13. Change .1844 to a common fraction, &c. 14. Change .0556 to a common fraction, &c. 15. Change .1216 to a common fraction, &c. 16. Change .2005 to a common fraction, &c. 17. Change .0015 to a common fraction, &c. 336. Common Fractions reduced to Decimals. Ex. 1. Change to a decimal. Suggestion. Multiplying both terms by 10, the fraction becomes 48. Again, dividing both terms by 5, it becomes 1. (Art. 191.) But 1.8, which is the decimal required. (Art. 311.) 5 Note.--Since we make no use of the denominator 10 after it is obtained, we may omit the process of getting it; for, if we annex a cipher to the numerator and divide it by 5, we shall obtain the same result. Operation. 5)4.0 .8 Ans. A decimal point is prefixed to the quotient to distinguish it from a whole number. 2. Reduce to a decimal. Ans. 0.625. 337. Hence, to reduce a Common Fraction to a Decimal. Annex ciphers to the numerator and divide it by the denominator. Point off as many decimal figures in the quotient, as you have annexed ciphers to the numerator. OBS. 1. If there are not as many figures in the quotient as you have annexed ciphers to the numerator, supply the deficiency by prefixing ciphers to the quotient. 2. The reason of this rule may be illustrated thus. Annexing a cipher to the numerator multiplies the fraction by 10. (Arts. 98, 186.) If, therefore, the numerator with a cipher annexed to it, is divided by the denominator, the quotient will obviously be ten times too large. (Art. 141.) Hence, in order to obtain the true quotient, or a decimal equal to the given fraction, the quotient thus obtained must be divided by 10, which is done by pointing off one figure. (Art. 131.) Annexing 2 ciphers to the numerator multiplies the fraction by 100; annexing 3 ciphers by 1000, &c., consequently, 'when 2 ciphers are annexed, the quotient will be 100 times too large, and must therefore be divided by 100; when three ciphers are annexed, the quotient will be 1000 times too large, and must be divided by 1000, &c. (Art. 131.) QUEST.-337. How are Common Fractions reduced to Decimals? 6 3. Reduce 17 to a mixed number. Reduce the following fractions to decimals: 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 ob›taining 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 wher it cannot be exactly expressed by a decimal, it is said to be inter minate, 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 annexcid ciphers, what is to be done? 339. What are periodic: 1 or circulating decimals? 340. When 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 con'tains any other prime factor besides 2 and 5, it will not terminate. Thus reduced to its lowest terms, becomes, and the prime factors of 20 are 2, 2, and 5; that is, 20=2X2X5. (Art. 165.) We also find that .05; it is therefore terminate. Again, ; and the prime factors of 15 are 3 and 5; that is, 15= 3×5; and≈.0666666, &c. ; it is therefore interminate. Hence, 5 5 60 60 342. 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 mot terminate. 50 4 0 30 22. Will produce a terminate or interminate decimal? 23. Will 25 produce a terminate or interminate decimal? 24. Will produce a terminate or interminate decimal? 25. Will produce a terminate or interminate decimal? produce a terminate or interminate decimal? produce a terminate or interminate decimal? 28. Will produce a terminate or interminate decimal? 26. Will 27. Will 80 6 396 500 343. 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: 1.5; that is, the decimal terminates with one place; for, the denominator 2, is taken only once as 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=2×2; 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. 5 For the same reason for, 8=2×2×2. 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. will terminate with three places, and is equal to .125, 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, thè. 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, .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 : 3 6 51. How many decimal figures are required to express ? 52. How many decimal figures are required to express? 53. How many decimal figures are required to express T 54. How many decimal figures are necessary to express ? 3 25 ? Б 55. How many decimal figures are necessary to express 11? 56. How many decimal figures are necessary to express To24 ? 57. Reduce to a decimal. 59. Reduce to a decimal. 2 3 286 60. Reduce 37 to a decimal. Note.—For the method of finding the value of periodical decimals, or of re ducing them to Common Fractions, also of adding, subtracting, multiplying, and dividing them, see the next Section. CASE III. 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 of 240d. The question therefore resolves itself into this: reduce the fraction 142 to decimals. Ans. £.675. Hence, 240 346. 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. £.237+. 9. Reduce 2 fur. 2 ods to the decimal of a mile. 10. Reduce ¡¿ min. ɔû sec. to the decimal of an hour. 11. Reduce 3 hrs. 3 min. to the decimal of a day. QUEST.-346. How is a compound number reduced to the decimal of a higher denom lination ? |