TABLE VII. THE VALUE OF AN ANNUITY OF £1 OR $1, for a single LIFE. 13 per 13 per 4 per 14 per 5 per 16 per 10 19,87 18,27 16,88 15,67 14,60 12,80| This Table is formed by ascertaining from Bills of mortality the mean length of the lives of persons of a certain age, and then calculating the value of the annuity for the number of years they may thus be expected to live. This mean is called the Expectation of Life, at any given age, which is exhibited in the following table. TABLE VIII. EXPECTATION OF LIFE AT SEVERAL AGES. Age. Expec. Age. Expec. | Age. Expec. | Age. Expec. | Age. Expec.} 129,80 20 33,62 4221,65 62 11,68 82 3.31 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 number 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 47-3-4576, 9-73528 and 05463, &c. REDUCTION OF CIRCULATING DECIMALS. CASE I. 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 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. •·3=}=}; and ·324=334=14. Ans. } and 17. 969 2. Reduce 7 to its equivalent vulgar fraction. Ans. 3. Reduce 2.37 to its equivalent vulgar fraction. Ans. 2370 989. 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. *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 circulate ·1, and since 1 is the decimal equivalent to 1,2 will = =3,3=3, and so on till •9=3=1. Therefore every single repetend is equal to a vulgar fraction, whose numerator is the repeating figure and denominator 9. 1 Again, or being reduced to decimals, make 010101, &c. and 001001001, 99 990 &c. ad infinitum=01 and 001; that is, 701, and 15 =·001, consequently 3,—·02, '—-03, &c. and 53, 002, 535 003, &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 13 is divisible into the finite decimal ⚫1, and the repetend '03: but 15, and 03 would be equal to provided the circulation began immediately after the place of units; but as it begins after the place of tenths, it is 3 of 13, and so the vulgar fraction=13 is +=% +6% 3 , and is the same as by the rule. 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 an nex 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 2. What is the least vulgar fraction equal to 4123? Ans. 198 9990 3. Required the finite number equivalent to 45-78? Ans. 45. 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. Now find the least common multiple of these several sums, thus: 1, 1, 2, 1, 1, 1 and 2x3=6 units; therefore, the similar and conterminous repetends must contain 6 places.† Any given repetend whatever, whether single, compound, pure, or mixed, anay 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. also 79=7979-797, and so on, The learner may observe that the similar and conterminous repetends begin just so far from unity, as is the farthest among the dissimilar repetends; and it is so in all cases. 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 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. * In dividing 1.000, &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 999, &c. is less than 1000, &c. by 1, therefore 999, &c. divided by any number whatever, will, when the repeating figures are at their period, leave 0 for a remainder. : 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. Thus, let 390539053905, &c. be a circulate, whose repeating part is 3905. Now, every repetend (3905,) being equally multiplied, must give the same product: For although 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. 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 same; thus, '='09 ; and 4 fr or X 436, whence the number of places in each are alike. U u |