ADDITION OF CIRCULATING DECIMALS. OPERATIOX. Dissimilar. 303: Ex. 1. Add 2.763, 7.1667), 3.67i, .7, and .1728 together. Ans. 14.55436. Having made the given Similar and Conterminous. repetends similar and con-2.7 6 5 2.7 6 5 6 5 terminous (Art. 302), we 7.1 6 6 7 4 7.1 6 6 74 add as in addition of 3.6 7 i whole numbers, and obtain 3.6 136 14.55433. The right-hand . .79 77 figure of this result we in.i 72 g .1 7 2 8 i crease by such a number as would have been carAns. 1 4.5 5 4 3 6 ried, if the repetends had been continued farther to the right. In that case we should have had to carry 3 after finding the amount of the first left-hand column of the repetends continued. We therefore increase the sum as first found, and thus have the true amount as in the operation, 14.55436. RULE. Make the given repetends, when dissimilar, similar and conterminous. Add as in addition of finite decimals, observing to increase the repetend of the amount by the number, if any, to be carried from the left-hand column of the repetends. EXAMPLES 2. Add 3.5, 7.65i, 1.763, 6.173, 51.7, 3.7, 27.63i, and 1.003 together. Ans. 103.2591227. 3. Reduce }, }, and f to decimals, and find their sum. 4. Find the sum of 27.56, 5.632, 6.7, 16.356, 7i, and 6.1234. Ans. 63.1690670868888. 5. Add together .165002, 31.64, 1.06, .34634, and 13. 6. Add together .87, .s, and 1876. Ans. 2.644553. 7. Required the value of 3 +.45 +.45 +.35i +.6468 +.6468 +.6468, and .6468. Ans. 4.1766345618. 8. Find the value of 1.25 + 3.4 +.637 + 7.885 + 7.875 + 7.875 + 11.i. Ans. 40.079360724. 9. Add together 131.613, 15.00i, 67.134, and 1000.63. 10. Find the value of 5.16345 + 8.6381 +3.75. Ans. 17.55919120847374090302. SUBTRACTION OF CIRCULATING DECIMALS. OPERATION milar and Conterminous. Dissimilar 304. Ex. 1. From 87.1645 take 19.479167. Ans. 67.685377. Having made the repe tends similar and conter8 7.1 6 4 5 8 7.1 6 45 45 minous, we subtract as in 1 9.4 7 9 167 1 9.4 7 9 1 6 Ż whole numbers, regarding, however, the right-hand Ans. 6 7.6 8 5 3 7 7 figure of the subtrahend as increased by 1, since 1 would have been carried to it in subtracting, if the repetends had been continued farther to the right, as is evident from the circulating part of the subtrahend being greater than that of the minuend. Rule. - Make the repetends, when dissimilar, similar and conterminous. Subtract as in subtraction of finite decimals ; observing to regard the repetend of the subtrahend as increased by 1, when it exceeds that of the minuend. EXAMPLES. 2. From 7.i take 5.02. Ans. 2.08. 3. From 315.87 take 78.0378. Ans. 237.838072095497. 4. Subtract from. Ans. .079365. 5. From 16.1347 take 11.0884. Ans. 5.0462. 6. From 18.1678 take 3.27. Ans. 14.895i. 7. From 3.123 take 0.7i. Ans. 2.40595i. 8. From take it Ans. .246753. 9. From take 4. Ans. .i58730. 10. From take : Ans. .176470588235294i. 11. From 5.12345 take 2.3523456. Ans. 2.7711055821666927777988888599994. MULTIPLICATION OF CIRCULATING DECIMALS. OPERATION. 305.° Ex. 1. Multiply .36 by 25. Ans. 0929. We change the 36 28 23 .36 to given numbers 10 their equivalent com mon fractions, and, ti X Mon=.0929 Ans. multiplying, obtain which, reduced to its equivalent decimal, gives .0925, the answer required. 92 990 46 4969 Rule. - Change the given numbers to their equivalent common fractions. Multiply them together, and reduce the product to its equivalent decimal. EXAMPLES. 2. Multiply 87.32586 by 4.37. Ans. 381.6140338. 3. Multiply 582.347 by .03. 4. Multiply 3.145 by 4.297. Ans. 13.5169533. 5. What is the value of .285714 of a guinea ? Ans. 8s. 6. What is the value of .461607142857 of a ton? Ans. 9cwt. Oqr. 23-+lb. 7. What is the value of .284931506 of a year? Ans. 104d. DIVISION OF CIRCULATING DECIMALS. OPERATION. ;.15 306.° Ex. 1. Divide .54 by .15. Ans. 3.506493. We change 54 6 15 14 7 the given num.54 bers to their 99 11 10 90 f1 • 25 equivalent com339 3.506493 Ans. mon fractions, and, dividing obtain 272, which, reduced to its equivalent decimal, gives 3.506493, the answer required. RULE. Change the given numbers to their equivalent common fractions. Divide, and reduce the quotient to its equivalent decimal. EXAMPLES. 2. Divide 345.8 by .6. Ans. 518.83. 3. Divide 234.6 by 7. 4. Divide 13.5i69533 by 3.145. Ans. 4.297. 5. Divide 381.6140338 by 4.37. Ans. 87.32586. 6. Divide .428571 by .625. 7. Find the value of 2.370 • 4.923076. Ans. .48i. 8. Find the value of .09 230769. Ans..39. 9. Find the value of 316.31015 • .3. 10. Find the value of 100006 : ... 11. Divide .36 by .25. Ans. 1.422924901185770750988i. CONTINUED FRACTIONS. 307. A CONTINUED FRACTION is a fraction having for its numerator 1, and for its denominator a whole number plus a fraction whose numerator is 1, and whose denominator is a whole number plus a fraction, and so on. Thus, 1 4+ , is a continued fraction. The partial fractions composing the parts of a continued fraction are called its terms. Thus, in the fraction given above, }, }, }, &c. are its terms. 308. Continued fractions are used in obtaining, in smaller numbers, the approximate values of fractions whose terms, when reduced to their simplest forms, are expressed in numbers inconveniently large. 309.° To transform a common fraction into a continued fraction, and to find, in smaller numbers, its approximate values. Ex. 1. Transform to into a continued fraction, and find its several approximate values. OPERATION. 1 6 + } 19) 60(3 19 57 Hence, 3 + 1 1 1st approx. value, 1)3(3 3 3 Ans. 3 1 x 6 6 2d 0 (3 X 6) +1 19' approx. value, (6 X 3)+1 19 the original value, (19 X 3) + 3 60' Dividing both terms of 1: by the numerator, which operation will not change the value expressed (Art. 217), the fraction becomes 1 ; the denominator of which being between 3 and 4, the value of 3_3 19 3 3 + 19 the given fraction must be between f and ; and neglecting the fraction 18, for the present, in the denominator, we have } for the first approximate value. This approximation, however, is greater than the true value, since the denominator, 3, is less than the true denominator 33. We therefore divide both terms of the lo by its nu 1 merator, and it becomes which is between £ and 1. By neglecting 63" the fraction f in the denominator, and taking the į instead of the 1 1 we have is for the second approximate value of 31 the given fraction; which approximation is too small, since in the denominator, instead of us, we used ], which is greater than the If we now include in the calculation the remaining partial fraction 1 1 }, we have bi, the original fraction, 34 +438 By the processes of the operation it will be seen that the first approximate value sought was obtained by disregarding all the partial fractions after the first, the second approximate value by disregarding all the partial fractions after the second, &c. Rule. Diride the greater term of the given fraction by the less, and the divisor by the remainder, and so on, as in finding the greatest common divisor. The quotients thus found will be the denominators of the several terms of the continued fraction, and the numerator of each will be 1. For the First approximation, take the first terms of the continued fraction. For the second approximation, multiply the terms of the first approcimate fraction by the denominator of the second term of the continued fraction, adding 1 to the product of the denominators. For each sucCEEDING approximation, multiply the terms of the approximation last found by the denominator of the next term of the continued fraction, and add the corresponding terms of the preceding approximation. NOTE 1. – When the fraction given is improper, the true approximations will be the reciprocals of the fractions found by the rule. NOTE 2. — In a series of approximations the first is larger, the second smaller, and so on, every odd fraction being larger, and every even one smaller, than the given fraction.. Each successive approximate fraction, however, approaches more nearly than the one preceding it to the value of the given fraction. When the continued fraction indicates many approximations, it is generally sufficient for ordinary purposes to find only from three to six of them. NOTE 3. — A continued fraction may, for convenience, be expressed by writing its terms one directly after another, with the sign plus (+-) between the denominators; thus, the continued fraction equivalent to ai may be written ++1 |