| XXX | (398) | XXX |
59.3 A R I T H M E T I C K. and .0^99 = .1; and .0099 = .or. and 44.^9 = 45. We nou' refolve fa :h denoiflinator5 iftto their cdmpo- I; Hence may be afcertained the value of an infinite nertt parts, and divide the numerator by' one of thefe parts, and then divide the quot by the other. Thus, feries1 decreafmg in a deekple pioporrion. Thus, T%. + tbu *1" tuW) — 'J* And -j^. -f- -f C 15-5X3. 5)4.0( 8 and 3).8(.2^ If the denominator of a vulgar fradlion be neither 2 40 6 nor 5, ftor any power of 2 or 5, nor any produdl of
their powers ; nor 3, nor 9, nor any prodddl "&f 3 into (o) 20 2 or c, or into fome of their powers, or product of 18 their powers, the decimal feftlhing from fuch a vulgar fraction will circulate. ~V) like repeaters, are of two forts,- viz. pure The number of places in the finite part of a mixt re- andCirculates, A pure circulate coofills .of the figures of peater may be afcertained from the number of units in the mix-t. circle only; as .09, 09, <6r. or 18, 18, A the index of the powers of 2 or 5. circulate has a finite part betwixt the decimal point And, univerfally, to find the number of places in the mixt and the figure that begins the circle; as .0, 4;, 45, finite part of fuch fractions, divide the denominator firfl
- or .32, 142857, 143857, drr. Some ebufe to
fey'3, and thefi divide the quot by 2, 5, or 10, till the diftinguifh the finite part from the circle, and one circle laft quot be 1, and o remain; and the number of divi- from by a comma, as above. Others dafii the fors, excluding 3, fhows the number of places in the fi- find andanother, lad figure of the circle. It is likewile ufuaJ to nite part. mark the remainder where the new circle begins, by afRepeating decimals are ufualiy marked by a dafh an afteriilc. through the right-hand figure, as in the examples a- fixing Examp. V. Reduce TV to a decimal. bove: But fome chufe to mark them by a point fet over The denominator it gives a pare 1 i)».oo(.09.<>9, 99 the repeating figure, thus, .3, .26. The remainder circle of two figures. Where the repetition begins is commonly marked with
- 100
an afterifk. Becaufe arty quotient multiplied by the divifor reproduces the dividend, it follows, that any decimal multiplied by the denominator of the vulgar fraftiofl from which it refulted, will reproduce the numerator with the jV := -(>3> ^3* t*t=;.i8, 18, -72, 8l, 72, TT4 — -27, 27, TT= annexed ciphers. Thus, if .75, the decimal of -J-, be TfT= .Bl, multiplied by 4, it will reproduce the numerator 3 and tbs v r — •36> 36, two annexed ciphers,. tt — *45» 4S» 44 = .90, 90, Now, fuppofe the given decimal to be a repeater } A—•54>I4> to perceive, that if any of the vulgar fracfach as .3-, refilling from the vulgar fraflion if the tionsIt isin eafy the above fpecimen have both its numerator and repeating decimal be multiplied by the denominator 3, denominator by 9, there wilt arife a new vulit will, by carrying at 9.0a the right hand, reproduce gar fradtion ofmultiplied fame value, whofe numerator will be the numerator 1 with the annexed cipher. In like man- the figures of the circle, and its denomirator the like ner, if the repeater .# — 4* multiplied by 3, it will, number of 9’s. theThus, by carrying at 9 cm the right hand, reproduce the nu3X9 , , an<,.7X9 „, merator 2 with the annexed cipher.. Again, if the repeater be multiplied by the denominator 9, it will, by iO<9^’ iiX9^ carrying at 9 on the right hand, reproduce the numera- As the denominator 11, whereof 90 is a multiple, tor 1 with the annexed cipher. And, if the mixt re- gives a pure circulate of two places, fo any denominapeater .2^ = ■/(-> be multiplied by the denominator 15, tor, whereof 999, or 9999, or 99999, <5rc. are multiit will, by carrying at 9 on the right hand, reproduce ples, will give a pure circulate of three, four, five, &c. the numerator 4 with the two annexed ciphers. places; that is, of as many places as there are p’s si From thefe remarks we may conclude, that the right- the multiple. And fuch denominators are all the prime hand figure of every repeating decimal is ninth-parts: numbers, except 2, 3, and 5,. viz. 7, 11, 13, 17, 19, and the fame truth may be evinced by refotving the de- 23> 29, 31,, 37, 41, &c.) alfd their products into 3, cimal into its conftituent parts,, in the following manner. viz. 21, 33, 39, 51-, 57, 69, &c. Such too are all The vulgar fraction reduced to a decimal gives the powers of 3, except 3 and 9, viz. 27, 81, 243, ^7, 6^.; and this repeater refolved into decimal confti- 729, 2187,. 6c. tueftt parts, becomes -xo +-rJo-+-rouo > to infinity. The reafen is plain : for if any divifor, as 37, divide But if we efteem the right-hand figure to be ninth-parts, 999, without a remainder, it will alfo divide 1000, and we have To- + °f-r?> —tV =+ -ps5 = leave a remainder of 1, to begin a new circle. the ves given vulgar fradtion. And as the vulgar fradlion To find how many places the circle will coofifi cf, divide a competent number of 9’s by any of the above '4univerfally, - g> -i7’ afoferies
- 8ives
'99
tbat is, .$9 = £= 1. And, of nines infinitely continued is equal dfenominators,. continuing the operation till ©.remain ; and to auity in the place on the left hand; thus, .$99=: the number of o’s ufed will fliow the. number of places. Thus,,
