EXAMPLES.

1. Required the least vulgar fractions equal to 6 and 123.

6==; and 123=12345 Ans.

4. Required the least vulgar fraction equal to '769230.

Ans. 19.

CASE II.

To reduce a mixed repetend to its equivalent vulgar fraction.

RULE.*

1. To as many nines as there are figures in the repetend, annex as many cyphers as there are finite places, for a denominator.

2. Multiply the nines in the said denominator by the finite part, and add the repeating decimal to the product, for the

numerator.

3. If the repetend begin in some integral place, the finite value of the circulating part must be added to the finite part.

In like manner for a mixed circulate; consider it as divisible into its finite and circulating parts, and the same principle will be seen to run through them also: thus, the mixed circulate '16 is divisible into the finite decimal '1, and the repetend '06; but

1, and 06 would be, provided the circulation began immediately after the place of units; but as it begins after the place of

tens, it is of, and so the vulgar fraction ='16 is + 6 +%=15, and is the same as by the rule.

9

909

EXAMPLES.

1. What is the vulgar fraction equivalent to '138? 9x13+8=125= numerator, and 900= the denomina

2. What is the least vulgar fraction equivalent to '53 ?

Ans..

3. What is the least vulgar fraction equal to '5925?

Ans.

16

4. What is the least vulgar fraction equal to '008497133?

Ans.

5. What is the finite number equivalent to 31-62?

CASE III.

8 3 9763

Ans. 31.

To make any number of dissimilar repetends similar and con

terminous.

RULE.*

Change them into other repetends, which shall each consist of as many figures as the least common multiple of the several numbers of places, found in all the repetends, contains units.

*

Any given repetend whatever, whether single, compound, pure, or mixed, may be transformed into another repetend, that shall consist of an equal or greater number of figures at ple

ure: thus 4 may be transformed to 44,

444 or

be

or 44, &

Also '57=5757="5757-575; and so on; which is too evident to need any farther demonstration. M

VOL. I.

2. Make 3, 27 and 045 similar and conterminous.

3. Make ·321, ‍8262, '05 and '0902 similar and conterminous.

4. Make 5217, 3643 and 17'123 similar and contermin

ous.

CASE IV.

To find whether the decimal fraction, equal to a given vulgar one, be finite or infinite, and of how many places the repe

tend 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.

* In dividing 1'0000, &c. by any prime number whatever, except 2 or 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 exactly the same number, when the dividend is any other number whatever. For the product of any circulating number, by any other given ember, will consist of the same number of repeating figures as dia.. el Thus, let 507650765076, &c. be a circulate, whose rething part is 5076. Now every repetend (5076) being equally

ltiplied, must produce the same product. For though these roducts will consist of more places, yet the overplus in each,

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 9s used will show 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.

1. Required to find whether the decimal equal to 210 be finite or infinite; and if infinite, how many places the repetend will consist of.

3

2 2 2

1110=217|8|4|2|1; therefore the decimal is

finite, and consists of 4 places.

2. Let be the fraction proposed.. 3. Let be the fraction proposed.

4. Let

5. Let

be the fraction proposed.

be the fraction proposed.

ADDITION OF CIRCULATING DECIMALS.

RULE.*

1. Make the repetends similar and conterminous, and find their sum as in common addition.

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 oth. er number whatever.

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. * These rules are both evident from what has been said in reduction.

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, when it has not so many places as the repe

tends.

3. Carry the quotient of this division to the next column, and proceed with the rest as in finite decimals.

EXAMPLES.

1. Let 3'6+78'3476+735*3+375+27+1874 be added together.

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+18721+42965+2178496

+42176+523+58'30048 be added together.

Ans 5974'10371.

3. Add 9'814+1 5+8726+083+124'09 together.

4. Add 162+134

+814 together.

Ans. 222'75572390.

+2′93+97′26+3*769230+99′083+1'5

Ans. 501'62651077.