the several numbers of places, found in all the repetends,

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 17123 similar and conter minous.

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.

RULE.*

1. Reduce the given fraction to its least terms, and divide the denominator by 2, 5 or 10, as often as possible.

..

2. If

..

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= 5757-5757-575; 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

then

N

1

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 os used will shew the number of places in the repetend; which will begin after so many places of figures, as there were 10s, 2s or 5s, used in dividing.

EXAMPLES.

210

1120

1. Required to find whether the decimal equal to be finite or infinite; and, if infinite, how many places the repetend will consist of.

210

2|8|4|2| 1; therefore the decimal

is finite, and consists of 4 places.

2. Let

than 10000, &c. by 1, therefore 9999, &c. divided by any number whatever will leave o for a remainder, when the repeating figures are at their period. Now whatever number of repeating figures we have, when the dividend is I, 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 continue alike. And the same will hold for any other number what

ever.

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='90, and, or X3=27, where the number of places in each is alike, and the same will be true in all cases.

I

5. Let 8344 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 column, and proceed with the rest as in finite decimals.

1. Let 3.6+78 3476+735-3+375+27+187-4be add.

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.

2. Let 5391-357+72.38+187-21+4-2965+217-8496

+42.176+523+58.30048 be added together.

Ans. 5974-10371.

Ans. 222-75572390.

3. Add 9.814+15+87-26+083+124.09 together.

..

4. Add 162+134-09+2.93+97-26+3.769230+99.083 +1.5+814 together.

Ans. 501-62651077.

SUBTRACTION of CIRCULATING DECIMALS.

RULE.

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 unis ty, than it would be, if the expressions were finite.

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.

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.