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the several numbers of places, found in all the repetends, contains units.
87.26666666 •083 •08333333
124:09 124:09090909 2. Make '3, 27 and .045 similar and conterminous.
3. Make .321, •8262, .05 and *0902 similar and conterminous.
4. Make •5217, 3:643 and 17.123 similar and conterminous.
CASE IV. To find whether the decimal fraction, equal to a given vulgar
one, be finite or infinite, and of how many places the repetend will consist.
1. Reduce the given fraction to its least terms, and divide the denominator by 2, 5 or 10, as often as possible.
consist of an equal or greater number of figures at pleasure : thus •4 may be transformed to ·44, or •444, or ·44, &c. Also ·57= '57575-5757='575 ; and so on; which is too evident to need any further demonstration.
* In dividing 1'0000, &c. by any prime number whatever, except 2 or 5, the figures in the quotient will begin to repeat over again as soon as the remainder is 1. And since 9999, &c. is less
2. If the whole denominator vanish in dividing by 2, 5 or 10, the decimal will be finite, and will consist of so many places as you perform divisions.
3. If it do not so vanish, divide 9999, &c. by the result, till nothing remain, and the number of gs used will shew the number of places in the repetend ; which will begin after so many places of figures, as there were tos, 2s or 58, used in dividing
1. Required to find whether the decimal equal to 10 be finite or infinite ; and, if infinite, how many places the repetend will consist of.
116=27= 181 412 | 1; therefore the decimal is finite, and consists of 4 places.
than 10000, &c. by 1, therefore 9999, &c. divided by any number whatever will leave o for a remainder, when the repeating fig. ures are at their period. Now whatever number of repeating figures we have, when the dividend is 1, there will be exactly the same number, when the dividend 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. Thus, let '507650765076, &c. be a circulate, whose repeating part is 5076. Now every repetend (5076) being equally multiplied, must 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 con. tinue alike. And the same will hold for any other number what
Now hence it appears, that the dividend may be altered at pleasure, and the number of places in the repetend will still be the same : thus 1 ='90, and ir, or 4x3='27, where the number of places in each is alike, and the same will be true in all cases.
2. Let be the fraction proposed.
ADDITION of CIRCULATING DECIMALS.
RULE,* 1. Make the repetends similar and conterminous, and find their sum as in common addition.
2. Divide this sum by as many nines as there are places in the repetend, and the remainder is the repetend of the sum ; which must be set under the figures added, with cyphers on the left hand, when it has not so many places as the repetends.
3. Carry the quotient of this division to the next cola umn, and proceed with the rest as in finite decimals.
EXAMPLES. 1. Let 36+783476+735+3+375+:27+1874be add ed together.
Dissimilar, Similar and conterminous.
3.6 3.6666666 78.3476 = 78.3476476. 7353
1380·0648193 the sum. In this question, the sum of the repetends is 2648191, which, divided by 999999, gives 2 to carry, and the remainder is 648193.
* These rules are both evident from what has been said in reduction,
2. Let 5391-357+72-38+187:21 +4:2965+217-8498 +42:176+:523+58:30048 be added together.
Ans. 5974.1037i. 3, 4dd 9:814+1.5+87:26+083+124:09 together.
Ans. 222-75572390. 4. Add 162 +134:09+2.93+97-26+3.769230+99-083 +15+814 together.
SUBTRACTION of CIRCULATING DECIMALS..
Make the repetends similar and conterminous, and subtract as usual ; observing, that, if the repetend of the subtrahend be greater than the repetend of the minuend, then the right-hand figure of the remainder must be less by uni, ty, than it would be, if the expressions were finite.
EXAMPLES 1. From 85:62 take 13876432.
71-80193 the difference. 2. From 476-32 take 8454697. Ans. 391-552.4. 3. From 3.8564 take •0382.
MULTIPLICATION of CIRCULATING DECIMALS.
1. Turn both the terms into their equivalent vulgar fractions, and find the product of those fractions as usual.
2. Turn the vulgar fraction, expressing the product, into an equivalent decimal, and it will be the product required.
I. Multiply ·36 by '25.
•25 => Ar=='0929 the product. 2. Multiply 37*23 by •26.
Ans. 9'928. 3. Multiply 8574*3 by 87.5. Ans. 750730'518. 4. Multiply 3973 by 8.
Ans. 31'791. 5. Multiply 49640°54 by •70503. Ans. 34998-4199003. 6. Multiply 3*145 by 4*297.
Ans. 13:5169533. .
Dirision of CIRCULATING DECIMALS.
RULE. 1. Change both the divisor and dividend into their equivalent vulgar fractions, and find their quotient as usual.
2. Turn the vulgar fraction, expressing the quotient, into its equivalent decimal, and it will be the quotient required.
1. Divide 36 by '25.
-25=! *+}}=***=*=1133=1.4229249011857707509881 the quotient.
2. Divide, 319.28007112 by 764'5. Ans. '4176325.