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 a r r II M E T I C K. 4°4 integers, where the method of dividing by 9,59,9^9, &c. Example I. is explained. L. d. L. Hence it follows, that if we add the cirFrom 48 6 =48.525 cles as they (land, without minding any carSub. 8 12 8^=18.634375 riage from the left, and to the fum add the excrefcent "figure on the left of the decimal Rem.. 29 17 94 = 29:890625 point, we (hall have the full fum of the Exam p l e II. 2.110628 circles, both as to the integral and frac2 tional part, as jn the margin. 67. lb. C. r rom 54 2 21 = 54.6875 2.110630,, Sub. 36 3 . 14 = 36.875 Pure repeaters, being the numerators of vulgar fractions, whofe denominator is 9 as often taken as the digit Rem. 3 17 = 17.8125 is repeated, may be added in the fame manner as circles. But in examples clear of circulates, the method prefcriApproximates. bed in Cafe II. is preferable. approximates, negleft the right-hand fi.857142, .6666, In adding circles and pure re- gureIn offubtra&ing as uncertain; but an unit bor.571428, .6666, peaters by the method now ex- rowed onthetheremainder, right .857142, .6666, plained, it will fometimes happen ples following. muft»be repaid, as in the two examthat the fra&ional part of the .714285, Ex. 1. Ex. 2. 1.9999, fum will be a feries of nines, From 783.0625 From 549,4643 as in the margin: And in this Sub. 495.28571 42.999999, Sub. 78.0875 cafe, the numerator of the fraction being the fame with the de287.7767 certain. Rem. 471.376 certain. 3.000000 nominator, its value will be uni- Rem. II. If one of the given decimals is a repeater, ty; and accordingly 1 muft be andRule the finite'decimal, give the repeater one place added to the integral part. But in adding pure repeaters more thanothertheafinite decimal, and in fubtraiting borrow by the method in Cafe II. this cannot happen. 9 on the right hand. By way of proof, we fhall here add all the vulgar if both major and minor repeat, give them an efractions in Examp. I. and reduce their fum to a mixt qua!Butnumber of places, and then fubtraft as above. number, continuing the divifion to a decimal. Ex. 1. Ex. 2. Ex'. 3.’ 4 + t =+ tt + Ir = tVtYt 4* '4rff + AVtt 4" From •7i4583'3 .525 .9989583Sub. .634375 Trrrr ■2333 .0291666 14553)30716(2.110630, Rem. .0802083' .1916 .9697916 29106 In Ex. x. and 2. you give the repeater one place more than the finite decimal, and by this means you obtain the repeating figure of the remainder. Butin Ex. 3. you give the two repeaters an equal number of places. In Ex. 2. and 3. you borrow 9 on the right hand. Rule III. If both the given decimals be circulates, make the circles conterminous, and work as in integers; only if, in the left-hand column of the circles, you forefee, that, in fubtradting the figure of the minor from that of the major, one muft be borrowed, in,this cafe add Here the dividual being the I to the right-hand figure of the minor, and then fubfame with the fecond, a new tradl. circle begins. If one of the given decimals be a circulate, and the other a repeater, give the repeater the form of a conter*16100 minous circulate, and then fubtradt as above. If one of the given decimals be a circulate, and the IV. Svbtraftion of Decimals. other a finite decimal, extend the finite part of the cirRule I. Place the minor under the major, fo that culate to as many places as there are figures in the finite that the points may be in one column; and then, if the decimal, and then fubtradt. given decimals be finite or approximate, work as in fub- Examp. I. From ^=.6,428571, =.64,2857x4, traftioir of integers. Sub. -4^=.17,857142*=.17,857142, If the major and minor have not the fame number of places, imagine the void places to be filled up with ciRem. .46,428571, phers. In this example, becaufe, in the left-hand column of the