CIRCULATING, OR REPEATING DECIMALS. Definition 1. When the denominator of a vulgar fraction, in its lowest terms, is not compounded of 2 or 5, or both, the decimal produced from such a vulgar fraction will be infinite; it is called a repetend, or circulating decimal, from a continual repetition of the same figures.

2. A single repetend is a decimal, where only one figure repeats, as 222, &c. or 3333, &c. and these may be expressed by putting a mark over the first figure. Thus 222, &c. may be denoted by 2', and ·3333, &c. by ·3′.

3. A compound repetend has the same figures circulating alternately, as 575757, &c. or 57235723, &c. and these may be distinguished by marking the first and last repeating figure. Thus, 5757, &c. may be written '5'7', and 57235723, &c. 5'723'.

4. Pure repetends are such as have no figures in them but what belong to the repetend, as 3′, 5′, 4'73', &c.

5. Mixed repetends are such as have ciphers or significant figures, between the repetend and the decimal point, or such as have whole numbers to the left hand of the decimal point, as '04', '07'53', '473', 357'3', 6·5', 4·3'75', &c. 6. Dissimilar repetends are such as begin at different places from the decimal points, as 2'53', 475'2', &c.

7. Similar repetends are such as begin at an equal distance from the decimal points, as 35'4', 2.75'34', &c. 8. Conterminous repetends are such as end at the same distance from the decimal points, as 125', 3'54′, &c.

9. Similar and conterminous repetends are such as begin and end at the same place after the decimal points, as 53.27'53′, 4·63′25', and ·46'32′, &c.

REDUCTION OF CIRCULATING DECIMALS. Proposition 1. To reduce a pure repetend to its equivalent vulgur fraction.

α

RULE I.

Make the given decimal the numerator, and let the denominator be a number consisting of so many nines as there are figures in the repetend. The terms of this fraction, divided by their greatest common measure, will give the least equivalent vulgar fraction required.

Prop. 2. To reduce a mixed repetend to its equivalent vulgar fraction.

RULE.

From the given mixed repetend subtract the finite figures for a numerator, and to the right hand of so many nines as there are pure repetends, annex so many ciphers as there are finite decimals for a denominator. Then reduce this fraction to its lowest terms.

Note 1. Any finite decimal may be considered as infinite by making ciphers to recur; thus •35=3500000, &c.

2. If any circulating decimal have a repetend of any number of figures, it may be considered as having a repetend of twice or thrice that number of figures, or any multiple thereof. The number 2·35′7′, having two repetends, may be considered as having a repetend of 4, 6, 8, 10, &c. places. Thus, 2·35′7′=2·35′757′=2.35′75757= 2.35'7575757′, &c. Hence any number of dissimilar repetends may be made similar and conterminous.

3. If any circulating decimal have a repetend of more than one figure, it may be transformed into another decimal, having a repetend of the same number of figures; thus, 5'7′=57′5=•575'7', and 3.47′85′-3-478′57′—3·4785′78′-3.47857′85'.

4. When any circulating decimal has a repetend of more than one figure, it cannot be transformed into another decimal, having a greater or less number of figures at pleasure; but the new repetend must always contain either the same number of places as the original repetend, or some multiple thereof. Thus, 5'7'57′5′=·575′7′ =•5757′5′= •5757′575′: Or, •5′7′=•5′757′=•5′75757', according to the second note. But, 5'7' never can be equal to 5'75', for then would be equal to 575, which evidently is not the case. The truth of any of the preceding notes may be examined by turning the given repetends into their equivalent vulgar fractions, and comparing them together by the latter part of Note 2, Prop. 12, Vulgar Fractions.

999'

5. Any series of nines, infinitely continued, is equal to unity, or one, in the next left hand place. Thus, 999, &c. ad infinitum,=1; 0999, &c=1 ; also 00999, &c.=01, and 5.999, &c.=6.

Any number may be multiplied by 9, 99, 999, &c. by annexing so many ciphers to the right hand of it as there are nines, and then subtracting it from itself, thus increased. Thus,

147x9=1470-147=1523, 147 x 99=14700-147-14553, and 147x999-147000-147=146853.

7. Any number, divided by 9, 99, 999, &c. will be equal to the sum of the quotients of the same number continually divided by 10, 100, 1000, &c. Thus,

425 42.5 4.25 425
+-
·+·

10 10 10

=
&c. ad infinitum,
+

10

425

=47.2′, and

9

hence, it appears that every recurring decimal is a geometrical series, decreasing, ad infinitum, and the equivalent vulgar fraction to every recurring decimal is equal to the sum of such a series.

8. If any number be divided by another prime to it, and the division continued on indefinitely, the number of repeteuds in the quotient will always be less than the number of units in the divisor.

9. If two or more numbers, that have repetends of equal places, be added together, the sum will have a repetend of the same number of places; for every column of periods will amount to the same sum.

10. If any circulating number be multiplied by any given number, the product will be a circulating number, containing the same number of figures in the repetend as before, for every repetend will be equally multiplied, and consequently must produce the same product.

Prop. 3. Having a vulgar fraction given, to find whether its equivalent decimal will be finite or infinile, and how many places the repetend will consist of.

RULE.

Reduce the given fraction to its lowest terms, and divide the denominator by 10, 2, or 5, as often as possible: then divide 9999, &c. by this result, till nothing remains, and the number of nines made use of will be equal to the number of figures in the repetend. The repetend will always begin after so many places of figures as you perform divisions by 10, 2, or 5; and, if the whole denominator should vanish after these divisions, the decimal will be finite.

Examples to Proposition 1.

(1.) Required the least equivalent vulgar fraction to ⚫3', and '1'35'.

First, 33, and •1'35'=}}}=77.

(2.) Required the least equivalent vulgar fractions to 6', 1'62', 7'69230', '9'45', and '0'9'.

(3.) Required the least equivalent vulgar fractions to 5'94405', '3'6', and '1'42857.

Examples to Prop. 2.

(4.) Required the least equivalent vulgar fractions to 2.41'8', '59'25', '0084'97133', and '53'.

(5.) Required the least equivalent vulgar fractions to ∙138′, 7·54′3′, ·043′54'′, 37·54′, ·67′5′, and •75′4347′. (6.). Required the least equivalent vulgar fractionsto ·75', '43'8', '093', 4·75'43', '0098'7', and '45'.

Examples to Prop. 3,

(7.) Required to find whether the decimal equivalent

to 4 249 be finite or infinite; if infinite, how many

places the repetend will consist of, whether there will be any finite decimals to the left hand of the repetend, and how many?

First,

249 83 29304 9768

; then +2
9768

1

2 <2 48342442

=1221, and 1221)999999(719: here are 6 nines made use of before nothing remains, and the denominator has been divided by 2 three times, and cannot be abridged any more: therefore the decimal will be infinite, and will consist of three finite decimals and six pure repetends. Thus 080084'97133

210

(8.) Whether is the decimal equivalent to finite or infinite?

(9.) Whether is the decimal equivalent to or infinite?

80 72

finite

(10.) Let 122, 133, 34, and 2 be proposed. 7351 232

ADDITION OF CIRCULATING DECIMALS.

RULE.

Make the repetends similar and conterminous, and to the right hand thereof set two or three of the first repeating figures, which add together as whole numbers, and carry the tens contained in the left-hand-row to the right-hand row of the conterminous repetends: collect these together into one sum, like finite decimals, for the

answer.

Note. The sum of the repetend, found by the preceding rule, will sometimes, though very rarely, consist of a number of nines; whenever that is the case, reject them, and make the next left-hand figure an unit more. If the decimals to be added contain only single recurring figures, after having made them end together, the sum of the right-hand row may be increased by as many units as it contains nines, instead of carrying the repetend out.

(1.) Add '125', 4∙1'63′, 1·7′143′, and 2.5'4', together. Similar. Similar and conterminous.

Dissimilar.

(2.) Add 67.34'5'+9.6′51'+2′5′+17·47'+'5', together.

(3.) Add 4'75'+3.754′3′+64.7′5′+5'7 +17'88', together.

(4.) Add '5'+4.37′+49.45'7'+49'54'+7'345', toge

ther.

(5.) Add 175'+42.5'7'+37′53'+59'45 +3.75'4', together.

(6.) Add 165.1'64′ +147·0'4' + 4′9′5′ + 94°37′ + 4.7'123456', together.

SUBTRACTION OF CIRCULATING DECIMALS.

RULE.

Make the repetends similar and conterminous and subtract, as if they were finite decimals; only observe, that if the repetend of the subtractor be greater than the repetend of the subtrahend, the right-hand figure of the remainder must be less by unity than it would be, if the expressions were finite.

Note. If either the subtrahend or subtractor be finite decimals, they must be made similar and conterminous with ciphers.