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 K, A R I r H M 2 2 I4 397 = io X2 ; and the fum .of the indices, 2+4 2 or 5, or the produft of fome of their powers. See = 6, gives the number of decimal places, viz. -rvav — Chap. XII. and Algebra, Chap. III. The powers of numbers are fometimes exprefled by .000625. And, in general, to find what number of decimal indices or exponents placed at the corners of the num-3 places any fuch vulgar fraflion will give, divide the de1 bers. Thus, 22 fignifies the fecond power of 2^ and 5 by 2, 5, or 10, till the lafl: quotient be i, and fignifies the third power of 5; and 104 fignifies the fourth nominator the remainder o; and the number of divii'ors (hews power of 10, &c. The index of the root or firlt power the number of decimal places. Thus, -s*^ gives a d.eciis feldom exprefled. 2)2)2) Any power of 2 multiplied into the like power of 5 gives a product equal to the fame power of 10; as ap- mal of four places; for 2)16(8(4(2(1. And gives pears from the following fpecimen of the powers of 2, a decimal of three places; for 5)125(25(5(1., s)s) And 5, and 10. a 1 TtRrtr decimal of three places; for io)xooo 2 X y= 10 = 10)10) 2 r= s 4 X 2*=I03 = 100 532= And -rg-Vs- gives a decimal of fix places ; 12*= 104 =: 1000 (100(10(1. 10) S = 12* 168 XX 62*=io 2) 2)2)2) 24s — l6 5*— ! s = 10000 for 10)1600(160(16(8(4(2(1. 62* 2 = 32 * = 3125 32X 3125=: 106 = 100000 If the denominator of a vulgar fraflion be neither 2 2®— 64 *®= 1562* 64 X 1*625= 10 = 100000b nor 5, nor any of their powers, nor produfl of their 27 = I28 *7 = 78i2* 128X7812*= to7 = IOOOOOOO powers, fuch a denominator Will not divide the nume6-c. &c. rator with annexed ciphers without a remainder; and The produfl: of two different powers of 2 and 5, is e- the decimal thence refulting is called infinite, or interqual.to the product that will arife by raifing 10 to the minute. Of infinite or interminate decimals, there are two power denoted by the lefler given index, and then multi- forts. fome conftantly repeat the fame figure; and plying this power of 10 into that power of the other num- are calledForrepeating repeaters, or fingle repeber which is denoted by the difference of the two given tends. Others repeatdecimals, a circle of figures ; and on that exponents. Thus, <s , , 4 account are called circulating decimals, circulates, or 2 X5 =64X 25 = IO X22 =4 100X 16 = 1600 repetends. 2* X *® = 4 X i*625 = io X* = 100X625 = 62500 compound Examp. III. Reduce to a decimal. From thefe remarks it is eafy to perceive, that 2 or Here the remainder being ftill the fame, 3)l-o(..3’ 5, or any of their powers, or product of their powers, viz. 1, the fame figure will conftanfly be re-. 9 will meafure 10 or its powers, v/z. 100, 1000, &c. peated in the quot. or their multiples, fuch as, 20, 200, 2000, <&c. 30, 300, 3000, lire.; and fuch every numerator becomes Repeating decimals are of two kinds: viz. fomeU)conby having ciphers annexed; and therefore 2 or 5, or fid only of the repeating figures, fuch as the examples their powers, or produdt of their powers, ufed as a de- above; and thefe are called pure repeaters; others have nominator, will divide" any numerator with a competent one or more digits or ciphers betwixt the decimal point number of ciphers annexed, and leave no remainder ; and the repeating figure; and thefe are called mixt reand confequently the decimal thence refulting will be peaters ; and the digits or ciphers on the left of the refinite. peating figures are called the finite part of fuch deciIf the numerator of the vulgar fraftion be unity, and mals. the denominator any Angle power of 2 or 5, there will Pure repeaters take their rife from vulgar fraflions be as many decimal places in the quot as there are4 units whofe denominator is 3, or its multiple 9; and are but in the index of the given power. =:Thus, i6 = 2 gives few in number. a decimal of four places, viz. tV- *0625 ; and, 125 Mixt repeaters derive their origin from vulgar frac= *3 gives a decimal of three places, viz. x-rT=.oo8. tions whofe denominator is the produfl of 3 into 2 or When the denominator is the product of like powers 5, or into fome of their powers, or produfl of their of 2 and 5 ; in this cafe,, fuch a produft being equal to powers ; and fuch denominators may be confidered as the like pov, er of 10, and any power of 10 being equal the produfl of two component parts, whereof one is 2 to 1, with as many ciphers annexed as there are units in or 5, or fome of their powers, or produfl of their powers ; the index, it follows, that there will ftill be as many de- and hence the finite part. The other component part is cimal places in the quot as there are units in the 3index,3 3 ; and hence the repeating figure. either3 of 2, of 5, or 10. Thus, 8 X 125 = 2 X * Examp. IV. Reduce -fiT to a decimal. =io =iooo, gives a decimal of three places, viz. Here the repeater is mixt, the finite part 15)4.o(.2$ being 2, and the repeating figure 6. 30 'TSotS — .OOI. When the denominator is the product of different powers of 2 or y, find what power of 10, and what power of 2 or 5, upon being multiplied, will give the 90 fame produfl, as is taught above; and the fum of the indices' (hews the number of decimal places ; thus, (10) Vol. I. No. 17. 3 5H