REDUCTION OF CIRCULATING DECIMALS.

CASE 1.

To reduce a simple repetend to its equivalent vulgar fraction.

RULE.*

1. Make the given decimal the numerator, and let the denominator be a number, consisting of as many nines as there are recurring places in the repetend.

2. If there be integral figures in the circulate, as many cyphers must be annexed to the numerator, as the highest place of the repetend is distant from the decimal point.

EXAMPLES.

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

•6; and 125-133 Ans.

999

2. Reduce 3 to its equivalent vulgar fraction.

41 3 33

Ans..

* If unity, with cyphers annexed, be divided by 9 ad infinitum, the quotient will be 1 continually; i. e. if be reduced to ifbe

a decimal, it will produce the circulate i; and since i is the decimal equivalent to 1, 2 will=3, 3=3, and so on till 9=9

Therefore every single repetend is equal to a vulgar fraction, whose numerator is the repeating figure and denominator 9. Again,, or, being reduced to decimals, makes ⚫010101, &c. or 001001, &c. ad infinitum='01 or 001; that is, 01, and001; consequently 02, 3=03, &c. and 535-002, 53, 003, &c. and the same will hold universally.

99

3. Reduce 1.62 to its equivalent vulgar fraction.

Ans. 1620

999

4. Required the least vulgar fraction equal to 769230.

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 denomina.or.

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.

EXAMPLES.

1. What is the vulgar fraction equivalent to •138?

9x13+8=125 numerator, and 900 the denomina

* 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 11, 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 frac5, and is the same as by the

tion

rule.

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?

5. What is the finite number equivalent to 31.62?

CASE 3.

Ans. 31.

To make any number of dissimilar repetends similar and

conterminous.

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 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.

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

3. Make ·321, •8262, 05 and 0902 similar and conter

minous.

4. Make 5217, 3.643 and 17.123 similar and contermi

nous.

CASE 4.

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.

* In dividing 10000, &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 O 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 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 whatever.

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.

1120

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.

2 2 2

1130=2176 | 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

1

be the fraction proposed.

5. 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.

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 09, and, or 1×3=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.