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 foot, 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 these again into 3 equal parts, called barley corns. (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 divided into 12 parts, be divided into 10 parts, or tenths of a foot, and each of these again into 10 parts, which would be tenths of tenths, 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 towards the left, and of decrease towards the right, is the same, those figures which express parts of an unit may obviously be managed precisely in the same manner as those which denote integers, or whole numbers. But to prevent confusion it is customary to separate the figures expressing parts from the integers by a point, called a separatrix. 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 should write 12.3ft. for 8 feet and 46 hundredths, 8.46ft. 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 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 100 Millions. -Millions. 100 Thous. Hundredths. 10 Thousandths. coTenths. coTens. 10 Millionths. From this table it will be seen, that the names of the pla ces, 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 by 1 tenth part of a thing? 2 tenths? 3 tenths? &c. 4. How would you write twenty-five hundred and twenty-five hundredths? One and 2. What is meant by 1 hun-six hundredths? One hundred, dredth? 5 hundredths? 35 and four ten thousandths? hundredths? 5. How would you express the following numbers in words: 0.1, 0.3, 0.01, 0.05, 0.35, 0.04, 0.7, 0.17, 0.02, 0.008, 0.401. 3. How would you write 4 tenths in figures? 7 tenths? 17 hundredths? 2 hundredths? 8 thousandths? 401 thou- | 0.000001, 700.007, 25.25, 1.06, sandths? 1 millionth? 7 thou- | 100.0004. sand and 7 thousandths? 116. Ciphers on the right of decimals do not alter their value for while each additional cipher indicates a division into parts ten times smaller than the preceding, it makes the decimal express 10 times as many parts, (113) Thus 5 tenths denot~ 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, by removing each significant figure one place towards the right, (115.) In the decimals, 0.5, 0.05, 0.005, the second is only I tenth part as much as the first, and the third 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. What is the sum of 4 tenths of a foot, 75 hundredths of a foot, and 9 hundredths of a foot? 0.4 0.75 0.09 Ans. 1.24 ft. We first write 0.4; then as .75 is 0.7 and 0.05, we write 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 place of hundredths. We then add the hundredths, and find them to be 0.14, equal to one 1 tenth and 4 hundredths; we 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 8, and 4 are 12; but 12 tenths of a foot are equal to foot and 2 tenths; we therefore write 2 in the place of tenths, and place the 1 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. 4. What is the sum of 21.3, 312.984, 918, 2700.42, 3153, 27.2, and 581.06? Ans. 4564.117. 5. What is the sum of 37, and 8 hundred and twentyone thousandths, 546 and 35 hundredths, eight and four tenths, and thirty-seven and three hundred twenty-five thousandths? Ans. 629.896. 6. What is the sum of six thousand years and six thousandths of a year, five hundred years and five hundredths of a year, and forty years and four tenths of a year? Ans. 6540.456 yrs. 7. Twelve +7.5+0.75 + 1.304, are how many? 8. Seventeen +0.1+0.11, +0.111+0.7707, are how many? MULTIPLICATION OF DECIMALS. ANALYSIS. 119. 1. How much butter in 3 boxes, each containing 4 pounds and 75 hundredths of a pound? 4.75 4.75 4.75 Ans.14.25 lb. The method of solving this question 4.75 3 Ans. 14.25 lb. By Addition. by Addition, must be sufficiently obvi- ByMultiplication ous, (117). In doing it by Multiplication, we proceed as at the right hand, saying, 3 times 5 are 15; and as the 5 are hundredths of a pound, the product is obviously hundredths; but 0.15 are 0.1 and 0.05, we therefore write 5 in the place of hunredths, 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 pound are equal to 2 pounds and 2 tenths of a pound. We therefore write the 2 tenths in the place of tenths, and reserve the 2 lbs. to be united with the pounds. Lastly, we say, 3 times 4 lbs. are 12 lb. to which we join the 2 lb. 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 25 215 Having written the numbers as at the left hand, we 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 is multiplied, were units, the product, 15, would be tenths, (119). But since the 5 is only tenths of units, the product, 15, can be only 10ths of 10ths, or 100ths of units; but as 0.15 are 0.1 and 0.05, we Ans. 10.75 miles. write 5 in the place of hundredths, reserving the 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 reserevd, we have 21 tenths, equal to 2.1 miles; we therefore write 1 in the place of tenths, and 2 in the place of 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. ? 0.5 0.5 1 foot, multiplied by itself, gives a square, measuring 1 foot on each side. 0.5 ft. by 0.5 gives a square, measuring 0.5 ft. equal to foot, on each side. But Ans. 0.25 ft. the latter square, as shown by the diagram, is only 0.25, or of the former; hence 0.25 ft. 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. 1 foot. 0.5 4. If you multiply 0.25 ft. by 0.25 ft. what will be the product? 0.25 .0125 .050 1 foot. 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 hundredths; 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, inay be shown by a diagram. A square foot being the area of a square Ans. .0625 ft. 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 is evident from the 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 (0.062516-1 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. RULE. I foot. 0.25 122. 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 87.64 lb. what will 9 such boxes weigh? 87.64 6. What will be the weight of 13 loads of hay, each weighing 1108.124 lb.? Ans. 14405.612 lb. 7. What is the product of 5 Ans. 1. Ans. 788.76 lb. by 0.2? |