Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/481

 m A R r T H M E T I C K. Thus, 7)999^99 fix places. Thus, 27)992tliree Piacts- circle. Thus, to);050o(:o5o(2C>5v4i 142^57of figures in. a circle, . ' when37 feme power (2439, and c remains. So you may conclade, that die The number of 3 is the denominator^ may alfa be found thus: Di- finite part will confift of three places, and the circle of vide the given denominator by 9, and the number of u- Univerfally, any vulgar fraction being given, v.e may nits in the cjuot will be equal to the number of figures in determine whether the decimal thence teluking will be the circle. Thus, 9)27(3 places. Thus, 9)81(9 finite or infinite; and if infinite, whether pure or mist; places, <&c. of places, foe. in the’ following manner. If 3 divide a repeater whofe repeating figure is not a with the number the given vulgar fra&ion to its lowed terms, multiple of 3, the quot will be a pure circulate of three thenReduce divide the denominator 5, or 2, as often as places. Thus, 3).(i'xi(-037, and 3)-^55(-i85, and poifible; and if the lad quotbybejo,unity, without any re3)-/77(-2S9., not being a mainder, the decimal is finite, amd the number of diviIf 3 divide a pure circulate, the. circle the number of decimal places. multiple of 3, the quot will be a pure circulate of thrice forsIffhews ladquotbe 3, or any power of 3, the refulting as many places as the circle of the dWidend. 1 has, decimalthewill be a mixt repeater, the number of whole fi?).o37,03:7,037(012345679. will be equal to the number of divifors. Mfxt circulates take their rife from fractions whole niteIfplaces the lad quot cannot be divided by 2, 5, 10, or 3, denominators are the prime numbers 7, u, *3, 17, 19, die refulting decimal will be a mixt circulate; and the 23, 29, &c. multiplied into 2, 5, or 10, or into fome of way of finding the number of places, both in the finite their powers, or protkrdt of 9their powers. part and circle, is taught above. Examp. VI. Reduce ^ T to a decimal. the denominator of the given vulgar fraction can 28)9.o(.32,142857,14 The denominator 7% — T'X be Ifdivided, neither by 2, 5, nor 10, the refulting deci2X2, gives a mixt civculate, 84 will be a pure repeater, or a pure circulate, accorconfiding of the finite part 32, mal as the denominator is 3 or 9, or fome of the prime and a circle of fix figures or ding 60 7, ir, 13, foe.; as has been already explained. places, whofe fum is equal to numbers, 56 Every vulgar frafticn may be reduced to a decimal, the produft of 9 into half the finite that is, to a finite decimal, to- a renumber of figures; that is, 9X3 peater,or orinfintte; a circulate. For if the denominator divide =27. the numerator with ciphers annexed, fo as to leave no remainder, the refulting decimal is finite. If the remaining figure be always die fame, the refoiting decimal will be a repeater. If neither of thefe he die cafe, yet, becaufe die divifor is a finite number, the remainder at go laft mull either be the fame with the numerator of the 56 vulgar fraction, or the fame with fonae preceding remainder, and then a new circle begins ; and confequently the 240 refalting decimal will be a circulate. 324 Becaufe in circulates the circle runs on fometimes ta 16* 18, 22*. 28, 8i, 243, foe. places, and becaufe, in 160 decimals of every fort, the finite part runs fometimes on. 140 to many places, fuch circulates, or finite parts, may, without any fenfible error, be limited at five or fix places, 200 and ufed as finites: for live decimal places* divide the 196 integer into 100,000 equal parts, and all the lofs that can be occafioned by fuch limitation, is Ids than one hun*40 dred thoufandth part of the integer. And in moft cafes,, 28 the decimal may be limited at three places, which divide the integer into 1000 equal parts. 120 Circulates, or finite parts, thus limited, are called ^/>113 proximate decimals; and are fometimes marked with + or — annexed, according as the right-hand figure is taor greater than juft : for in limiting the decimal, The number of places, both in the finite part and in kenyoulefs forefoe that the fucceeding figure of the quot the circle, may be afeertained thus: Divide the deno- ifwould be 6 or 7, or any figure above 5, you lefien the minator of the vulgar fra£tion by 10, 5, or‘2, as often error by increafing as poffihle,. and the number of divifors will fnow the mate by unity. the right-hand figure of the approxinumber of places in the finite part; make the lad qnot a divifor, and the dividend any competent number of 9’s; Pros. II. To reduce the parts of coin,, foe. to decontinue die operation till o remain, and the number of cimals. 9’s ufed will be equal to the number of places in the Rule. Convert the given part or parts to a vulgarfraftiun
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