APPENDIX. ON CIRCULATING DECIMALS. WHEN in reducing a vulgar fraction to a decimal, by an x ing ciphers to the numerator, and dividing by the denomninator, a remainder appears which has before appeared, the quotient can never terminate, but will consist of the same figure or series of figures in regular succession for ever, beginning with the figure next after the remainder which had before occurred, either in the production of the vulgar fraction, or in the operation of reducing it to a decimal. Thus if we attempt to reduce & to an equivalent decimal fraction, we shall find that the figure 6 will repeat in endless succession, and hence we shall see that cannot be expressed in a finite decimal. DEFINITIONS. 1. All decimals in which one or more figures repeat, are ter med circulating decimals, or recurring decimals. 2. A decimal consisting throughout of one repeating figure, is called a pure repetend; as, 111 &c., or 333 &c. 3. A decimal consisting throughout of a series of figures repeating in regular order, whatever may be the number of figures in the series, is called a compound repetend; as, 47474747 &c., or 325325325 &c. 4. When a decimal has in it other figures than those which repeat, it is styled a mixed circulate or mixed repetend; as, •42999 &c., or '130101 &c. NOTATION. A pure repetend is denoted by writing the repeating figure once, with a dot over it; as, 2, the decimal arising from the vulgar fraction; or 3, the decimal equivalent to }. A compound repetend is expressed by writing the series that repeats, and by placing a dot over the first and last figures of the series; as, .09, the decimal equal in value to, and signifying that the two figures 09 repeat for ever: in like manner 285714 will be the expression for converted to a decimal. A mixed circulate has the repeating figure or figures marked according as the repetend is pure or compound; and the figures 83 which do not recur, are left undotted; as, 138, the decimal arising from; 008497133, the decimal equivalent to and so of others. 9768 NOTE. It sometimes happens in compound repetends, that part of the circulate is integral. REDUCTION OF CIRCULATING_DECIMALS. PROBLEM I To reduce a pure repetend to its equivalent vulgar fraction. RULE. Write the figure or figures of the repetend for a numerator, and under this for denominator set as many nines as there are repeating figures in the repetend: then reduce, this vulgar fraction to its lowest terms. EXAMPLES. 1. Required the vulgar fraction equivalent to 3. Writing 3 for numerator, and setting one 9 under it, we have for Answer. *If instead of 3, the given repetend had been 333, then the first fraction would have been 333 as before. 999 2. Required the vulgar fraction equal in value to 7. Ans. Ļ 3. Required the vulgar fraction equal in value to '66. PROBLEM II. Ans. 66. To reduce a compound repetend to its equivalent vulgar fraction. RULE. Write the significant figures of the repetend, for numerator; and under them set as many nines, as there are figures in the repetend: but if part of the repetend be integral, then as many ciphers must be annexed to the numerator, as there are integral figures in the repetend. EXAMPLES 1. Required the vulgar fraction equivalent to 18. Writing 18 for numerator, and setting two nines under, for denominator, because there are two figures in the repetend, we have for Answer. 2. Required the vulgar fraction equal in value to 3∙17. Here one of the figures of the repetend being integral, we annex one cipher to them for numerator, and then set three nines under these for denominator. Hence the fraction required is 3. Required the vulgar fraction equivalent to 450.9. Ans. 409000501000 4. Required the vulgar fraction equivalent to 6·4321 PROBLEM III. 7777 Ans. 643210 $5995 To reduce a mixed circulate to its equivalent vulgar fraction. RULE. Write the whole decimal as an integer, and from this integer subtract the figures that do not repeat, considered also as an integral number: the remainder will be the numerator of the fraction required. Then for denominator set as many nines as the given repetend contains repeating figures, and to them anmex as many ciphers as there were figures not repeating. EXAMPLES. 1. Required the vulgar fraction equal in value to '79. +7979-7, the fraction required. 2. Required the fraction equivalent to 1694. 1694 1694 169, the fraction required. - 9600 3. Reduce 007039 to an equivalent vulgar fraction. 007039-7039-7, the equivalent fraction. 4. It is required to reduce 0138 to a vulgar fraction. Ans. 5. Reduce 2418 to a vulgar fraction in lowest terms. 6. What vulgar fraction is equal to 4239581. Ans. 153 Ans. 1900 4'239539 Ans. Ans. 207-5 2443 8. Reduce 8497133 to a vulgar fraction. 9. Reduce 48.71492 to an equivalent whole number and vulgar traction. Ans. 4871421 ADDITION OF CIRCULATING DECIMALS. CASE I. To add pure and compound circulates together. 99900 RULE. Continue the recurring places of the several decimals till they all end at the same distance from the decimal point. Then add as in common addition of decimals, but remark the number of integers carried beyond the point in the sum. Lastly, add this "number remarked" to the last figure of the decimals, and dot the first and last decimal figure in the new sum. EXAMPLES. 1. Required the sum of 18.3, 6.0724, 39, 40′i4, 6'2, and 156.05, Here if all the decimals be carried out to four places, they will be conterminous; that is, the repeating periods of each will terminate at the same distance from the decimal point. wherefore it will be, 18.3533 6'0724 *3939 40.1414 2. Required the sum of 9, 21.843, 72′4i, 2′904, 6′g, and '92. Here the decimals must be carried to six places to make ther conterminous; and it will be, 3. Required the sum of 129238, 147, 98.6174, 16-23, and 100.42. Ans. 242.9756. 4. It is required to add the following circulates together: 133, 14.915, 19°23, 11·201621, 91°8, 208 1234, and 75'102042. Ans. 553-79753701535¿ CASE II. To add mixed circulating Decimals together. RULE. Convert the given decimals into vulgar fractions by the rules of reduction: then add the reduced fractions together and lastly bring the sum back to a decimal. When in all the given circulates the repeating part of the decimal begins at the same distance from the decimal point, addition may be performed without reducing the decimals to their equivalent vulgar fractions: and this is done as follows: Make all the decimals conterminous, as directed in the FIRST CASE: then add, as in common addition of decimals, noting the number carried from the repeating to the finite decimals. Lastly, add this “number noted" to the last figure of the sum, and dot the figures in the sum, under the first and last of the repeating columns. NOTE. Pure repetends may be supposed to begin to circu late at any place we find most convenient, and with attention to this, many operations may be shortened. EXAMPLES. 1. Required the sum of 14623, 169°45012, 7:37432186, and 206'0434. In this example the given mixed circulates all begin to repeat at the same distance from the point, viz. in the third decimal place; wherefore addition may be performed without reducing the decimals to vulgar fractions: thus, 14.62333333 169 45012012 7.37432186 206 04342434 Add 1 carried from the repeaters 1 Second sum 397 49120966 Answer. 2. Required the sum of 14°213, 172′5, 2·84928, 46°24, and 219'7. Here addition may be performed without reducing the given numbers to vulgar fractions, because the several repetends may be made conterminous, as follows: 14.21333 172-55555 2.84928 46.24242 219-77777 First sum 455 63835 Add 2 carried from the repeaters 2 Second sum 455-63837 Answer 3. Required the sum of ·002, 4·1, 53·9i¿, ·0427, and eg. Ans. 64-2904198). |