DECIMAL FRACTIONS. LXVIII. 1. A DECIMAL FRACTION1 differs from a vulgar fraction only in respect to its denominator, being uniformly either 10 or 100 or 1000, &c., and therefore it need not be, and seldom is, expressed. 2. The numerator then is written alone with a point before it, to distinguish it from whole numbers; this point is thence called a separatrix, and sometimes the decimal point. 100 10009 10000 • 3. Thus .3 is; .34 is 34; .345 is 345; .3456 is 3450 4. UNITY then in decimals is first divided into 10 equal parts, which are therefore called TENTHS. 5. The TENTH is divided into 10 other equal parts, making 100 equal parts of unity, which are thence called hundredths. 6. The HUNDREDTH is divided into 10 other equal parts, making 1000 equal parts of unity, which are thence called THOUSANDTHS; and so on, as in the following 10 TABLE I. make 1 unit. make 1 tenth. make 1 hundredth. make 1 thousandth. make 1 ten-thousandth. make 1 hundred-thousandth. 7. Since one decimal figure has for its denominator 1 with one cipher, as .5=5; two decimal figures, 1 with two ciphers, as .2512; three decimal figures, 1 with three ciphers, as .12512%, and so on; therefore, 25 100 8. A DECIMAL FRACTION is that fraction whose denominator is always understood to be a unit, or 1, with as many ciphers annexed as the given decimal has places of figures. 8 100 100 125 10000 9. Thus, .8 is ; .08 is 18; .35 is 35; .0125 is 10 10. When the numerator has not so many decimal places as the denominator has ciphers, we must prefix ciphers enough to the numerator to make as many. 4 5 11. Thus is written .05; 10000 = .0045; Toooo 5 100 .0000. 12. Since .5=15, .05=150, .005, then .05 is 10 times less in value than .5, and .005 is 10 times less than .05: LXVIII. Q. How does a decimal fraction differ from a vulgar one? 1. flow is the numerator written? 2. What does .3, .34, .345, .3456, with the point before each number, mean? 3. How is unity divided and sub-divided in decimals? 4, 5, 6. Repeat the table of these divisions. What is the denominator of one, two, or three decimal figures? 7. What then is a Decim d Fraction? 8. What is the denominator for .8? See 9. What is the denoninator for .08?—for .35?-for .0125? 9. When are ciphers to be prefixed to the numerator? 10. How do you write decimally 4 5 Too, or 100009 or 10000? 1 A DECIMAL FRACTION is so called from the Latin word decimus, signifying tenth because it increases and decreases in a tenfold proportion 13. For 10 times 5 is 50 15%, and 10 times 1510& 1000 1000 1009 14. Hence, a cipher placed on the left of any decimal decreases its value in a tenfold proportion, by removing it farther from the separatrix or decimal point. 15. But a cipher on the right of a decimal merely changes its name without altering its value. 100 16. For .5 is; so .50 is 51, and .500 is 5001, but the first is read 5-tenths, the second, 50-hundredths, the third, 500thousandths. 17. Decimals then increase from the right to the left like whole numbers, and of course their decrease from right to left is in the same proportion. 18. Hence every removal of any figure one place further towards the right decreases its value in a tenfold proportion. 5 5 5 19. Thus 555.555 is really 500, 50, 5, 10, 100, T005• 20. Our system of notation, then, which begins in whole numbers, is carried out by means of decimals, so as to embrace as many places below units as above or beyond them, even millions and millionths, billions and billionths, as in the following 21. The first decimal figure is 5 tenths; the second is 5 hundredths, or the first two 55 hundredths; the third is 5 thousandths, or the first three 555 thousandths; the fourth is 5 ten-thousandths, or the first four 5555 ten thousandths, &c. RULE FOR NUMERATING AND READING DECIMALS. 22. Begin on the left and say, tenths, hundredths, thousandths, ten-thousandths, &c., as in the table. Q. What is the difference in value between .5 and .05?-.05 and .005? (See 12.) What is the effect of the cipher? 14. What is the use of a cipher on the left of a decimal? 15. How is a decimal figure affected by changing its place? 18 Why? 17. What is the value of each figure in 555.555? 19. Describe our sys tem of notation. 20. Repeat the decimal part of the table beginning with "tenths" and ending with "billionths." Suppose each decimal place to be filled with the figure 5, what would be the value of the first 5?-of the second?---third?-fourth? &c. 21. What is the rule for numerating decimals? 22. 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. 5 18 50. So 5.62 is 562 hundredths, and 3.005 is 3005 thousandths. 51. In the examples 5 and .252, if we annex a cipher to .5 it becomes .50=5, having the same denominator with .25, therefore, 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-hun dredths 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 reducea to de.mals 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 315, 17.8, and .212, to thousandths. 5 8 A. 3.500; 17.800; .212. 55. Reduce the following numbers to decimals having the same denominators. 8.5; 3,2; 10; 1000; 756.3; 100000; 9811'. 56. Answers. 8.50000; 3.21000; .05000; .00800; 756.30000; .00009; 981.10000. 1009 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 1000 REDUCTION OF DECIMALS. LXIX. 1. Reduction of Decimals is the changing of their forms, without altering their value. CASE I. 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 nundredths. 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. 5 2. Reduce .5 to a vulgar fraction. .5 is CASE II. To reduce a vulgar fraction to a decimal. 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 × 10=20 tenths.) which brings it into tenths, (as 5 5 4. Then as many times as the denominator is contained in the nume ator, so many 10ths are contained in the fraction, (as 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 1 7. Reduce, 2, 1, 2, and, to decimal fractions. 2) 1.0 4)3.00 8)7.000 25)1.00 200)1.00 A. .5 A. .75 A. .875 A. .04 A. .005 13 to decimal fractions. A. .255; .0208. 625 8. Reduce and 200 8 2 25 10. Reduce to decimals, 40, 1, 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. 8 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 |