2. Any pure periodical may be considered as mixed, by taking the given period for the terminate part, and making the given period the interminate part. Thus .46.46+.0046, &c. 3. A single period may be regarded as a compound periodical. Thus .3 may become .33, or .333; so. .63 may be made .6333, or .63333, &c. 4. A single period may also be made to begin at a lower order, regarding its higher orders as terminate decimals. Thus .3 may be made .33, or .3333, &c. 5. Compound periods may also be made to begin at a lower order. Thus .36 may be changed to .363, or .36363, &c.; or by extending the number of places .479 may be made .47979, or 4797979, &c.; or making both changes at once .532 may be changed to .5325325, &c. Hence, 360. To make any number of dissimilar periodical decimals similar. Move the points, so that each period shall begin at the same order as the period which has the most figures in its terminate part. 21. Change 6.814, 3.26, and .083 to similar and conterminous periods. Operation. 6.814 6.8148148i 3.26 3.26262626 .083 0.08333333 Having made the given periods similar, the next step is to make them conterminous. Now as one of the given periods contains 3 figures, another 2, and the other 1, it is evident the new periodical must contain a number of figures which is some multiple of the number of figures in the different periods; viz: 3, 2, and 1. But the least common multiple of 3, 2, and 1 is 6; therefore the new periods must at least contain 6 figures. Hence, 361. To make any number of dissimilar periodical decimals, similar and conterminous. First make the periods similar; (Art. 360;) then extend the figures of each to as many places, as there are units in the least common multiple of the NUMBER of periodical figures contained in each of the given decimals. (Art. 176.) 27. Change 46.162, 5.26, 63.423, .486, and 12.5, to similar and conterminous periodicals. Operation. 46.162 46.16216216 5.26262626 63.423-63.42342342 5.26 .486 12.5 0.48666666 12.50000000 The numbers of periodical figures in the given decimals are 3, 2, 3, and 1; and the least common multiple of them is 6. Therefore the new periods must each have 6 figures. 23. Make .27, .3, and .045 similar and conterminous. ADDITION OF CIRCULATING DECIMALS. Ex. 1. What is the sum of 17.23+41.2476+8.61+1.5+ 35.423? Operation. Dissimilar. Sim. & Conterminous. 17.23 =17.2323232 41.2476=41.2476476 8.61 = 1.5 = 8.6161616 35.423 35.4232323 = First make the given decimals similar and conterminous. (Art. 361.) Then add the periodical parts as in simple addition, and since there are 6 figures in the period, divide their sum by 999999; for this would be its denominator, if the sum of the periodicals were expressed by a common fraction. (Art. 355.) Setting down the remainder for the repeating decimals, carry the quotient 1 to the next column, and proceed as in addition of whole numbers. Hence, Ans. 104.0193648 362. We derive the following general RULE FOR ADDING CIRCULATING DECIMALS. First make the periods similar and conterminous, and find their sum as in Simple Addition. Divide this sum by as many 9s as there are figures in the period, set the remainder under the figures added for the period of the sum, carry the quotient to the next column, and proceed with the rest as in Simple Addition. OBS. If the remainder has not so many figures as the period, ciphers must be prefixed to make up the deficiency. 2. What is the sum of 24.132+2.23+85.24+67.6 ? 3. What is the sum of 328.126+81.23+5.624+61.6? 4. What is the sum of 31.62+7.824+8.392+.027? 5. What is the sum of 462.34+60.82+71.164+.35? 6. What is the sum of 60.25+.34+6.435+.45+45.24 ?. 7. What is the sum of 9.814+1.5+87.26+0.83+124.09 ? 8. What is the sum of 3.6+78.3476+735.3+375+.27+ 187.4? 9. What is the sum of 5391.357+72.38+187.2i+4.2965+ 217.8496+42.176+.523+58.30048 ? 10. What is the sum of .162+134.09+2.93+97.26+3.769230 +99.083+1.5+.814 ? SUBTRACTION OF CIRCULATING DECIMALS. Ex. 1. From 52.86 take 8.37235. Operation. 52.86=52.86868 8.37235= 8.37235 44.49632 We first make the given decimals similar and conterminous, then subtract as in whole numbers. But since the period in the lower line is larger than that above it, we must borrow 1 from the next higher order. This will make the right hand figure of the remainder one less than if it was a terminate decimal. Hence, 363. We derive the following general RULE FOR SUBTRACTING CIRCULATING DECIMALS. Make the periods similar and conterminous, and subtract as in whole numbers. If the period in the lower line is larger than that above it, diminish the right hand figure of the remainder by 1. OBS. The reason for diminishing the right hand figure of the remainder by 1, if the period in the lower line is larger than that above it, may be explained thus: When the period in the lower line is larger than that above it, we must evidently borrow 1 from the next higher order. Now if the given decimals were extended to a second period, in this period the lower number would also be larger than that above it, and therefore we must borrow 1. But having bor rowed 1 in the second period, we must also carry one to the next figure in the lower line, or, what is the same in effect, diminish the right hand figure of the remainder by 1. 2. From 85.62 take 13.76432. Ans. 71.86193. 3. From 476.32 take 84.7697. 4. From 3.8564 take .0382. 9. From .634852 take .021. 10. From 8482.421 take 6031.035. MULTIPLICATION OF CIRCULATING DECIMALS. Ex. 1. What is the product of .36 into .25? RULE FOR MULTIPLYING CIRCULATING DECIMALS. First reduce the given periodicals to common fractions, and multiply them together as usual. (Art. 219.) Finally, reduce the product to decimals and it will be the answer required. OBS. If the numerators and denominators have common factors, the operation may be contracted by canceling those factors before the multiplication is performed. (Art. 221.) 2. What is the product of 37.23 into .26? Ans. 9.928. 3. What is the product of .123 into .&? 4. What is the product of .245 into 7.3? 5. What is the product of 24.6 into 15.7? 7. What is the product of 8574.3 into 87.5? 8. What is the product of 3.973 into 8? 9. What is the product of 49640.54 into .70503? 10. What is the product of 7.72 into .297 ? DIVISION OF CIRCULATING DECIMALS. Ex. 1. Divide 234.6 by .7. · Operation. Now 2847=zg4×4=6330 We first reduce the divisor and dividend to common fractions; (Art. 358;) and divide one by the other; (Art. 229;) then reduce the quotient to a decimal. (Art. 337.) Hence, 365. We derive the following general RULE FOR DIVIDING CIRCULATING DECIMALS. Reduce the divisor and dividend to common fractions; divide one fraction by the other, and reduce the quotient to decimals. OBS. After the divisor is inverted, if the numerators and denominators have factors common to both, the operation may be contracted by canceling those factors. (Art. 232.) 2. Divide 319.28007112 by 764.5. Ans. 0.4176325. 3. Divide 18.56 by .3. 4. Divide .6 by .123. 5. Divide 2.297 by .297. 6. Divide 750730.518 by 87.5. 7. Divide 54 by .15. 8. Divide 13.5169533 by 4.297. 9. Divide 24.081 by .386. 10. Divide .36 by .25. |