Note. In order to prevent mistakes in the proof of examples in Division of Decimals, always contrive in the process to separate 10, 100, &c. in the two fractions from the other figures, as in the above examples; and be sure never to effect the multiplication if there be tens left in the denominator; nor, if there be tens left in the numerator, to effect it until the last step of the operation. Ex. Divide 172.9 by 142 to three places of decimals. •142) 172-900000 (1217-605 142 309 284 250 142 1080 994 860 852 800 710 Here we must affix 5 cyphers to 172.9; for if we affix two according to the rule, the division up to that point will give the integral part of the quotient only, and therefore as the quotient is to be obtained to three places of decimals, we must affix three cyphers more, that is, we must affix five altogether. 1. Divide, (proving the truth of each result by Fractions): (1) 10.836 by 5'16, and 34.96818 by 381. (5) 31.5 by 126, and 5.2 by 32. (6) 3217 by 0625, and 03217 by 6250. (11) 130'4 by 0004 and by 4, and 46·634205 by 4807 65. (17) 816 by 0004, and 0019610652875 by 2.38645. (20) 684-1197 by 1200-21, and also by 0120021. 2. Divide to four places of decimals each of the following, and prove the truth of the results by Fractions: (1) 32.5 by 87; :02 by 17; 1 by 013.. (2) 009384 by '0063; 51846·734 by 1·02. (3) 7380 964 by 023; 6.5 by 3:42; 25 by 19. (4) 176432 76 by 01257; 7457·1345 by 6535496*2. 3. Find the quotient (verifying each result) of (1) 0029202 by 157, and by 1.57. (2) 5005 by 1953125; of 50.05 by 195-3125; of 05005 by ⚫0001953125. (3) (7 of 1+1) by 0005; of 31-008 by 8 of 11 of 52536; 7575 18190; by 16. 93. Certain Vulgar Fractions can be expressed accurately as Decimals. RULE. Reduce the fraction to its lowest terms; then place a dot after the numerator and affix cyphers for decimals; divide by the denominator, as in division of decimals, and the quotient will be the decimal required. There is one decimal place in the dividend and none in the divisor; therefore there is one decimal place in the quotient. Note. In reducing any such fraction as or to a decimal, we may proceed in the same way as if we were reducing ; taking care however in the result to move the decimal point one place further to the left for each cypher cut off. ='006, for in fact, we divide by 5, and then by 10, 100, &c., according as the divisor is 50, 500, &c. Ex. 4. Convert &+31+2+6 into a decimal. 40 125 Note. 10 is sometimes called the first power of 10, 247 17 11 (10) + 1512 + +2000+ 62.5 10 × 10.. 10 x 10 x 10.. second power of 10, ............ ... third power of 10, 10 x 10 x 10 x 10 x 10......... fifth power of 10, and so on; similarly of other numbers. 94. We have seen that, in order to convert a vulgar fraction into a decimal, after reducing the fraction to its lowest terms and affixing cyphers to the numerator, we have in fact to divide 10, or some multiple of 10 or of its powers, by the denominator of the fraction: now 10=2×5, and these are the only factors into which 10 can be broken up; therefore, when the fraction is in its lowest terms, if the denominator be not composed solely of the factors 2 and 5, or one of them, or of powers of 2 and 5, or one of them, then the division of the numerator by the denominator will never terminate. Decimals of this kind, that is, which never terminate, are called indeterminate decimals, and they are also called CIRCULATING, REPEATING, or RECURRING DECIMALS, from the fact that when a decimal does not terminate, the same figures must come round again, or recur, or be repeated: for since we always affix the same figure to the dividend, namely a cypher, whenever any former remainder recurs, the quotient will also recur. Now when we divide by any number, the remainder must always be less than that number, and therefore some remainder must recur before we have obtained a number of remainders equal to the number of units in the divisor. 95. PURE CIRCULATING DECIMALS are those which recur from the beginning: thus 3333..., 272727..., are pure circulating decimals. |