described in the preceding article,) are called Decimals. Numbers which are made up of integers and decimals, are called mixed numbers. NUMERATION OF DECIMALS. 115. It must be obvious from the two preceding articles, that the figures in decimals, as in whole numbers, have a local value, called the name of the place,(74) which depends upon their distance from the separatrix, or the place of unity, each removal of a figure one place towards the right diminishing its value ten times. (73) "The names of the places, both of integers and decimals, are expressed in the following TABLE. Decimals. From this table it will be seen, that the names of the places, each way from that of units are the same, excepting The termination th, or ths, which is added to the name of the last, or right hand place, in the enunciation of decimals. EXERCISES. 1. What do you understand 4. How would you write by 1 tenth part of a thing? 2 | twenty-five hundred and tenths? 3 tenths? &c. twenty-five hundredths? One 2. What is meant by 1 hun- ( and six hundredths? One dredth? 5 hundredths? 35 hundred, and four ten thouhundredths? sandths? 3. How would you write 4 5. How would you express tenths in figures? 7 tenths? | the following numbers in 17 hundredths? 2 hundredths? | words? 0.1, 0.5, 0.01, 0.05, 8 thousandths? 401 thou- | 0.35, 0.04, 0.7, 0.17, 0.02, sandths? 1 millionth? 7 thou- 1 0.008, 0.401, 0.000001, 25.25, sand and 7 thousandths? 700.007, 1.06. 100.0004. 116. Ciphers on the right of decimals io not alter their value; for while each additional cipher indicates a division into parts ten times smaller than the prereding, it makes the decimal express 10 times as many parts, (113), Thus 5 tenths denotes 5 parts of a unit, which is divided into 10 parts; 50 hundredths denotes 50 parts of a unit, which is divided into 100 parts, and so on: but as 5 is half of 10, and 50 half of 100, the value of each is the same, namely, one half a unit. On the contrary, each ciphér placed at the left hand diminishes the value of a decimal 10 times, by removing each significant figure one place towards the right:(115) In the decimals, 0.5, 0.05, 0.005, the second is only 1 tenth part as much as the second; and they are read, 5 tenths, 5 hundredths, and 5 thousandths. ADDITION OF DECIMALS. ANALYSIS. 117. 1. What is the sum of 4 tenths of a foot, 75 hundredths of a foot, and 9 hundredths of a foot? We first write 0.4; then as 75 is 0.7 and 0.05, we write 0.4 0.75 0.7 under 0.4, and place the 5 at the right hand in the place of hundredths; and lastly, we write 9 under the 5 in the 0.09 place of hundredths. We then add the hundredths, and find ihem to be 0.14, equal to one 1 tenth and 4 hundreilths; we Ans. 1.24 ft. therefore reserve the 0.1, to be united with the tenths, and write the 4 under the column of hundredths. We then say, 1 to 0 is 1, and 7 are 3, and 4 are 12; but 12 tenthis of a foot are equal to 1 foot and 2 tonths; we therefore write 2 in the place of tenths, and place the I foot on the left of the separatrix in the place of units. Thus we find the sum of 0.4, 0.75, and 0.09 of a foot, to be 1.24 ft. RULE. 118. Write down the whole numbers, if any, as in Simple Addition, and place the decimals on the right in such manner that tenths shall stand under tenths, hundredths under hundredths, and so on, and draw a line below. Begin at the right hand, and add up all the columns, writing down and carrying as in Simple Addition. Place the decimal point directly under' those in the numbers added. QUESTIONS FOR PRACTICE. 2. What is the sum of 25.41 3. What is the sum of six rods, 16.05 rds. and:8.842 rds? thousand years and six thou25.4 sandths of a year, five hun16.05 dred years and five hun8.842 dredths of a year, and forty years and four tenths of a Ans. 50.292 rods. year? Ans. 6540.456 yrs. 4. What is the amount of 6. What is the sum of 37, seventeen pounds and seven and 8 hundred and twentytenths, eight pounds and six- one thousandths, 546 and 35 ty-six hundredths, and one hundredths, eight and four pound and seven hundredths? tenths, and thirty-seven and 17.7 three hundred twenty-five 8.66 thousandths ? Ans.629.896. 1.07 7. Twelve +7.5 +0.75+ 1,304, are how 5. What is the sum of 21.3, 8. Seventeen +0.1+0.11, 312,984, 918, 2700.42, 3.153, +0.111 + 0.7707, are how 27.2, and 581.06? many? Ans. 4564,117. many? 3 3 MULTIPLICATION OF DECIMALS, ANALYSIS, 119, 1. How much butter in 3 boxes, each containing 4 pounds and 75 hundredths of a pound? The method of solving this question By Addition, by Addition, must be sufficiently obvi- By Multiplication. ous,[117] In doing it by Multiplica- 4.75 4.75 tion, we proceed as at the right hand, 4.75 saying, 3 times 5 are 15; and as the 5 are hundredths of a pound, the product Ans. 14.25 lb. Ans. 14.25 lb. is obviously hundredths; but 0.15 are 0.1 and 0.05, we therefore write 5 in the place of hrindredths, and reserve the 1 to be joined with the tenths. We then say; times 7 are 21, which are so many tenths, because the 7 are tenths, and to these we join the 1 tenth reserved, making 22 tenths; but 22 tenths of a pound are equal to 2 pounds and 2 tenths of a pound. We therefore write the 2 tenths in the place of tens, and reserve the 2 lbs. to be united with the pounds. Lastly, we say, 3 times 4 lbs. are 12 lbs. to which we join the 2 lbs. reserved, making 14 pounds, which we write as whole numbers on the left hand of the separatrix. From this example it appears, that when one of the factors contains decimals, there will be an equal number of decimal places in the product. 120. 2. If a person travel 4.3 miles per hour, how far will he travel in 2.5 hours? 4.3 Having written the numbers as at the left hand, we 2.5 say 5 times 3 are 15. Now as the 3, which is multi plied, is tenths, it is evident, that if the 5, by which it 2.15 is multiplied, were units, the product, 15, would be 8.6 tenths,(119) But since the 5 is only tenths of units, the product, 15, can be only 10ths of 10ths, or 100ths Ans. 10.75 miles. of units; but as 0.15 are 0.1 and 0.05, we write 5 in the place of hundredths, reserving the 1 to be joined with the tenths. We then say 5 times 4 are 20, which are tenths, because the 5 is tenths; joining the 0.1 reserved, we have 21 tenths, equal to 2.1 miles; we therefore write 1 in the place of tenths, and 2 in the place of 'I foot. 0.25 units. We then multiply by 2, as illustrated in article 119, and write the product, 8.6, under the corresponding parts of the first product, and, adding the two partial products together, we have 10.75 miles for the distance travelled in 2.5 hours. 121. 3. What is the product of 0.5 ft. multiplied by 0.5 ft.? 1 foot, multiplied by itself, gives a square, 0.5 1 foot. measuring 1 foot on each side. 0.5 ft. by 0.5 foot on each side. But the latter square, as Ans. 0.25 ft. shown by the diagram, is only 0.25, or of the former; hence 0.25 is evidently the product of 0.5 by 0.5 ft. Here we perceive that multiplication by a decimal diminishes the multiplicand, or, in other words, gives a product which is less than the multiplicand. 4. If you multiply 0.25 ft. by 0.25 ft. what will be the product? 0.25 Here the operation is performed as above; but since tenths multiplied by tenths, give hundredths,(120) the 5 at the left hand of the second partial product is evidently hunAns..0625 ft. dredths; it is therefore necessary to supply the place of tenths with a cipher. Or the necessity of a cipher at the left of the 6, in the answer, may be shown by a diagram. A square foot being the area of a square which measures 1 foot on each side, a square 0.25, or quarter, of a foot, is a square measuring 0.25 of a foot on each side; but such a square, as 18 evident from the 1 foot. 0.25 diagram, is only one sixteenth part of a square foot. Hence to prove that the decimal 0.0625 ft. is equal in value to one sixteenth part of a square foot, we have only to multiply it by 16 (0.0625x16=1 ft.) and the product is 1 foot. In like manner it may be shown that every product will have as many decimal places as there are deoimal places in both the factors. RULE. 122. Write the multiplier under the multiplicand, and proceed in all respects as in the multiplication of whole numbers. In the product, point off as many figures for decimals as there are decimal places in both the factors counted together. Note. If there be not so many figures in the product as there are decimal places in the factors, make up the deficiency by prefixing ciphers. QUESTIONS FOR PRACTICE. 5. If a box of sugar weigh 7. What will be the weight 87.64 lb, what will 9 such box of 13 loads of hay, each weighes weigh? ing 1108.124 Ib.? 87.64 Ans. 14405.612 lb. 9 8. Multiply 0.026 by 0.003. Prod. 0.000078, Ans. 788.76 lb. 6. What is the product of 9. Multiply 125 by 0.008. by 0.2? Ans. 1. Prod. 1, I foot. 0.25 ! 10. Multiply 25.238 by 13. Seven +117 X 1.024 12.17. Prod. 307.14646. how many? 11. Multiply 5 thousand by 14. 128.75 + 144.25 X 0.06 5 thousandths. Prod. 25. =16.38 Ans. 12. Twenty-five X 0.25 are 15. 0.004+0.0004X0.00002 how many? =0.000000088 Ans. 1234 SUBTRACTION OF DECIMALS. ANALYSIS. 123. , 1. What is the difference between 43.25 rods and 22.5 roas We write down the numbers as for Addition, with the .25 Gargest uppermost. As there are no hundredths in the 22.5 subtrahend, we bring down the 5 hundredths. ProceedAns. 20.75 rods. ing to the 10ths, we are unable to take 0.5 from 0.2; we therefore borrow a unit from the 3 units, which being, 19 tenths, we join 10 to the 2, making 12 tenths; from which we take 5 tents, and write the remainder, 7 tenths, in the place of tenths below the line 'The rest of the operation must be obvious. From 24 hours take 18.75 hours, what remains? 24. Here, as we cannot take 5 from nothing, we borrow 0.10 from the 4 units, or 400 hundredths; then taking 5 (=0.05) 18.75 from 0.10, the remainder is 0.05. The 400 hundredths bas Ans. 5,25 h. now become 390 hundredths, or 39 tenths, or 3.9; then 0.7 from 0.9 leaves 0.2, and so on. RULE. 124. Write down the numbers as in Addition of Decimals, observing to place the largest number uppermost. Beginning at the right, subtract as in Simple Subtraction,(99) and place the decimal point in the remainder directly under those in the given numbers. NOTE 1.-When the numbers are all properly written, and the resulte correctly pointed, the decimal points will all fall in one vertical column, or directly under one another, both in Subtractior ind Addition. NOTE 2.-In numbers given for Addition or Subtraction, the decimal places may all be made equal by annexing ciphers to a part of them,(116) without altering their value, and then all the decimals will express similar parts of a unit, or be of the same denomination. QUESTIONS FOR PRACTICE. 3. A person bought 27.63 lb. 4. From 468.742 rods, take of cinnamon, and sold 19.814 76.4815 rods, lb. how much had he left? Rem. 392.2605. 27.63 5. From 9 ft. take 0.9 ft. 19.814 what remains ? Ans. 8.1 ft. 6. From 2,73 take 1.9185. Ans. 7.816 lb. Rem, 0.8115. |