TABLE V. The annuity which 1. or D.I will purchase for any number of years to come, from 1 to 40. ys. 4 per cent. 4 per cent. | 5 per cent. | 5 per cent. | 6 per cent. 1.06 ⚫534. ⚫5378 54162 •54544 CIRCULATING DECIMALS ARE produced from Vulgar Fractions, whose denominators do not measure their numerators, and are distinguished by the continual repetition of the same figures. 1. The circulating figures are called repetends; and, if one figure only repeats, it is called a single repetend: As·1111 &c. •6666 &c. 2. A compound repetend has the same figures circulating alternately: As 010101, &c. 379379379, &c. 3. If other figures arise before those which circulate, the decimal is called a mixed repetend; thus, •375555 &c. is a mixed single repetend, and 378123123, &c. a mixed compound repetend. 4. A single repetend is expressed by writing only the circulating figure with a point over it; thus, 1111, &c. is denoted by ·1, and 6666, &c. by 6. 5. Compound repetends are distinguished by putting a point over the first and last repeating figures; thus, 010101, &c. is written 01, and .379379379, &c. thus, •379.. 6. Similar circulating decimals are such as consist of the same num ber of figures, and begin at the same place, either before or after the decimal point; thus, 3 and 5 are similar circulates; as are also 3.54 and 7.36, &c. 7. Dissimilar repetends consist of an unequal number of figures, and begin at different places. 8. Similar and conterminous circulates are such as begin and end at the same place; as 47-34576, 9.73528 and 05468, &c. REDUCTION OF CIRCULATING DECIMALS« 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 so many nines as there are recurring places in the repetend. 2. If * If unity, with cyphers annexed, be divided by 9 ad infinitum, the quotient will be 1 continually; that is, if be reduced to a decimal, it will produce the circu 19 late 1,and fince 1 is the decimal equivalent to 4, 2 will = 3,3 3, and fo on till 9 == 1. Therefore every fingle repetend is equal to a vulgar fraction, whose = numerator is the repeating figure and denominator 9. Again, 1 or すすす 5 being reduced to decinfals, make 010101, &c.and •001001001, &c. ad infinitum — Ol' and 001; that is, 501, and 5001, confequently = 2. If there be integral figures in the circulate, so 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 3 and ⚫324. 4. Required the least vulgar fraction equal to 384615. Ans. CASE II. To reduce a mixed Repetend to its equivalent Vulgar Fraction. RULE.*-1. To so many nines as there are figures in the repetend, annex so many cyphers as there are finite places, (that is, as there are decimal places before the repetend) for a denominator. 2. Multiply the nines in the said denominator by the finite part, and add the repeating decimals to the product for the numerator. 3. If the repetend begins 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 153? There being 1 figure in the repetend, and 2 finite places, I annex 2 cyphers to 9 for a denominator, viz. 900; then I multiply the 9 in the denominator by the two figures in the finite part, and add the repeating figure for a numerator; thus, 9x15+3=138 numerator. 2. What is the least vulgar fraction equal to 4123? 3. Required the finite number equivalent to 45-78? Ans. 455. CASE • In like manner for a mixed circulate; confider it as divisible into its finite and circulating parts, and the fame principle will be feen to run through them alfo ; thus the mixed circulate ⚫13 is divifible into the finite decimal ·1, and the repetend 08: but 1= and 03 would be equal to 3 provided the circulation began immediately after the place of units; but as it begins after the place of tenths, it is CASE III. To make any number of dissimilar repetends similar and conterminous; that is, of an equal number of places. RULE.* Change them into other repetends, which shall each consist of so many figures, as the least common multiple of the sums of the several numbers of places, found in all the repetends, contains units. EXAMPLES. 1. Make 6-317; 3.45; 52-3; 191-03; 057; 5.3 and 1.359 similar and conterminous. Here, in the first repetend, there are three places, in the second, one, in the third, none, in the fourth, two, in the fifth, three, in the sixth, one, and in the seventh, one. 3 Now find the least common multiple of these several sums, thus: 3, 1, 2, 3, 1, 1 _) 1, 1, 2, 1, 1, I and 2x3=6 units; therefore, the similar and con terminous repetends must contain 6 places.† Dissimilar made similar and conterminous. 2. Make 531, 7348, 07 and 0503 similar and conterminous. CASE IV. To find whether the decimal fraction, equal to a given vulgar one, be finite or infinite, and how many places the repetend will consist of. RULE.-1. Reduce the given fraction to its least terms, and divide the denominator by 2, 5 or 10, as often as possible. 2. Divide Any given repetend whatever, whether fingle, compound, pure, or mixed, may be transformed into another repetend, which shall consist of an equal or greater number of figures at pleasure; thus, 3 may be transformed into 33, or 333, &c. alfo 79=7979=797, and so on. The learner may obferve that the fimilar and conterminous repetends begin juft fo far from unity, as is the fartheft among the diffimilar repetends; and is fo in all cafes. In dividing 1000, &c. by any prime number whatever, except 2 or 5, the figures in the quotient will begin to repeat over again as foon as the remainder is 2...P 1: and 2. Divide 9999, &c by the former 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, divided by. 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. EXAMPLES. 2800 1. Required to find whether the decimal equal to 47 be finite or infinite, and if infinite, how many places that repetend will consist of. First 25 Then, 2800 112 7)999999 ; therefore, because the denominator 112 did not 142857 vanish in dividing by 2, the decimal is infinite, and, as six 9s were used, the circulate consists of 6 places, beginning at the fifth place, because four 2s were used in dividing. 2. Let be the fraction proposed. 3. 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. 2. Divide this sum (of the repetends only) by so many nines as there are places in the repetend, and the remainder is the repetend of their 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 infinite decimals. EXAMPLES. 1: and fince 999, &c. is lefs than 1000, &c. by 1, therefore 999, &c, divided by any number whatever, will, when the repeating figures are at their period, leave O for a remainder. Now, whatever number of repeating figures we have, when the dividend is 1, there will be exactly the fame number, when the dividend is any other number whatever. Thus, let 390539053905, &c. be a circulate, whofe repeating part is 3905. Now, every repetend (3905,) being equally multiplied, must give the fame product: For although thefe products will confift 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 confequently every four places will continue alike. And the fame will hold for any other number. Now from hence it appears that the dividend may be altered at pleasure, and the number of places in the repetend will still be the fame; thus, 09; and or X4=36, whence the number of places in each are alike. 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