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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.


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

·6=;=}; and ·123=;Ans.


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4. Required the least vulgar fraction equal to 769230.

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To reduce a mixed repetend to its equivalent vulgar fraction.


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


2. Multiply

Again,, or, being reduced to decimals, makes '010101, &c. or *001001, &c. ad infinitum ='01 or 'oot; that is, oi, and; consequently


03, &c. and

002,003, &c. and the same will hold universally.

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


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.


1. What is the vulgar fraction equivalent to 138 P 9X13+8=125= numerator, and 900 the de

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

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


Ans. 1.


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

8 Ans. 93.

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



Ans. 3133.

To make any number of dissimilar repetends similar and con



Change them into other repetends, which shall each consist of as many figures as the least common multiple of


tens, it is of %, and so the vulgar fraction 16 is t +, and is the same as by the rule.



* Any given repetend whatever, whether single, compound, pure or mixed, may be transformed into another repetend, that shall


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

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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 conter



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.

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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

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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 10000, &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 9s 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.



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.


11 20

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| 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 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 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 '90, and, or ×327, where the number of places in each is alike, and the same will be true in all cases.

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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′4 be adde ed together.

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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

* These rules are both evident from what has been said in re-. duction.

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