tient of 7258 6319. As the dividend would require 4 decimals, and the divisor that we should suppress 3 figures, we place one cipher on the right of 7258; and, proceding as before, we point off, in the quotient, 4 figures for decimals. The true quotient being 1,148599+, the quotient obtained by contraction is within not only a ten-thousandth, but a millionth of a unit. 248. When there are decimals in one or both of the given numbers, we may render them integral, (226 and 238,) and operate as in the preceding example. Thus, to find the quotient of 562,49,3927, within a thousandth, we have (238) 56249003927. Then, as the dividend would require 3 decimals, and the divisor the suppression of 3 figures, the numbers remain as they are. As the figure 3, increased by a unit, and consequently too large, is contained in the remainder 23 nearly 6 times, we place 6 in the quotient with the sign The above is easily applied to the reduction of a vulgar fraction to a decimal having a limited number of figures. Thus, if we would have the value of 35 within a ten-thousandth, as the dividend requires 4 ciphers, and the divisor the suppression of two figures, we divide 37500 by 629, as usual, and have ,5962 for the quotient. 249. By the suppression of its units figure, the divisor, each time, becomes of a ten fold higher order, and requires the quotient to be of a ten fold lower order to give an invariable product. Each new product and new dividend is therefore of the same order; that is, all the products and dividends are of the order of the right-hand figure of the first-prepared dividend. The division of the suppressed part of the dividend by the divisor, could not give a unit in the quotient; that is to say, the quotient cannot, on account of this suppression, be a unit too small. Again, the suppression of the figures of the divisor tends to increase the quotient; that is, to counteract the error occasioned by the suppression of part of the dividend. Also, as we take the divisor within half a unit, increasing, when necessary, its right-hand figure; the error, which cannot exceed about half a unit of the next order on the left of the denominating, or right-hand order, is sometimes plus and sometimes minus. Hence, we infer, that for all ordinary purposes, where only a small number of decimals is required, the method, as an expedite approximation, may be relied on. Required the quotient of ,527694735,23748, within a ten-thousandth. Removing the comma 5 places to the right, we have 52769,473523748, (237:) then, as the dividend is a number of ten-thousandths, and the divisor requires the suppression of 4 figures, we divide thus: 23748) 52769 (2,2220 523 24 49 The divisor 24 being too great and the last fraction, we may, if we please, write the quotient thus: 2,2211. The true quotient is 2,222059+. 250. When the denominator of a fraction, which we would reduce to a decimal, is the product of two or more convenient factors, we divide as in article 219; except that here we can carry the division, where it is not exact, to any proposed degree of approximation, without noticing the final remainders. For example, to find the decimal value of 120, within a ten-thousandth, we proceed thus: 11) 120,0000 11) 10,9090 ,9917 + The quotient,9917 is (37) within a ten-thousandth. 4 Again, to have the value of 118, within a ten-thousandth, as it is easy to perceive, from what has been said on the properties of numbers, that 4955 X 9 X 11, we proceed thus: 5) 119,0000 9) 23,8000 11) 2,6444 ,2404 As the next figure would be a cipher, the quotient ,2404 is not only within a ten-thousandth, but within a hundredthousandth. 1. 2. 3 Examples. ,5; 1,25; 1,75; 1,125. ,375;=,625; 7,875;=,0625. 3.,1875; 5,3125; 6,4375. = 16 4.,5625; 1,6875; 1,8125; 18,9375. The above fractions being of frequent recurrence in business, the scholar should remember their decimal equivalents. 5. 88,998347107438016+. =,998737373+. Let the following divisions be performed according to the method of contraction, art. 248 and following: 7. Required the value of 31 8991 within a ten-thousandth. Answer, ,0345 —. 8. Required the value of 161, within a hundred-thousandth. 8999 Answer, ,17909-. Answer, ,10055. 9. Required the value of 18, within a hundred-thousandth. 10. Required the value of 868, within a ten-thousandth. 11. Required the value of sandth. 8999 Answer, ,9655. 87012, within a hundred-thouAnswer, ,99832. 671327 12, within a hundred-thou 12. Required the value sandth. of 113 Answer, ,00168. SECTION XII. REPEATING DECIMALS. 251. WHEN the value of a fraction cannot be exactly expressed in decimals, we find, by sufficiently extending the division, a periodical return of the same figures in the quotient without end, whence they are often called periodical, circulating, or infinite decimals. Thus, in finding the value of 1, we have, 6363, &c.; where, in continuing the division, the figures 6 and 3 will succeed each other without end. The reason of this will appear evident as follows: the remainder must be one of the numbers 1, 2, 3, &c., but cannot equal the divisor: therefore, if we have a different remainder at each division, we cannot perform as many divisions as there are units in the divisor, without falling upon some remainder that we have had before, in which case the quotient figures will return in the same order. 252. The figures which repeat, constitute what is called the period. Thus, in the expression,6363, &c., the period is 63. In the expression ,568568, &c., the period is 568. To distinguish the period with facility, if it contains only one figure, a point is placed over this figure: thus,,1111, &c. is written,Ỉ: and 333, &c. is written,3. If it contains more figures than one, it is distinguished by two points, one of which is placed over the first figure, and the other over the last, thus: ,6363, &c. is written,63; and ,568568, &c. is written,568. 253. As a unit, in any place towards the left, is composed of a number expressed by as many nines as there are places on the right of it, plus 1; it is plain that in dividing a unit by 9,99,999, &c., as the remainder will always be a unit, the only significant figure in the continued quotient will also be a unit. Thus we see that =,11111, &c. ;=,0101, &c.; } = 999 ,001001, &c. = 63 In Now we may consider the periodical decimal fraction,6363, &c. ; or,63 as the product of ,0101, &c., or,01 multiplied by 63. But,,01; therefore, ,631 × 63 = §§. the same manner we find that any periodical decimal fraction is equivalent to the vulgar fraction of which the numerator is the period, and the denominator as many nines as there are 148 FRACTIONAL EQUIVALENTS OF REPEATING DECIMALS. places of figures in the period. Thus, 568-568;,016 9997 254. It frequently happens that the period does not commence with the first decimal figure, in which case, we consider the figures on the left of the period as units. On the right of these units, we place the vulgar fraction equivalent to the period, and reduce the mixed number, thus formed, to an improper fraction. Lastly, on the right of the denominator we place as many ciphers as there are decimal places between the period and the comma. For example, if we would reduce,325656, &c., or ,3256 to its equivalent vulgar fraction, as the period is equal to 58, the fraction,3256 is,325§. 99 Suppressing the comma, and reducing 3255 to an improper fraction, we have 3224 But, in suppressing the comma, we render the number 100 times its former value; we must therefore divide 3224 by 100, which is done (55 and 165) in placing two ciphers on the right of the denominator 99. We have therefore,3256-3331. 99 It is easy to see that when the figures between the comma and the period are ciphers, we may proceed at once to express the period by its equivalent fraction, and place those ciphers on the right of its denominator: also that, in many cases, we may reduce the fractional equivalents to lower terms. For exam ple,,002475 2475 225 25 999900 08350 X 1000=101· Let this example be proved by reducing to a decimal. 2.,03727, and,0037 = 240 4.,729 = 27, and ,00764,357 37 99000 5.,0099- =T&T, and,0002475=1010 255. When two figures repeat, the denominator of the fractional equivalent (253) is 99. But 999 X 11; wherefore, if the two figures which repeat be a multiple of 9, as the fac |