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 amd 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 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. 575 9999 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. 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, 147 x9=1470-147-1323, 147 × 99-14700-147-14553, and 147 × 999—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, 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 divi sion continued on indefinitely, the number of repetends 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 infinite, 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, 3, and 1'35'-13 (2.) Required the least equivalent vulgar fractions to •6′, ∙l′62', •7′69230′, ·9′45′, and ∙0′9′. (3.) Required the least equivalent vulgar fractions to 594405', '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 fractions to 75', '43'8', '093', 4.75'43', '0098'7', and '45'. Examples to Prop. 3. (7.) Required to find whether the decimal equivalent to 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 293049768 then ÷2 9768 48342442 =1291, 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, 0084′97133. (8.) Whether is the decimal equivalent to or infinite? 83 210 TI20 TT finite (9.) Whether is the decimal equivalent to finite er infinite? 80 12 72 (10.) Let T5, 133, 344, and 25, be proposed. 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. Examples. (1.) Add 125', 41'63', 1.7143', and 2.5'4', together. (2.) Add 67.34′5′49·6′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', together. (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. (2.) From 47°53′ take 1.7'57'. (9.) From 1.49′37′ take ·1475. MULTIPLICATION OF CIRCULATING DECIMALS. GENERAL RULE. Turn the decimals into their equivalent vulgar fractions, and find the product of these fractions: then turn the vulgar fraction, expressing the product, into an equivalent decimal fraction, and it will be the product required. OR Proposition 1. When the right hand figure of the multiplicand is a single repetend, and the multiplier a finite number. Rule. In multiplying, increase the right-hand figure of each resulting line by as many units as there are nines in the product of the first figure in that line; and the righthand figure of each line will be a repetend: make them all end at the same place, and then add them together. Prop. 2. When the multiplicand is a compound repetend, and the multiplier a finite number. Rule. Set the repeating figures in the multiplicand twice over, multiply the second period mentally, and carry the tens, contained in the product of the left-hand figure, to the product of the right-hand figure of the first period; then multiply the rest of the figures in the multiplicand as in common multiplication. Proceed thus with each figure in the multiplier, and every product will contain a repetend of the same number of places as the repetend in the multiplicand; lastly, make every product conterminous towards the right-hand before you add them together. Note. It is possible for the product of the repetend to consist of a number of nines; if ever this should happen, increase the product of the right-hand figure of the first period by an unit. L |