DECIMALS. ciphers where necessary to give each significant figure its true local value. II. Place the decimal point before the first figure. RULE FOR DECIMAL NUMERATION. I. Numerate from the decimal point, to determine the de nominator. II. Numerate towards the decimal point, to determine the numerator. III. Read the decimal as a whole number, giving it the name or denomination of the right hand figure. EXAMPLES FOR PRACTICE. 1. Write 425 millionths. 2. Write six thousand ten-thousandths. 3. Write one thousand eight hundred fifty-nine hundredthousandths. 4. Write 260 thousand 8 billionths. 5. Read the following decimals: 6. Write five hundred two, and one thousand six millionths. 7. Write thirty-one, and two ten-millionths. 8. Write eleven thousand, and eleven hundred-thousandths. 9. Write nine million, and nine billionths. 10. Write one hundred two tenths. Ans. 10.2. 11. Write one hundred twenty-four thousand three hundred fifteen thousandths. 12. Write seven hundred thousandths. Second? Rule for numeration, first step? Second? Third? 121 REDUCTION. CASE I. 149. To reduce decimals to a common denominator. 1. Reduce .5, .375, 3.25401, and 46.13 to their least common decimal denominator. OPERATION. .50000 .37500 3.25401 46.13000 ANALYSIS. The given decimals must contain as many places each, as are equal to the greatest number of decimal figures in any of the given decimals. We find that the third number contains five decimal places, and hence 100000 must be a common denominator. As annexing ciphers to decimals does not alter their value, (144., 3) we give to each number five decimal places by annexing ciphers, and thus reduce the given decimals to a common denominator. Hence, RULE. Give to each number the same number of decimal places, by annexing ciphers. NOTES. 1. If the numbers be reduced to the denominator of that one of the given numbers having the greatest number of decimal places, they will have their least common decimal denominator. 2. A whole number may readily be reduced to decimals by placing the decimal point after units, and annexing ciphers; one cipher reducing it to tenths, two ciphers to hundredths, three ciphers to thousandths, and so on. EXAMPLES FOR PRACTICE. 2. Reduce .17, 24.6, .0003, 84, and 721.8000271 to their least common denominator. 3. Reduce 7 tenths, 24 thousandths, 187 millionths, 5 hundred millionths, and 10845 hundredths to their least common denominator. 4. Reduce to their least common denominator the following decimals: 1000.001, 841.78, 2.6004, 90.000009, and 6000. What is meant by the reduction of decimals? Case I is what? Give explanation. Rule. CASE II. 150. To reduce a decimal to a common fraction. 1. Reduce .75 to its equivalent common fraction. OPERATION. .75=7%= ANALYSIS. We'omit the decimal point, supply the proper denominator to the deci mal, and then reduce the common fraction Hence, thus formed to its lowest terms. RULE. Omit the decimal point, and supply the proper denominator. EXAMPLES FOR PRACTICE. 2. Reduce .125 to a common fraction. CASE III, 151. To reduce a common fraction to a decimal. 1. Reduce to its equivalent decimal. FIRST OPERATION. 300 1=188=156.75, Ans. SECOND OPERATION. 4) 3.00 ANALYSIS. We first annex the same number of ciphers to both terms of the fraction; this does not alter its value. We then divide both resulting terms by 4, the significant figure of the denominator, to obtain the decimal denominator, 100. Then the fraction is changed to the decimal form by omitting the denominator. If the intermediate steps be omitted, the true result may be obtained as in the second operation. .75 2. Reduce to its equivalent decimal. Case II is what? Give explanation. Rule. Explain first operation. Second. Case III is what? THIRD OPERATION. 16) 1.0000 .0625, Ans. ciphers, there must be 4 places in the required decimal; hence we prefix 1 cipher. This is made still plainer by the following operation; thus, 10000 = 160000 1838.0625. From these illustrations we derive the following RULE. I. Annex ciphers to the numerator, and divide by the denominator. II. Point off as many decimal places in the result as are equal to the number of ciphers annexed. NOTE. A common fraction can be reduced to an exact decimal when its lowest denominator contains only the prime factors 2 and 5, and not otherwise. NOTE. The sign, +, in the answer indicates that there is still a remainder. 14. Reduce to a decimal. Ans. .513513+. NOTE. The answers to the last two examples are called repeating decimals; and the figure 3 in the 13th example, and the figures 513 in the 14th, are called repetends, because they are repeated, or occur in regular order. Third operation. Rule, first step? Second? When can a common fraction be reduced to an exact decimal? ADDITION. 152. 1. What is the sum of 3.703, 621.57, .672, and 20.0074? OPERATION. 3.703 621.57 .672 20.0074 645.9524 ANALYSIS. We write the numbers so that figures of like orders of units shall stand in the same columns; that is, units under units, tenths under tenths, hundredths under hundredths, &c. This brings the decimal points directly under each other. Commencing at the right hand, we add each column separately, and carry as in whole numbers, and in the result we place a decimal point between units and tenths, or directly under the decimal point in the numbers added. From this example we derive the following RULE. I. Write the numbers so that the decimal points shall stand directly under each other. II. Add as in whole numbers, and place the decimal point, in the result, directly under the points in the numbers added. 4. Add 1152.01, 14.11018, 152348.21, 9.000083. 2.5 Amount, 415.65703 Ans. 153523.330263. Ans. 5038.4514. 5. Add 37.03, 0.521, .9, 1000, 4000.0004. 6. What is the sum of twenty-six, and twenty-six hundredths; seven tenths; six, and eighty-three thousandths; four, and four thousandths? Explain the operation of addition of decimals. Second. Ans. 37.047. Give rule, first step. |