 phers. you cannot, but 8 from 13 and 4 remains; 1 borrowed, We now proceed to multiplication. Rule I. If both fadtors are finite or approximate, and 2, make 3, fcc. work exadly as in multiplication of integers. Example II. Ex. 1. Ex. 2. From .785 .125 Sub. ■} —4 =.{£,666666, •75 -2> Rem. .2,619047, 3925 625 In the above example the repeaters ape given in the 5495 250 form of conterminous circulates. Example III. .58875 .03125 From -/T=.38461 $, = .384,615384, In Ex. 2. the produdt not affording fo many deeim d Sub. -y = .125 =.125 places as are in the multiplicand and multiplier, the defedt is fupplied by prefixing ciphers. Rem. .259,615384, The reafon of giving as many decimal, places to-the In the lafl: example the finite part of the circulate is ex- product as are in both fa&ors, appears by'cotifidering tended to as many places as there are figures in the fi- that the operation is the fame here as in multiplication of nite decimal, by which means iijce things come to be vulgar fractions. fubtradted, and you obtain the exadt circle of the re- t-qoJss = • 5 88 7 5. Thus, .785 X .75= ToVa X— mainder. To multiply by 10, 100, 100, move the decimal , point fo many places toward the right hand as there are V. Multiplication of Dec'unals, ciphers in the multiplier. And thus: In multiplication and divifion there may happen nine .4375 XThus io- = 4.375 6.875X10=68.75 varieties, arifing from the different 'nature of the numbers that may occur in the operation ; and thefe are of • .4375X100 = 43.7$ 6.875X100 = 687.5 three forts, viz. integers, mixt numbers, and pure deApproximates. cimals. approximates, the certain places of the Now, fince the multiplier or divifor may be of three In multiplying may be determined by one or other of the two kinds, and the multiplicand or dividend of as many, produdt viz. there muft of confequence be nine varieties; which are rules1. following, If, both fadtors are approximates, the uncertain thefe following. 4 places of the produdt will be one more than the number Tan integer, pf places in the longeft fadtor, An integer may multiply or divide ^aa mixt ur number, 2. If one of the fadtors be finite, and the other ap^ CC anPinteger, e decimal. proximate, the uncertain places of the produdt will be A mixt number may multiply or divide -< a mixt number, one more than the number of places in the finite fadtor. Ex. 1, Ex. 2. Ca pure decimal, 245.118— .2105264C an integer, A pure decimal may multiply or divide < aa mixt number, .3529+ 2.875 C pore todecimal, ■ Of thefe.varieties, the firft belongs' properly vulgar 2206062 1052630 arithmetic, the other eight occur in decimal operations. 490236 1473682 But in multiplication and divifion of decimals, there 1225590 1684208 will occur other nine .varieties, arifing likewife from the 421052 735354 nature of the mvnbers; which may either be finite, repeating, or circulating. 86.5021422 .605262250 And fince the multiplier or divifor may be of three Ex. 1. the integral part of the produdt, vrzi 86, forts, and the multiplicand or dividend of as many, there is Incertain, and all the decimal places on the right arc muft of courfe be nine varieties; 1 and thefe are fo obIn Ex. 2. only four places on the left, viz. vious, that it would be lofing time here to enumerate Uncertain. .6052^ are certain, and all the other places uncertain. them. The reafon of Rule 1. is plain. For if in Ex. 1. we Before entering on multiplication, we fhall lay down make the longeft fadtor the multiplier, and the total a rule for pointing the prodiidf, which is of a general na- produdt will be the fame either way, it is'obvious,••that ture, and extends to decimals of every fort, whether fi- in this cafe we (hall have fix part(:ular produdts, in each nite, repeating, or circulating.; and is as follows. Of which the right-hand figure will be uncertain, and Vol. I. Numb. 17. g 5K confequently