This is real value in our currency does not vary essentially from $4 84. the value assigned to it by act of Congress, in calculating ad valorem duties in our custom-houses on goods imported from England, which are invoiced in sterling money. Therefore, multiply sterling by $4.84 and we shall have the custom-house and market par value of sovereigns or pounds sterling. N. B. — $4.444 never represented the true value of the pound sterling in the United States currency. Under the act of Congress of the 2d of April, 1792, establishing the mint and regulating the coins of the United States, the value of the pound sterling was $4.56+ 1371 the By the act of Congress of the 28th of June, 1834, called the Gold Bill, the value of the pound or sovereign was $4.871869. By the act of Con gress of the 18th of January, 1837, supplementary to the act of 1834, value of the pound sterling becomes $723303 += $4.86723303— ereigns are usually valued at $4.85 at the banks. 3520000 474742 Sov SECTION XXX. INFINITE OR CIRCULATING DECIMALS.* DEFINITIONS. 1. DECIMALS produced from Vulgar Fractions, whose denominators do not measure their numerators, and distinguished by the continual repetition of the same figure or figures, are called infinite or circulating decimals. 2. The circulating figures, that is, those that are continually repeated, are called repetends. If only the same figure is repeated, it is called a single repetend, as .11111 or .5555, and is expressed by writing the figure repeated with a point over it. Thus .11111 is denoted by .1, and .5555 by .5. 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 3.478. * Infinite or circulating decimals being less important for use than many other rules, and somewhat difficult in their operation, the student can omit them until he reviews the Arithmetic. 6. Dissimilar repetends begin at different places; as and .4625. .986 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 one continually; that is, if be reduced to a decimal, it will produce the circulate .1; and since .1 is the decimal equivalent to, .2 will be equivalent to,.3 to , 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,=.01 and .001; that is, = .01, and = .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. 2. Reduce .3 to its equivalent vulgar fraction. Ans. Ans. 1. 3. Reduce 1.62 to its equivalent vulgar fraction. Ans. 137. 4. Reduce .769230 to its equivalent vulgar fraction. CASE II. Ans. 19. To reduce a mixed repetend to its equivalent vulgar fraction. 1. What vulgar fraction is equivalent to 138 ? OPERATION. 17 .138=b+80= !!? +680 = £25 = √ Ans. As this is a mixed circulate, we divide it into its finite and circulating parts; thus .138.13, the finite part, and .008 the repetend or circulating part; but .13; and .008 would be equal to, if the circulate began immediately after the place of units; but, as it begins after the place of hundreds, it is & of Too 980. Therefore .138 16% +880 = 38z+ 980=138 = Ans. Q. E. D. 36 = ច 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. 3. What is the least vulgar fraction equivalent to .5925 ? Ans. 1. 4. What is the least vulgar fraction equivalent to .008497133? Ans.. 5. What is the finite number equivalent to 31.62? CASE III. Ans. 313. To make any number of dissimilar repetends similar and conterminous. 1. Dissimilar made similar and conterminous. 14.6 3.165 12.432 OPERATION. = 12.43243243 Any given repetend whatever, 9.1679.61767676 whether single, compound, pure, or = 14.60000000 mixed, may be transformed into an3.16555555 other repetend, that shall consist of an equal or greater number of figures at pleasure; thus .4 may be changed into .44 or .444; and 29 into .2929 or 2929. And as some of the circulates in this question consist of one, some of two, and others of three places; and as the least common multiple of 1, 2, and 3 8.181 = = 8.18181818 1.307 1.30730730 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. 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 3.671, 1.0071, 8.52, and 7.616325 similar and con terminous. 3. Make 1.52, 8.7156, 3.567, and 1.378 similar and conter minous. 4. Make .0067,.141414, and 887.1 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 divisions. 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 Ŏ 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 the same will be true in all cases. EXAMPLES. 1. Required to find whether the decimal equal to 20% be finite or infinite; and if infinite, of how many places the repetend will consist. 3 (2) (2) (2) f12% = 2)16 2)16=8=4=2=1; therefore, because the denominator vanishes in dividing, the decimal is finite, and consists of four places; thus, 16)3.000. 2. Required to find whether the decimal equal to 75 be finite or infinite; and, if infinite, of how many places that repetend will consist. 475 2800 (2) (2) (2) 1428 2)112=56=28=147. Thus, 799999 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. 1. Let 3.5+7.651+1.765+6.173+51.7+3.7+27.63i and 1.003 be added together. OPERATION. Dissimilar. Similar and Conterminous. 3.5 = 7.651 = 3.5555555 1.765 1.7657657 = = = 999999 Having made all the numbers similar and conterminous by Sect. XXX., Case III., we add the first six columns, as in 6.173 = 6.1737373 Simple Addition, and find the sum to 51.7 51.7777777 be 3591224 35912243.591227. 3.7 = 3.7000000 The repeating decimals .591227 we 27.63127.6316316 write in their proper place, and carry 3 1.0030030 to the next column, and then proceed as in whole numbers. 103.2591227 1.003 |