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.003=ph, and so on. This correspondence exists universally; and, therefore, any circulate-not containing an integer- is equal to a vulgar fraction, whose numerator is the circulating figures, and whose denominator is denoted by as many 9s as there are places in the circulate.
RULE. Make the repetend the numerator, and for the denominator take as many 98 as there are figures in the repetend.
When there are integral figures in the repetend, a number of ciphers equal to the number of integral figures must be annexed to the numerator.
1. Reduce .6 to a vulgar fraction.
2. Reduce .037 to a vulgar fraction; giving the fraction in its lowest terms.
3. Reduce.i23 to a vulgar fraction.
CASE II. To reduce a mixed infinite decimal to a vulgar fraction.
Observe, that a mixed infinite decimal consists of two parts—the finite part, and the repeating part. The finite part may be reduced as shown in Art. xi, Case 1; and the repeating part, as shown in the first case of this article; observing, however, to reckon the value of the fraction obtained from the repeating part ten times less for every place occupied by the finite figures. For example, the decimal .26 is divisible into the finite decimal .2, and the repetend .06. Now .2=to, and .. would be=f, if the circulation began immediately after the place of units; but since it begins after the place of tenths, it is of to =%. Then, .26 is equal to to +6=+=.
To as many 98 as there are figures in the repetend, annex as many ciphers as there are finite places, for a denominator. Then, multiply the same number of 98 by the finite part of the decimal, and add the repetend to the product, for the numerator.
7. What is the least vulgar fraction equal to .13?
CASE III. To make any number of dissimilar repetends, similar and conterminous.
Observe, that a single repetend may be represented either as a compound repetend or as a mixed decimal ; thus, ..=.666=.66666.
Also, a compound repetend may be represented as a mixed decimal; thus, .243 = 24824 5.24824824. Also, a finite decimal may be represented as a mixed infinite decimal, by annexing ciphers as repetends; thus, .39=.390=3900 .390000 Hence, two or more decimals, whether repetends, circulates, or mixed decimals, may be expressed with circulating figures beginning and ending together.
RULE. Find the least common multiple of the several numbers of decimal places in the several repetends; extend the repetend which begins lowest to as many places as the multiple has units, and make all the other repetends to conform thereto.
13. Make 6.317, 3.45, 52.3, 191.03, .057, 5.3 and 1 359 similar and conterminous. 6.317= 6.31731731
The first repetend has
3 places; the second, 2; 3.45 3.45555555
the fourth, 2; the fifth, 3; 52.3 52.30000000 the sixth, 1; the seventh, 191.03 =191.03030303 1. The least common
.057 .05705705 multiple of 3, 2, 2, 3, 1, 5.3 5.33333333
1, is 6; therefore the simi
lar and conterminous repe1.359= 1.35999999
tends have 6 places. 14. Make 9.814, 1.5, 87,26, .083 and 124.09 similar and conterminous.
15. Make .32i, .8262, .05, .0902 and 6 s milar and conterminous.
16. Make .531, .7348, .Öï .0503 and .749 similar and conterminous.
CASE IV. To find whether a given 'vulgar fraction is equal to a finite, or infinite decimal; and, of how many figures the repetend will consist.
If we divide unity with decimal ciphers annexed (1.0000, &c.] by any prime number, except the factors of 10, [2 and 5], the figures in the quotient will begin to repeat as soon as the remainder is 1. And since 9999, &c. is less than 10000, &c. by 1, therefore, 9999, &c. divided by any number whatever will leave 0 for a remainder, when the repeating figures are at their period. Now, whatever number of repeating figures we have, when the dividend is 1, there will be the same number, when the divider.d is any other number whatever: for the product of any circulating number, by any other given number, will consist of the same number of repeating figures as before. Take, for instance, the infinite decimal .386738673867, &c. whose repeating part is 3867. Now every repetend (3867] being equally multiplied, nest produce the same product: for though these products will consist of more places, yet the overplus in each, being alike, will be carried to the next, by which means each product will be equally increased, and consequently every four places will continue alike. From these observations it appears, that the dividend may be altered at pleasure, and the number of places in the repetend will still be the same: thus, in=.09, and it ori'i X3=.27.
RULE. Reduce the vulgar fraction to its lowest terms, and divide the denominator by 10, 5, or 2, as often as possible. If the whole denominator vanish in dividing, the decimal will be finite, and will consist of as many figures as there are divisions performed.
If the denominator do not vanish, then by the last quotient divide 9999, 8c till nothing remains: the num
ber of 98 used, will show the number of places in the repetend; which will begin after so many places of figures as there were 10s, 5s, or 2s used in dividing.
17. Is the decimal equal to ila finite, or infinite- and if infinite, how many places has the repetend ? 2 112
Since the denominator does 2 56
not vanish in dividing by 2, the decimal is infinite: and, as six
9s are used, the repetend will 2 14
consist of six figures; beginning 7)999999 at the fifth place, because four 142857
2s were used in dividing 18. Examine the fraction in, as above directed. 19. Examine the fraction ? as above directed. 20. Examine the fraction
as above directed. 21. Examine the fraction 3544, as above directed. 22 Examine the fraction }, as above directed.
ADDITION OF INFINITE DECIMALS.
RULE. Make the repetends similar and conterminous, and add them together. Divide this sum by as many 98 as there are places in the repetend; denote the remainder as the repetend of the sum, filling out its places with ciphers when it has not as many places as the repetends added; and carry the quotient to the next column.
23. What is the sum of 3.6 +78.3476+735.3+ 375. + 27 +187.4? 3.6 3.6666666
The sum of the repe78.3476= 78.3476476
tends is first found to be
2648191. This sum is then 735.3 =735.3333333
divided by 999999, and it 375. = 375.0000000
gives a quotient of 2, which .27 .2727272
we carry to the column of 187.4 =187.4444444 tenths, and a remainder 1380.0648193
of 643193, which we de
note as a repetend. 24. What is the sum of 5391.357 +75.38+187.21 + 4.2965 +217.8496 +42.176+ 523 +58.30048 ?
25. What is the sum of 9.814 +1.3+87.267.083 +124.09 ?
26. What is the sum of .162 +134.09 +2.93.+ 97.26+3.769230+99.083 +1.5+.814?
SUBTRACTION OF INFINITE DECIMALS. RULE. Make the repetends similar and conterminous, and subtract as usual; observing, that, if the repetend of the subtrahend be greater than that of the minuend, the right hand figure of the remainder must be less by 1, than it would be, if the expression were finite. 27. Subtract 13.76432 from 85.62.
Here, the whole repetend 85.62 =85.62626
of the subtrahend is greater 13.76432-=13.76432
than that of the rninuend, and 71.86193 the last figure in the remain
i der is diminished by 1.
MULTIPLICATION OF INFINITE DECIMALS.
99 23 90
RULE. Change the factors to vulgar fractions, multiply these fractions together, and reduce their product to a decimal. 31. What is the product of .36 x.25 ? .36=16
ix= =.0929 Ans. 32. What is the product of 27.23 X.26 ? 33. What is the product of 8574.3 X 87.5? 34. What is the product of 3.973 X 8? 35. What is the product of 49640.54 x .70503 36. What is the product of 3.145 X 4.297 ? 37. What is the product of 8.3 X 4.6 x7.09 ? 38. What is the product of .3 x.09 x 8.2 x 9?