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3. Reduce 17 to a mixed number. Ans. 1.0625.
338. It will be seen that the last two examples cannot be exactly reduced to decimals; for there will continue to be a remainder after each division, as long as we continue the operation.
In the 20th, the remainder is always 4; in the 21st, after obtaining three figures in the quotient, the remainder is the same as the given numerator, and the next three figures in the quotient are the same as the first three, when the same remainder will recur again. The same remainders, and consequently the same figures in the quotient, will thus continue to recur, as long as the operation is continued.
339. Decimals which consist of the same figure or set of figures continually repeated, as in the last two examples, are called Periodical or Circulating Decimals; also, Repeating Decimals, or Repetends.
OBs. When only a single figure is repeated, it is more accurate to call them repeating decimals; but when two or more figures recur at regular intervals, they are very properly called periodical, or circulating decimals.
340. When a common fraction can be exactly expressed by a decimal, the decimal is said to be terminate, or finite ; but when it cannot be exactly expressed by a decimal, it is said to be interninate, or infinite.
OBs. It seems to be incongruous to call a fraction infinite. (Art. 180.) The term infinite, however, does not refer to the value of the fraction, but to the number of decimal figures required to express its value.
QUEST.--Obs. When there are not so many figures in the quotient as you have annexed ciphers, what is to be done ? 339. What are periodical or circulating decimals ? 340. Wheu is a decimal terminate ? When interminate ?
341. If the denominator of a common fraction when reduced to its lowest term, contains no prime factors but 2 and 5, its equivalent decimal will terminate ; on the other hand, if it contains any other prime factor besides 2 and 5, it will not terminate.
Thus ó reduced to its lowest terms, becomes to, and the prime factors of 20 are 2, 2, and 5; that is, 20=2X2X5. (Art. 165.) We also find that on=.05; it is therefore terminate. Again, p=ts; and the prime factors of 15 are 3 and 5; that is, 15= 3X5; and *=.0666666, &c.; it is therefore interminate. Hence,
342. To ascertain whether a common fraction can be exactly expressed by decimals.
Reduce the given fraction to its lowest terms, and then resolve its denominator into its prime factors. (Art. 341.)
Obs. The truth of this principle is evident from the consideration, that annexing ciphers to the numerator, multiplies it successively by 10; but 2 and 5 are the prime factors of 10, and are the only numbers that can divide it without a remainder. (Art. 165. Obs. 2.) But any number that measures another, must also measure its product into any whole number; (Art. 161. Prop. 14;) consequently, when the denominator contains no prime factors but 2 and 5, the division will terminate ; but when it contains other factors, the division can not terminate,
22. Will to produce a terminate or interminate decimal ? 23. Will is produce a terminate or interminate decimal ? 24. Will 37 produce a terminate or interminate decimal ? 25. Will to produce a terminate or interminate decimal ? 26. Will produce a terminate or interminate decimal ? 27. Will A produce a terminate or interminate decimal ? 28. Will 3% produce a terminate or interminate decimal ?
343. When the decimal is terminate, the number of figures it contains, must be equal to the greatest number of times that either of the prime factors 2 or 5, is repeated in its denominator, when the given fraction is reduced to its lowest terms.
OBs. The truth of this principle may be illustrated thus: \5.5; that is, the decimal terminates with one place ; for, the denominator 2, is taken only onceas a factor in 10, and therefore only one cipher is required to be annexed to the numerator to reduce it. Again, 1=.25, which contains two decimal places. Now the denominator 4=2X2; and since 2 is contained only once as a factor in 10, it is evident that 10 must be repeated as many times as a factor in the numerator, as 2 is taken times as a factor in the denominator, in order to reduce the fraction.
For the same reason } will terminate with three places, and is equal to .125; for, 8=2X2X2. So =.2; that is, the decimal terminates with one place; for, since its denominator 5, is taken only once as a factor in 10, it is necessary to add only one cipher to its numerator in order to reduce it. In like manner it may be shown that the number of figures contained in any terminate decimal, is equal to the greatest number of times that either of the prime factors 2 or 5, is repeated in the denominator of the given fraction.
The same reasoning will evidently hold true when the numerator is 2, 3, 4, 5, &c., or any number greater than 1. In this case the decimal will be as many times greater, as the numerator is greater than 1.
344. The number of figures in the period must always be one less than there are units in the denominator ; for, the number of remainders different from each other which can arise from any operation in division, must necessarily be one less than the units in the divisor. For example, in dividing by 7, it is evident, the only possible remainders are 1, 2, 3, 4, 5, and 6; and since in reducing a common fraction to a decimal, a cipher is annexed to each remainder, there cannot be more than six different dividends; consequently, there cannot be more than six different figures in the quotient. Thus, 7=.142857,142857, &c.
When the decimal is periodical or circulating, it is customary to write the period but once, and put a dot, or accent over the first and last figure of the period to denote its continuance. Thus, .46135135135, &c., is written .46135, and .633333, &c.,.63,
Reduce the following fractions to circulating decimals : 31. B. 36. b. 41. .
37. 4. 42.
45. . 50..
decimal figures are required to express Tib? 54. How many decimal figures are necessary to express othe?
55. How many decimal figures are necessary to express ? 56. How
decimal figures are necessary to express Tott? 57. Reduce to a decimal. 59. Reduce go to a decimal. 58. Reduce ts to a decimal. 60. Reduce 7 to a decimal.
Note. For the method of finding the value of periodical decimals, or of reducing them to Common Fractions, also of adding, subtracting, multiplying, and dividing them, see the next Section.
345. Compound Numbers reduced to Decimals. Ex. 1. Reduce 13s. 6d. to the decimal of a pound.
Analysis.-13s. 6d.=162d., and £1=240d. (Art. 281.) Now 162d. is H1 % of 240d. The question therefore resolves itself into this : reduce the fraction 20% to decimals. Ans. £.675. Hence,
346. To reduce a compound number to the decimal of a higher denomination.
First reduce the given compound number to a common fraction; (Art. 295;) then reduce the common fraction to a decimal. (Art. 337.).
2. Reduce 4s. 9d. to the decimal of £1. Ans. £.2375.
9. Reduce 2 fur. 2 rods to the decimal of a mile. 10. Reduce 15 min. 30 sec. to the decimal of an hour. 11. Reduce 3 hrs. 3 min. to the decimal of a day. 12. Reduce 5 lbs. 4 oz. to the decimal of a cwt. 13. Reduce 7 oz. 8 drams to the decimal of a pound. 14. Reduce 3 pks. 4 qts. to the decimal of a bushel. 15. Reduce 4 qts. 1 pt. to the decimal of a peck. 16. Reduce 4 qts. 1 pt. to the decimal of a gallon. QUEST.–346. How is a compound number reduced to the decimal of a higher denom iuation ?
347. Decimal Compound Numbers reduced to whole ones. 1. Reduce £.387 to shillings, pence and farthings.
Operation. Multiply the given decimal by 20, because £.387 20s. make £1, and point off as many figures
20 for decimals, as there are decimal places in shil. 7.740 the multiplier and multiplicand. (Art. 324.) 12 The product is in shillings and a decimal
of a shilling. Then multiply the decimal 4 of a shilling by 12, and point off as befar. 3.520 fore, &c. The numbers on the left of the Ans. 7s. 8d. 3 far. decimal points, viz: 7s. 8d. 3 far., form the
Hence, 348. To reduce a decimal compound number to whole numbers of lower denominations.
Multiply the given decimal by that number which it takes of the next lower denomination to make one of this higher, as in reduction, and point off the product, as in multiplication of decimal fractions. (Art. 324.) Proceed in this manner with the decimal figures of each succeeding product, and the numbers on the left of the decimal point of the several products, will be the whole number required.
2. Reduce £.725 to shillings, pence and farthings.
QUEST.—348. How are decimal compound numbers reduced to whole ones of a lower denomination ?