23. Then begin on the left and read, giving each figure the value assigned it in numerating. 24. Or numerate and read the entire decimal, as if it were a whole number, giving the name of the last right hand place to the whole. 25. Write on the slate the decimal figures expressing the following numbers, to be numerated and read at recitation. 26. Five tenths. 27. Seventy-six hundredths. 28. Nine tenths and two hundredths. 29. Three hundred and twenty-one thousandths. 30. Five tenths, two hundredths, and six thousandths. 31. Six tenths, two hundredths, three thousandths, and one ten thousandth. 32. Six thousand nine hundred and fifteen ten-thousandth. 33. Six tenths, 1 ten thousandth, and four millionths. 34. NOTE.-Supply all vacant places with ciphers. 35. Three tenths, five thousandths, and two millionths. 36. One hundred and one thousandths. 37. To express 5 hundredths, which has one vacant place, viz. tenths, we prefix 1 cipher [.05]; to express 5 millionths, which has five vacant places, we prefix 5 ciphers [.000005], and so on to any extent. 38. Hence, to express any number of hundredths or thousandths, &c.—Prefix as many ciphers as there are vacant places between it and the separatrix. 39. Write on the slate and recite as before the following numbers. 40. Seven hundredths. 41. Forty-five ten-thousandths. 42. Six hundred thousandths and one millionth. 43. Fifteen hundred-thousandths and fifteen billionths. 44. One thousandth, one millionth, and one billionth. 45. Nine hundredths, nine thousandths, and nine billionths. 46. Three hundred and sixty-five millionths. 47. One hundred and twenty-five trillionths. 48. When a whole number has a decimal annexed, they form a mixed decimal fraction, and may be read like decimals, giving the name of the last decimal figure to both. 49. Thus 45.2 is 45 or 452, that is, 452 tenths. 50. So 5.62 is 562 hundredths, and 3.005 is 3005 thousandths. 51. In the examples .5 and .25%, if we annex a cipher to .5 it becomes .50-50 100, having the same denominator with .25, therefore, 18 Q. What are both methods of reading decimals? 23, 24. How are 6-tenths, 1-ten-thousandth and 4-millionths written in one line? 34. How are 5-hundredths or 5-millionths written, and why? 37. What is the general direction? 38. What is a mixed decimal? 48. How may the following numbers be read, viz. 45.2, 5.62, and 3.005? [See 49, 50.] 52. Whole or mixed numbers and pure decimals are easily reduced to decimals having the same denominators by simply annexing ciphers. 53. Reduce 2.5, 8.1, and 7.05, each to hundredths. A. 2.50; 8.10; 7.05. 54. Reduce 3, 17.8, and .212, to thousandths. A. 3.500; 17.800; .212. 55. Reduce the following numbers to decimals having the same denominators. 8.5; 3,21; 1; 1000; 756.3; 100000; 98110. 56. Answers. 8.50000; 3.21000; .05000; .00800; 756.30000; .00009; 981.10000. 100 57. In decimals, no single expression, containing any number of figures whatever, can fully equal unity. 58. Thus, .9 is 1-tenth less than 10-tenths, which make one unit; so .999999 is .000001, or 1 millionth less than 1. 59. It is observable also, that of two decimal expressions, the greater one, (no matter of how many figures either consists) has the greater number of tenths, or if the tenths be equal, a greater number of hundredths, and so on. 60. Thus .4 is greater than .399999, or .3 with any number of 9s that can possibly be annexed. 61. For .4 is (by 52) =.4000000, or equal to .4 with any number of ciphers annexed; now, .400000 is obviously greater than .399999. 62. FEDERAL MONEY, by assuming the dollar, as the money unit, is perfectly adapted in all its inferior denominations, to the decimal notation. 63. For, as 10 dimes make one dollar; 10 cents 1 dime; and 10 mills one cent; dimes are 10ths of dollars; cents, 10ths of dimes or 100ths of dollars, and mills 10ths of cents, or 1,000ths of dollars. 64. Thus $3, 2 dimes, 4 cents and 5 mills are written decimally $3.245, that is, $3,245. REDUCTION OF DECIMALS. LXIX. 1. Reduction of Decimals is the changing of their forms, without altering their value. CASE 1. To reduce a decimal fraction to a vulgar one. RULE. 1. Write under the given decimal its proper denominator, and it Q. How may whole numbers, or decimals of different denominators, be reduced to a common denominator? 52. Reduce 2.5, 8, and 7.05, each to hundredths. 53. Is then any decimal expression fully equal to unity? What is the difference in value between unity and .9? Unity and .999999? Which is the greater decimal, 4 or .399999? 60. How do you ascertain it? 61. What similarity has Federal Money to decimals? 62. becomes a vulgar fraction, which may generally be reduced to lower terins. 2. Reduce .5 to a vulgar fraction. .5 is = RULE. 1. Annex a cipher to the numerator and divide by the denominator: if there be a remainder, annex another cipher and divide as before, and so on to any extent required. 2. The quotient will contain as many decimal places as there are ciphers annexed; but if there be not as many places, supply the defect by prefixing ciphers to the quotient. 3. For, annexing one cipher to the numerator multiplies it by 10, 2 x 10-20 tenths.) which brings it into tenths, (as 5 5 4. Then as many times as the denominator is contained in the nume rator, so many 10ths are contained in the fraction, (as 20 tenths =.4.) 5. Annexing another cipher brings the numerator into hundredths, then dividing by the denominator will show the hundredths contained in the fraction, and so on, (×100=100 hundredths=.25.) 6. When there are no tenths, hundredths, &c., the vacant places in the quotient must be filled with ciphers, to keep the significant figures of the quotient in their proper places. 3 7 7. Reduce, 2, 1, 2, and, to decimal fractions. 2) 1.0 4)3.00 8) 7.000 A. 5 A. .75 25) 1.00 A. .875 A. .04 8. Reduce and to decimal fractions. 51 20 9. Reduce and 13 625 to decimal fractions. 200) 1.000 A. .255; .0208. A. .125; .08. 3 10. Reduce to decimals, o, 3, 8000, 20000, 250, 40. Answers. .5625; .025; .375; .000625; .00005; .004; .075. 11. Reduce 14 to a decimal fraction. Reduce the fractional part separately, then annex it. A. 14.125. LXIX. Q. What is Reduction of Decimals? 1. CASE I. Q. What vulgar fraction is equal to .5?-to .75?-to .4?-to .25? What is the rule? 1. CASE II. Q. How is a vulgar fraction reduced to a decimal one? 1. How many decimal places must there be in the quotient? 2. Why is the cipher annexed? 3, 4. Give an example. Why annex two or more ciphers? 5. Why are ciphers in some instances to be prefixed to the quotient? 6. 12. Reduce to decimals 9}, 51, 67, 5, 745202000 and 1. A. 9.2; 5.125; 6.875; 5.0625; 74520.00005; .333333+. 13. Reduce to a decimal. =51=5.5, or 11.0÷2=5.5, recollecting that the quotient figure or figures, before ciphers are annexed, is, of course, a whole number. A. 5.5. 14. Reduce to decimals 1, 4251, 121, 14551, and 1o. A. 5.555; 850.2; 30.25; 582.04; 3.33333+. 15. In the last example, the decimal will repeat 33, &c., for ever, if we continue the operation. 16. Decimals which repeat one or more figures are called REPEATING DECIMALS, or REPETENDS. 17. REPEATING DECIMALS are also called INFINITE DECIMALS; those that terminate, or come to an end, FINITE DECIMALS.* Q. How is a mixed number reduced to a decimal? 11. How is an improper fraction reduced? 13. What decimal is equal to 51?—to 62?-to ? What are Repeating Decimals? 16. What other names have they, and when? 17. * REPEATING DECIMALS are also called CIRCULATING DECIMALS. When only one figure repeats, it is called a single repetend; but if two or more figures repeat, it is called a compound repetend: thus, .333, &c. is a single repetend, .010101, &c. a compound repetend. When other decimals come before circulating decimals, as .8 in .8333, the decimal is called a mixed repetend. It is the common practice, instead of writing the repeating figures several times, to place a dot over the repeating figure in a single repetend; thus, .111, &c. is written i; also over the first and last repeating figure of a compound repetend; thus, for .030303, &c. we write .03. The value of any repetend, notwithstanding it repeats one figure or more an infinite number of times, coming nearer and nearer to a unit each time, though never reaching it, may be easily determined by common fractions; as will appear from what follows. By reducing to a decimal, we have a quotient consisting of .1111, &c., that is, the repetend i; since is the value of the repetend 1, the value of .333, &c., that is, the repetend 3, must be three times as much, that is, and 43; 5 =5; and 9=3=1 or the whole. Hence we have the following RULE for changing a single repretend to its equal common fraction:-Make the given repetend a numerator, writing 9 underneath for a denominator, and it is done. 2 4 What is the value of.i? Of.2? Of.4? Of 7? Of.8? Of.6? A. 1, 3, 3, 7, 8, 8. By changing to a decimal, we shall have .010101, that is, the repetend .01. Then, the repetend .04, being 4 times as much, must be, and .36 must be 35, also, .455 If 99 be reduced to a decimal, it produces .001. Then the decimal .004, being 4 times as much, is 999, and .036-95. This principle will be true for any number of places. 999. Hence we derive the following RULE for reducing a circulating decimal to a common fraction:-Make the given repetend a numerator; and the denominator will be as many -9s as there are figures in the repetend. Change .18 to a common fraction. 4. 18 Change .72 to a common fraction. A. 77 3 8 18. In general, whether the decimal be finite or infinite, three or four places are sufficient for most practical purposes. 19. Change to a decimal fraction. to a decimal fraction. A. .1111+ or .1111. A. 8.062 A. $125.12, 124. A. 3.04m. 27. Reduce to single fractions first and then to decimals of; To reduce a simple number of a given denomination to a decimal of a higher denomination. RULE. 1. Divide as in Reduction of whole numbers, annexing ciphers and pointing off the places for decimals in each quotient, as in the last Case, and for the same reasons. Q. How many places of decimals are generally sufficient? 18. What are Circulating Decimals? [See reference from 17.] What are single, compound, and mixed repetends? Of the decimals .333, &c., .0101, &c., and .8333, &c., which are the repeating figures? What is the proper name for each of these decimals? How are repetends distinguished from other decimals? How is the value of the repetend .3 expressed, and why? What is the rule for it? What is the value of .2.7 and. .01? What is the rule for reducing a circulating decimal to a vulgar fraction? Reduce to a vulgar fraction .18, .72 and .003. Describe the process of finding the value of the mixed repetend .83. What is the rule? 8 In the following example, viz. Change .83 to a common fraction, the repeating figure is 3, that is, 3, and .8 is; then 3, instead of being of a unit, is, by being in the second place, of; then and added together, thus,+3 = 15 = 35, 30, Ans. Hence, to find the value of a mixed repetend-First find the value of the repeating decimals, then of the other decimals, and add these results together. Change .916 to a common fraction. A. 10%+6=335=11. Proof, 11÷12 =.918. Change .203 to a common fraction. A. 61 300 91 822 900 900 To know if the result be right, change the common fraction to a decimal again. If it produces the same, the work is right. Repeating decimals may be easily multiplied, subtracted, &c. by first reducing them to their equal common fractions. |