Obs. 1. It should be remembered that the units' place is always the right hand place of a whole number. The effect of annexing and prefixing ciphers to decimals, it will be perceived, is the reverse of annexing and prefixing them to whole numbers. (Art. 58.) 2. A whole number and a decimal expressed together, are called a mixed number. (Art. 108.) 184. To read decimal fractions. Beginning at the left hand, read the figures as if they were whole numbers, and to the last one add the name of its order. Thus, .5 is read 5 tenths. .25 25 hundredths. .324 324 thousandths. .5267 5267 ten thousandths. .43725 43725 hundred thousandths.. .735168 735168 millionths, &c. 66 66 Obs. In reading decimals as well as whole numbers, the units" place should always be made the starting point. It is advisable for young pupils to apply to every figure the name of its order, or the place which it occupies, before attempting to read them. Beginning at the units' place, he should proceed toward the right, thus—units, tenths, hundredths, thousandths, &c., pointing to each figure as he pronounces the name of its order. In this way he will very soon be able to read decimals with as much ease as he can whole numbers. Read the following numbers : (1.) (2.) (3.) (4.) .25 .5317 3.245 9.14712 .36 .1056 7.6071 1.06231 .451 .4308 4.3159 2.00729 .5675 .0105 3.87816 9.14051 .0007 5.91432 8.06705 Quest.--Obs. Which is the units' place? What is a whole number and a decimal written together, called ? 184. How are decimals read? Obs, In reading decimals, what should be made the starting point ? 300-700 Note...Sometimes we pronounce the word decimal when we come to the separatrix, and then read the figures as if they were whole numbers; or, simply repeat them one after another. Thus, 125.427 is read, one hundred twenty-five, decimal four hundred twenty-seven; or, one hundred twenty-five, decimal four, two, seven. Write the fractional parts of the following numbers in decimals : (9.) (10.) (11.) (12.) 3 12.567 100000 4560 2005 1000 100000 100000 13-1235 6 7 1000000 13. Write 49 hundredths ; 3 tenths ; 445 ten thousandths. 14. Write 36 thousandths ; 25 hundred thousandths ; 1 millionth. 15. Write 7 hundredths ; 3 thousandths ; 95 ten thousandths; 63 millionths ; 26 ten millionths. 185. Decimals are Added, Subtracted, Multiplied, and Divided in the same manner as whole numbers. 6 435 10000 10 10-534 4100 28100 6 4 5 1000 6,29 Obs. The only thing with which the learner is likely to find any difficulty, is pointing off the answer. To this part of the operation he should give particular attention. , ADDITION OF DECIMAL FRACTIONS. 186. Ex. 1. What is the sum of 2.5; 24.457; 123.4 and 2.369? Operation. Write the units under units, the tenths un der tenths, hundredths under hundredths, &c.; 2.5 then, beginning at the right hand or lowest 24.457 order, proceed thus : 9 (thousandths) and 7 123.4 (thousandths) are 16 (thousandths.)' Write 2.369 the 6 under the column added, and carry the 152.726 1 to the next column as in addition of whole numbers. 1 to carry to 6 (hundredths) makes 7 (hun QUEST.-Note. What other method of reading decimals is mentioned? dredths) and 5 are 12 (hundredths.) Set the 2 under the column, and carry the l as before. I to carry to 3 (tenths) makes 4, and 4 are 8 (tenths) and 4 are 12 (tenihs) and 5 are 17 (tenths) or 1 and 5 tenths. · Set ihe 7 under the column, and carry the 1 to the next col Finally, pluce the decimal point in the amount, directly under those in the numbers added. umn. 187. Hence, we deduce the following general RULE FOR ADDITION OF DECIMALS. Write the numbers so that the same orders may stand under each other, placing tenths under tenths, hundredths under hundredths, fc. Brgin at the right hand or lowest order, and proceed in all respects as in adding whole numbers. (Art. 29.) From the right hand of the amount, point off as many figures for decimals as are equal to the greatest number of decimal places in either of the given numbers. PROOF.— Addition of Decimals is proved in the same manner as Simple Addition. (Art. 28.) Note.--The decimal point in the answer will always fall directly under the decimal points in the given numbers. EXAMPLES. (2.) (3.) (4.) 31.25 15.263 20.13 7.0003 117.056 435 0.05 2185.05813 1393.9741 Ans. 85.306 620 30597 5. What is the sum of 2.5; 33.65 and 45.121 ? 6. What is the sum of 65.7; 43.09; 1.026 and 2.1765 ? Quest.–187. How are decimals added? How point off the answer ? How is addition of decimals proved ? 7. What is the sum of 6.15768; 1.713458 and .6573128 ? 8. What is the sum of .0256; 15.6941; 3.856 and .00035 ? 9. Add together 256.31 ; 29.7 ; 468.213; 5.6 and .75. 10. Add together 25.61;78.003; 951.072 and 256.3052. 11. Add together .567; 37.05; 63.501 ; 76.25 and .63. 12. Add together .005; 1.25; 6.456; 10.2563 and 15.434. 13. Add together 255.1 ; 10.15; 27.09; 35.560 and 2.067. 14. Add together 5.00257; 3.600701 and 2.10607. 15. Add together 5 tenths, 25 hundredths, 566 thousandths, and 7568 ten thousandths. 16. Add together 34 hundredths, 67 thousandths, 13 ten thousandths, and 463 millionths. 17. Add together seven thousandths, 63 hundred thousandths, 47 millionths, and 6 tenths. 18. Add together 423 ten millionths, 63 thousandths, 25 hundredths, 4 tenths, and 56 ten thousandths. SUBTRACTION OF DECIMAL FRACTIONS. Ex. 1. From 25.367 subtract 13.18. Operation. Having written the less number un25.367 der the greater, so that units may stand 13.18 under units, tenths under tenths, &c. 12.187. Ans. we proceed exactly as in subtraction of whole numbers. (Art. 40.) Thus, 0 (thousandths) from 7 (thousandths) leaves 7 thousandths. Write the 7 in the thousandıh's place. As the next figure in the lower line is larger than the one above it, we borrow 10. Now 8 from 16 leaves 8; &c. Finally, place the decimal point in the remainder directly under those in the given numbers. 189. Hence, we deduce the following general RULE FOR SUBTRACTION OF DECIMALS. Write the less number under the greater, with units under units, tenths under tenths, hundredths under hundredths, fc. Subtract as in whole numbers, and point off the answer us in addition of decimals. (Art 187.) PROOF.-Subtraction of Decimals is proved in the same manner as Simple Subtraction. (Art. 39.) Note.--When there are blank places on the right hand of the upper number, they may be supplied by ciphers without altering the value of the decimal. (Art. 183.) EXAMPLES. 2. From 15. take 1.5. Ans. 13.5. 3. From 256.0315 take 5.641. 4. From 15.7 take 1.156. 5. From 63.25 take 50. 6. From 201.001 take 56.04037. 7. From 1 take 125. 8. From 11.1 take .40005. 9. From .56078 take .325. 10. From 1.66 take .5589. 11. From 3.4001 take 2.000009. 12. From 1 take .000001. 13. From 256.31 take 125.4689301. 14. From 8960.320507 take 63.001. 15. From 57000.000001 take 1000.001. 16. From 75 hundredths take 75 thousandths. 17. From 6 thousandths take 6 millionths. 18. From 3252 ten thousandths take 3 thousandths. 19. From 539 take 22 thousandths. 20. From 7856 take 236 millionths. Quest.—189 How are decimals subtracted ? How point off the answer? How is subtraction of decimals proved ? |