DECIMALS AND FEDERAL MONEY. DECIMALS. 113. The method of forming numbers, and of expressing them by figures, has been fully explained in the articles on Numeration. (71, 72, 73) But it frequently happens that we have occasion to express quantities, which are less than the one fixed upon for unity. Should we make the fout, for instance, our unit measure, we should often have occasion to express distances which are parts of a foot. This has ordinarily been done by dividing the foot into 12 equal parts, called inches, and each of the e again into 3 equal parts, called barley corps. (38) But divisions of this nature, which are not conformable to the general law of Notation, (73) necessarily embarrass calculations, and also encumber books and the memories of pupils, with a great number of irregular and perplexing tables. Now, if the foot, instead of being dividest into 12 parts, be divided into 10 parts, or tenths of a foot, and each of these again into lli parts, which would be tenths of tenihs or hundredths of a foot, and so on to any extent found necessary, making the parts 10 times smaller at each division ;-then in recomposing the larger divisions from the smaller, 10 of the smaller would be required to make one of the next larger, and so on, precisely as in whole numbers. Hence, figures expressing tenths, hundredths, thousandths, &c. may be written towards the right from the place of units, in the same manner that tens, hundreds, thousands, &c. are ranged towards the left; and as the law of increase to wards the left, and of decrease towards the right, is the same, those figures which express parts of a unit may obviously be managed precisely in the same manner as those which denote integers, or whole numbers. But to prerent confusion, it is customary to separate the figures expressing parts from the integers by a point, called a separatrir. The points used for this purpose are the period and the comma, the former of which is adopted in this work; thus to express 12 feet and 3 tenths of a foot, we write 12.33 ft. for 8 feet and 46 bundredths, 8.46 feet. DEFINITIONS. 114. Numbers which diminish in value, from the place of units towards the right hand, in a ten fold proportion, (as described in the preceding article,) are called Decima's. Numbers which are made up of integers and decimals, are called mized numbers. NUMERATION OF DECIMALS. 115. It must be ohvious 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 in, 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? sandtbs? 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, 0.3, 0.01, 0.0.0, thousandths? 401 thou 0.35, 0.04, 0.7, 0.17, 0.02, sandths? 1 millionth? 7 thou-0.008, 0.401, 0.000001, 25.25, sand and 7 thousandths? 700.007, 1.06. 100.0004. 116. Cipher.; on the right of decimals o not alter their value; for whilo 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 cipher placed at the left hand diminishes the value of a decimal 10 times, hy 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 inuch as the second; and they are read, 5 tentiis, 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? 0.4 We first write 0.4; then as . 75 is 0.7 and 0.05, we writa 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 them to be 0.14, equal to one 1 tenth and 4 hundredths; we Ans. 1.24 ft. therefore reserve the 0.1, to be united with the tenths, and write the 4 under the colunin of hundredths. We then say, 1:100 is 1, and 7 are 8, and 4 are 12; but 12 tenths of a foot are equal to 1 foot and 2 tenths; 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 guin of 0.4, 0.75, and 0.09 of a foot, to be 1.24 ft. RULE. QUESTIONS FOR PRACTICE. sandths of a year, five hun- dred years and five hun8.842 dredths of a year, and forty ycars and four tenths of a Aps. 519.292 rods. year! Ans. 6510.456 yax. 4. What is the amount of 6. What is the sum of 37, seventeen pounds and seven and 8 hundred and twenty: tenths, 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 many? 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.c6? many? Ans. 4564.117. 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. 4.75 nus,[117] In doing it by Multiplica 4.75 4.75 tion, we proceed as at the right hand, 3 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 lh. is obviously hundredths; but 0.15 are 0.1 and 0.05, we therefore write 5 in the place of hnn. dredths, and reserve the 1 to be joined with the tenths. We then say, 3 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 povod 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. a: e 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 wili he travel in 2.5 hours? 4.3 Having written the numbers as at the left hand, we say 5 times 3 are 15. Now as the 3, which is ruulti plied, is tenths, it is evident, that if the 5, by which it 2.15 is multiplied, were units, the product, 15, would le 8.6 tenths,(119 But since the 5 is only trnihs of inits, thic product, 15, can be only 10ths of 10ths, or 100ths of units; but as 0.15 are 0.1 and 0.05, we write 5 in Ans. 10.75 miles. the place of hundredths, reserving the 1 to be joined with the tenths. We then say 5 times 4 are 20, which are tenths, becalise che 5 is tenths; joining the 0.1 reserved, we have 21 tenths, qual to 2.1 miles; we therefore write 1 in the place of tentus, and 2 in the place of 1 foot. S 0.25 units. We then multiply by 2, as illustrated in article 119, and write the 1 foot. 10.5 gives a squarc, measuring 0.5 ft. equal to } foot on each side. Br the latter square, as Ans. 0.25 ft. shown by the diagran, is only 0.25, or of the former; hence 0.25 is evidently th: 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 hundrelths,(120) the 5 at the left hanıl of the second partial product is evidently hur Ans. .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 sile; but such a square, as is evident from the 1 foot 0.25 diagrain, 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 Il 1 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 decimal places in both the factors. RÜLE. 122. Write the multiplier 'under the multiplicand, and proceed in all respects as is 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 weigla 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 11.? 87.64 .. Ans. 14405.612 lb. 9 8. Multiply 0.026 by 0.003. Ans. 788.76'lb. Prod. 0,000078. 6. What is the product of 9. Multiply 125 by 0,008. 5 by 0.2? Ans. 1. Prod. 1. I foot. 0.25 |