3. If the same figures circulate alternately, it is called a compound repetend; as .475475475, and is distinguished by putting a point over the first and last repeating figures; thus, .475475475 is written .475. 4. When other figures arise before those which circulate, it is called a mixed repetend; as .1246, or .17835. 5. Similar repetends begin at the same place; as .3 and .6; or 5.123 and 8.478. 6. Dissimilar repetends begin at different places; as .986 and .4625. 7. Conterminous repetends end at the same place; as .631 and .465. 8. Similar and conterminous repetends begin and end at the same place; as .1728 and .4987. REDUCTION OF CIRCULATING DECIMALS. CASE I. To reduce a simple repetend to its equivalent vulgar fraction. If a unit with ciphers annexed to it 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 .i; and since .i is the decimal equivalent to §, .2 will be equivalent to, .3 to 3, and so on, till .9 is equal to or 1. Therefore every single repetend is equal to a vulgar fraction, whose numerator is the repeating figure, and denominator 9. Again,, or, being reduced to decimals, makes .01010101, and .001001001 ad infinitum=.0i and .001; that is =.01, and s=.001; consequently -.02, and =.002; and, as the same will hold universally, we deduce the following RULE. Make the given decimal the numerator, and let the denominator be a number consisting of as many nines, as there are recurring places in the repetend. If there be integral figures in the circulate, as many ciphers 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 fraction equal to .6 and .123. Ans. 19. Ans. 18. 3. Reduce 1.62 to its equivalent vulgar fraction. CASE II. To reduce a mixed repetend to its equivalent vulgar fraction, 1. What vulgar fraction is equivalent to .138 ? OPERATION. .138=1+580117+8=128=3 Ans. As this is a mixed circulate, we divide it into its finite and circulating parts; thus .188.18 the finite part, and .008 the repetend or circulating part; but .18; and .008 would be equal to, if the circulation began immediately after the place of units; but, as it begins after the place of hundreds, it is of 100Therefore .138=+=+86-1883 Ans. Q. E. D. From the above demonstration we deduce the following 8 RULE. To as many nines as there are figures in the repetend, annex as many ciphers as there are finite places for a denominator; multiply the nines in the denominator by the finite part, and add the repeating decimal to the product for the numerator. If the repetend begins in some integral place, the finite value of the circulating, must be added to the finite part. 2. What is the least vulgar fraction equivalent to .53 ? Ans. 1. 3. What is the least vulgar fraction equivalent to .5925 ? Ans. 4. What is the least vulgar fraction equivalent to .008497133? Ans. 83 9768 5. What is the finite number equivalent to 31.62 ? Ans. 313 CASE III. To make any number of dissimilar repetends, similar and conterminous. 1. Dissimilar made similar and conterminous. OPERATION. 9.1679.16767676 Any given repetend whatever, whether 14.6 =14.60000000 single, compound, pure, or mixed, may be 3.1653.16555555 transformed into another repetend, that 12.432-12.43243243 shall consist of an equal or greater number 8.181= 8.18i81818 of figures at pleasure; thus .4 may be 1.307= 1.30730730 changed into 44 or .444; and .29 into .2929 or .2929. And as some of the circu lates in this question consist of 1, some of 2, and others of 3 places; and, as the least common multiple of 1, 2, and 3, is 6, we know, that the new repetend will consist of 6 places, and will begin just so far from unity, as is the farthest among the dissimilar repetends, which, in the present example, is the third place. Hence the following RULE. Change the given repetends into other repetends, which shall consist of as many figures, as the least common multiple of the several number of places, found in all the repetends, contains units. 2. Make 3.67i, 1.007i, 8.52 and 7.616325 similar and conterminous. 3. Make 1.52, 8.7156, 3.567 and 1.378 similar and contermin ous. 4. Make .0007, .141414 and 887.i similar and conterminous. CASE IV. To find whether the decimal fraction, equal to a given vulgar fraction, be finite or infinite, and of how many places the repetend will consist. RULE. Reduce the given fraction to its least terms, and divide the denominator by 2, 5 or 10, as often as possible. 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 division. If it do not vanish, divide 9999, &c., by the result till nothing remain, and the number of 9's used will show the number of places in the repetend; which will begin after so many places of figures, as there are 10's, 2's or 5's used in dividing. NOTE. In dividing 1.0000, &c. by any prime number whatever except 2 or 5, the quotient will begin to repeat as soon as the remainder is 1. And since 9999, &c. is less than 10000, &c. by 1, therefore 9999, &c. divided by any number whatever, will leave a 0 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 .378137813781, &c. be a circulate, whose repeating part is 3781. Now every repetend (3781) being equally multiplied, must produce the same product. For 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 whatever. Hence it appears, that the dividend may be altered at pleasure, and the number of places in the repetend will be still the same; thus,=.09, and = .27, where the number of places in each are alike, and same will be true in all cases. EXAMPLES. 210 1. Required to find whether the decimal equal to be finite or infinite; and, if infinite, of how many places the repetend will consist. 3 (2) (2) (2) 1120=2)18=8=4=2=1; therefore, because the denominator vanishes in dividing, the decimal is finite, and consists of four places; thus, 16)3.0000 .1875 2800 2. Required to find whether the decimal equal to be finite or infinite; and, if infinite, of how many places that repetend will consist. 475 (2) (2) (2) 7) 999999 2)112=56-28-14-7. Thus, 72299; therefore, because the denominator, 112, did not vanish in dividing by 2, the decimal is infinite; and as six 9's were used, the circulate consists of six places, beginning at the fifth place, because four 2's were used in dividing. 3. Let be the fraction proposed. 4. Let be the fraction proposed. SECTION XXV. ADDITION OF CIRCULATING DECIMALS. 1. Let 3.5+7.651+1.765+6.173+51.7+3.7+27.63i and 1.003 be added together. EXAMPLE. OPERATION. Dissimilar. Similar and Conterminous. 3.5 7.651 = 3.5555555 Having made all the numbers similar and 7.6516516 conterminous by Sect. XXV. Case III., we 1.765= 1.7657657 add the first six columns, as in Simple 6.173= 6.1737373 Addition, and find the sum to be 3591224= 51.7 =51.7777777 3591224. 224-3.591227. The repeating decimals 8.7000000 .591227 we write in its proper place, 27.631=27.6316316 carry 3 to the next column, and then pro1.003 1.0030030 ceed as in whole numbers. 3.7 103.2591227 Hence the following 999999 and RULE. Make the repetends similar and conterminous, and find their sum, as in common Addition. Divide this sum by as many 9's 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 ciphers on the left, when it has not so many places as the repetends. Carry the quotient of this division to the next column, and proceed with the rest as with finite decimals. 2. Add 27.56+5.632+6.7+16,356+.71 and 6.1234 together. Ans. 63.1690670868888. 3. Add 2.765+7.16674+3.67i+.7 and .1728 together. Ans. 14.55436. Ans. 17.559191208478740908ož. 4. Add 5.16345+8.6381+3.75 together. 5. Reduce the following numbers to decimals and find their sum; }, }, and {. Ans..587301. |