14. Given 123.45, 3.912, 3012, and 9.385, to make them 15. Make 12.384 and 2.34 similar and conterminous. Ans. 12.3843 and 2.3444. 16. Make .1234 and 5.043219 similar and conterminous. Ans. 12344444 and 5.04321943. 17. Make 3.5, 45, and 1.084, similar and conterminous. 253. To find whether the decimal equivalent to any given vulgar fraction be finite or infinite; and if infinite, to find how many places the repetend will consist of. RULE I. Reduce the given fraction to its lowest terms, then divide the denominator of the new fraction by 10, 5, or 2, as long as division by either can be made. II. If by this division the denominator be reduced to unity, the decimal will be finite, consisting of as many places as you performed divisions. III. But if after such division the denominator, viz. the last quotient, be greater than unity, divide 9999, &c. by the said last quotient till nothing remains; the number of nines made use of will be equal to the number of figures in the repetend, which will begin after as many places of figures as there were divisions by 10, 5, or 2. 210 1120 18. Is the decimal equivalent to finite or infinite? if infinite, how many places does the repetend consist of? (2) (2) (2) (2) Then, 16 8 4...2...1; therefore the given frac ... tion produces a finite decimal, consisting of four places, viz. finite, where does the repetend begin, and how many places does it consist of? six nines are used before the work terminates. Now since 3 divisions (by 10, 5, and 2) have taken place, there will be 3 finite places; and since there are six nines employed in the division by the last quotient 7, there will be six circulating figures, beginning at 1111 101 the fourth place of decimals. Thus, = 7700 700 = = .144285714. 23 11 113 60' 12' 1141 975 and 1050' 20. Are the decimals equivalent to finite or infinite, how many places does each consist of, and what are the particulars? 254. ADDITION OF CIRCULATING DECIMALS. RULE I. Make the repetends which are to be added together similar and conterminous, (Art. 252.) II. On the right hand of each repetend place two or three of the repeating figures, and add them together for the purpose of carrying. III. Carry the tens from the left of the sum of these figures to the right hand row of figures in the repetends, and add up the whole as in finite decimals; then mark as many figures of the sum for a repetend as there are in each repetend added'. The reason of this rule is sufficiently plain; for it is evident, that all the repetends to be added must be made similar and conterminous (if they are not so already) before the operation commences: and since these repetends may be continued indefinitely, and that the sum of the right hand figures of the first repetend would, in that case, be increased by the number carried from the left hand figures of the second, and the sum of the right hand figures of the second by the number carried from the right hand figures of the third, and so on; and that these carryings would be always the same, as each arises from the addition of the same figures; it follows, that, in order to have the true repetend in the sum, the right hand figure of that repetend must be increased by the number EXAMPLES. 1. Add 2.7 + 12.3456 + .45 + 456.7 +987. + .1234 toge The repetends being made similar and conterminous, the numbers marked fig. on the right are a few of the first figures of each repetend, and are added, only to find what is to be carried to the 4. 2. Add 78.3476 + 18.6 + 735.2 + 375.1 + 187.4 + 3.27 together. similar and Sum 1398.0537082 carry 2 to the 2. 3. Add 1.2 + .35 + 12.3 + 123.4 + 4.32 together. Sum 141.675. 4. Add 17.64 + 2.8 + 4.23 + 1.83 + 54.9 together. Sunt 81.604. carried from the left; or, (which is the same,) by carrying from the numbers marked fig. (as in the 1st and 2d examples) to the said right hand figure of the repetend. 5. Add 87.12 + 9.728 + 2.3 + 4.5+ 8.08 together. Sum 111.7873145. 6. Add .123 + 4.05 + 71.6 + 12.91 + 3.123456 together. Sum 91.8799125. 255. SUBTRACTION OF CIRCULATING DECIMALS. RULE I. Make the given repetends similar and conterminous, and place the less number under the greater. II. Subtract as in finite decimals; observing, that if the lower repetend be greater than the upper, the right hand figure of the remainder must be made less by 1 than it would be were the expressions finite ". 3. From 39.2178 take 17.68. Diff. 21.530949i. 4. From 1.3 take 1.0047. Diff. 2952. 5. From 85.62 take 13.86432. Diff. 71.86193. 6. From 10.0413 take .264. Diff. 9.7766948. m The reason of this rule will be plain from the preceding note. 256. MULTIPLICATION OF CIRCULATING DECIMALS. RULE I. Turn both terms into their equivalent vulgar fractions, and multiply those fractions together. II. Reduce the product to its equivalent decimal, and let the work be continued until the decimal figures repeat; then mark the first and last repeating figures, and it will be the product required". Explanation. I first turn the decimals into their equivalent vulgar fractions; I then mul tiply the fractions together, and reduce the product to a decimal, continuing the quotient until the figures recur. 2. Multiply 12.3 and 456 together. 3 111 37 9 = .456 = = 3 999 333 = 5.629 product. 3 333 999 6. Multiply .52, 3.1 and .235 together. Prod. .38440235 &c. 7. Multiply .23, 2.34, and 1.2 together. Prod. 1.14435839 &c. 8. Multiply .4, .4, and .0004 together. Prod. .0000790123456. Prod. .0929. Prod. 9.928. Prod..27613168 &c. " The reason of this operation will be plain; for if two quantities be respectively equal to other two, the product of the first two will equal the product of the last two; wherefore, in the present case, the product of the given decimals is evidently found when the product of their equivalent vulgar fractions is found. |