§ XX. DECIMAL FRACTIONS. ART. 173. A DECIMAL FRACTION is a fraction whose denominator is 10, 100, 1000, &c. ART. 174, Decimal fractions are commonly expressed by writing the numerator only with a point before it, called the decimal point or separatrix; thus, 99 = 100 .99 hundredths. 1000 ART. 175. By examining the foregoing fractions, it will be seen that .9 can occupy only one place while it remains a proper fraction; .99, only two places; and 99% = .999, only three places; for, if their numerators are increased by = .1,100 = .01, Tobo = .001, respectively, each fracTo tion becomes a unit or whole number. Hence, The first figure or place of any decimal on the right of the point is tenths, the second hundredths, the third thousandths, &c. NOTE. When a decimal place has no significant figure, it must be filled with a cipher. ART. 176. The denominator of .9 is 1 with one cipher annexed; the denominator of 99.99 is 1 with two ciphers annexed; the denominator of three ciphers annexed. Hence, 1000 .999 is 1 with The denominator of a decimal fraction is 1 with as many ciphers annexed as the numerator has places. ART. 177. Decimal fractions originate from dividing the unit, first, into 10 equal parts, and then each of these parts into 10 other equal parts, and so on indefinitely. Thus, 1÷10 = .1; ÷ 10 = 100 = .01; Too ÷ 10 = .001. Hence, = 1000 = The unit in decimal fractions is divided into 10, 100, 1000, &c., equal parts. QUESTIONS. Art. 173. What is a decimal fraction? Art. 174. How are decimal fractions commonly expressed? — Art. 175. What is the first figure or place of any decimal? The second? The third ? &c. Why? What must be done when a decimal place has no significant figure to fill it? — Art. 176. What is the denominator of a decimal fraction? - Art. 177. How do decimal fractions originate? ART. 178. If ciphers are placed on the left hand of decimal figures, they change their places, each cipher removing them one place to the right; thus, .3, but .03=18σ, and .003. Hence, Ciphers placed on the left hand of decimals decrease their value in a tenfold proportion. ART. 179. If ciphers are placed on the right hand of decimal figures, their places are not changed; thus, .3, and .30 30 100 = 1% .3. Hence, = Ciphers placed on the right hand of decimals do not alter their value. NUMERATION OF DECIMAL FRACTIONS. different orders ART. 180. The relation of decimals to whole numbers and to each other, and also the names of their and places, may be learned from the following QUESTIONS. Decimals. - Art. 178. What effect have ciphers placed at the left hand of decimals? Why? - Art. 179. What effect if placed at the right hand? Why? Art. 180. What may be learned from the table? - The preceding table consists of a whole number and decimal, which, taken together, are called a mixed number. The part on the left of the separatrix is the whole number, and that on the right the decimal. The value of the decimal is expressed in words thus: - Two hundred thirty-four millions five hundred sixty-seven thousand eight hundred ninety-three billionths. And the mixed number thus: Seven millions six hundred fifty-four thousand three hundred twenty-one, and two hundred thirty-four millions five hundred sixty-seven thousand eight hundred ninety-three billionths. From the table and explanations we have the following rule for numerating and reading decimals. RULE-Beginning at the left hand, name the order of each figure of the given decimal, and then read it as in whole numbers, giving the name of the last order to the whole. The pupil may write in words, or read orally, the following numbers: NOTATION OF DECIMAL FRACTIONS. ART. 181. By examining the decimal table, we perceive that tenths occupy the first place, hundredths the second, &c., and that each figure takes its value by its distance from the place of units; therefore, to write decimals, we have the following RULE. - Write the decimal figures in the place of the order denoted by their names, supplying with ciphers such places as have no significant figures. The pupil may write in figures the following numbers: — 1. Three hundred seven and twenty-five hundredths. 2. Forty-seven and seven tenths. Of what does it consist? QUESTIONS. What is the number called, when taken together? What is the part on the left hand of the separatrix ? The part on the right? What is the value of the decimal? What is the value of the mixed number? What is the rule for numerating and reading decimals? Art. 181. Upon what does the value of a decimal figure depend? What is the rule for writing decimals? 3. Eighteen and five hundredths. 4. Twenty-nine and three thousandths. 7. Seventy-five and nine tenths. 8. Two thousand and two thousandths. 9. Eighteen and eighteen thousandths. 10. Five hundred five and one thousand six millionths. 11. Three hundred and forty-two ten millionths. 12. Twenty-five hundred and thirty-seven billionths. ART. 182. It will be seen that decimals increase from right to left, and decrease from left to right, in the same ratio as simple numbers; hence they may be added, subtracted, multiplied, and divided in the same manner. ADDITION OF DECIMALS. ART. 183. Ex. 1. Add together 5.018, 171.16, 88.133, 1113.6, .00456, and 14.178. Ans. 1392.09356. OPERATION. 5.0 18 171.16 88.133 1113.6 We write the numbers, units under units, tenths under tenths, hundredths under hundredths, &c., and then, beginning at the right hand, add them .00456 as whole numbers, and place the decimal point in 14.178 the result directly under those above. 139 2.0 9 356 RULE. Write the numbers under each other according to their value, add as in whole numbers, and point off from the right hand as many places for decimals as there are in that number which contains the greatest number of decimals. Proof. The proof is the same as in simple addition. EXAMPLES FOR PRACTICE. 2. Add together 171.61111, 16.7101, .00007, 71.0006, and 1.167895. Ans. 260.489775. 3. Add together .16711, 1.766, 76111.1, 167.1, .000007, and 1476.1. Ans. 77756.233117. QUESTIONS. Art. 182. How do decimals increase and decrease? How may they be added, subtracted, multiplied, and divided Art. 183. How are decimals arranged for addition? What is the rule for addition of decimals? What is the proof? 4. Add together 151.01, 611111.01, 16.5, 6.7, 46.1, and .67896. Ans. 611331.99896. 5. Add fifty-six thousand and fourteen thousandths, nineteen and nineteen hundredths, fifty-seven and forty-eight ten thousandths, twenty-three thousand five and four tenths, and fourteen millionths. Ans. 79081.608814. 6. What is the sum of forty-nine and one hundred and five ten thousandths, eighty-nine and one hundred seven thousandths, one hundred twenty-seven millionths, forty-eight ten thousandths? Ans. 138.122427. 7. What is the sum of three and eighteen ten thousandths, one thousand five and twenty-three thousand forty-three millionths, eighty-seven and one hundred seven thousandths, fortynine ten thousandths, forty-seven thousand and three hundred nine hundred thousandths? Ans. 48095.139833. SUBTRACTION OF DECIMALS. ART. 184. Ex. 1. From 74.806 take 49.054. OPERATION. Ans. 25.752. Having written the less number under the greater, units 74.806 under units and tenths under tenths, &c., we subtract as 49.054 in whole numbers, and place the decimal point in the re25.752 sult, as in addition. RULE. Write the less number under the greater, units under units, tenths under tenths, &c.; then subtract as in whole numbers, and point off as many places for decimals as there are in that number which contains the greatest number of decimals. Proof. — The proof is the same as in simple subtraction. QUESTIONS.-Art. 184. What is the rule for subtraction of decimals? What is the proof? |