figures to express their value exactly in decimals, as to render them very inconvenient. There are many also, the value of which cannot be exactly expressed in decimals. In most calculations, however, it will be sufficient to use an approximate value. The degree of approximation necessary must always be determined by the nature of the case. For example, in making out a single sum of money, it is considered sufficiently exact if it is right within something less than 1 cent, that is, within less than of a dollar. But if several sums are to be put together, or if a sum is to be multiplied, mills or thousandths of a dollar must be taken into the account, and sometimes tenths of mills or ten-thousandths. In general, in questions of business, three or four decimal places will be sufficiently exact. And even where very great exactness is required, it is not very often necessary to use more than six or seven decimal places. A merchant bought 4 pieces of cloth; the first contained 283 yards; the second 342; the third 30; and the fourth 423 yards. How many yards in the whole? In reducing these fractions to decimals, they will be suffi ciently exact if we stop at hundredths, since ī¿o of a yard iз only about of an inch. 100 is exactly .6. If we were to continue the division of 4, it would be .28571, &c.; in fact it would never terminate; but .28 is within about one of of a yard, therefore sufficiently exact. is not so much as, therefore the first figure is in the hundredths' place. The true value is .6666, &c., but because is more than of, I call it .07 instead of .06. 7 is equal to .7777, &c. This would never terminate. Its value is nearer .78 than .77, therefore I use .78. 6 1000 When the decimal used is smaller than the true one, it is well to make the mark after it, to show that something more should be added, as .28+. When the fraction is too large the mark should be made to show that something should be subtracted, as 75 = .07 . The numbers to be added will now stand thus: From the above observations we obtain the following general rule for changing a common fraction to a decimal: Annex a zero to the numerator, and divide it by the denominator, and then if there be a remainder, annex another zero, and divide again, and so on, until there is no remainder, or until a fraction is obtained, which is sufficiently exact for the purpose required. Note. When one zero is annexed, the quotient will be tenths, when two zeros are annexed, the quotient will be hundredths, and so on. Therefore, if when one zero is annexed, the dividend is not so large as the divisor, a zero must be put in the quotient with a point before it, and in the same manner after two or more zeros are annexed, if it is not yet divisible, as many zeros must be placed in the quotient. 14783 Two men talking of their ages, one said he was 37, 3847 years old, and the other said he was 6421 years old. What was the difference of their ages? If it is required to find an answer within 1 minute, it will he necessary to continue the decimals to seven places, for 1 minute is 2560 of a year. If the answer is required only within hours, five places are sufficient; if only within days, four places are sufficient. it is evident that units must be subtracted from units, enths from tenths, &c. If the decimal places in the two numbers are not alike, they may be made alike by annexing zeros. After the numbers are prepared, subtraction is performed precisely as in whole numbers. Multiplication of Decimals. XXVII. How many yards of cloth are there in seven pieces, each piece containing 197 yards? 197 - 19.875 N. B. All the operations on decimals are performed in precisely the same manner as whole numbers. All the difficulty consists in finding where the separatrix, or decimal point, is to be placed. This is of the utmost importance, since if an error of a single place be made in this, their value is rendered ten times too large or ten times too small. The purpose of this article and the next is to show where the point must be placed in multiplying and dividing. In the above example there are decimals in the multiplicand, but none in the multiplier. It is evident from what we have seen in adding and subtracting decimals, that in this case there must be as many decimal places in the product, as there are in the multiplicand. It may perhaps be more satisfactory if we analyze it. 7 times 5 thousandths are 35 thousandths, that is, 3 hundredths and 5 thousandths. Reserving the hundredths, I write the 5 thousandths. Then 7 times 7 hundredths are 49 hundredths, and 3 (which I reserved) are 52 hundredths, that is, 5 tenths and 2 hundredths. I write the two hundredths, reserving the 5 tenths. Then 7 times 8 tenths are 56 tenths, and 5 (which I reserved) are 61 tenths, that is, 6 whole ones and 1 tenth. I write the 1 tenth, reserving the 6 units. Then 7 times 9 are 63, and 6 are 69, &c. It is evident then, that there must be thousandths in the product, as there are in the multiplicand. The point must be made between the third and fourth figure from the right, as in the multiplicand, and the answer will stand thus, 139.125 yards. Rule. When there are decimal figures in the multiphcand only, cut off as many places from the right of the product for decimals, as there are in the multiplicand. If a ship is worth 24683 dollars, what is a man's share worth, who owns 3 of her. ៖ = .375= 375 The question then is, to find 375 of 1000 1000 24683 dollars. First find of it, that is, divide it by 1000. This is done by cutting off three places from the right (Art. XI.) thus 24.683, that is, 24 683, because 683 is a remainder and must be written over the divisor. In fact it is evident that 75 of 24683 is 24683 But since this fraction is thousandths, it may stand in the form of a decimal, thus 24.683. 1000 24,683 1000 It is a general rule then, that when we divide by 10. 100, 1000, &c. which is done by cutting off figures from the right, the figures so cut off may stand as decimals, because they will always be tenths, hundredths, &c. Tooo of 24683 then is 24.683 and 375 of it will be 375 times 24.683. Therefore 24.683 must be multiplied by 375. This result must have three decimal places, because the multiplicand has three. The answer is 9256 dollars, 12 cents, and 5 mills. But the purpose was to multiply 24683 by .375, in which case the multiplier has three decimal places, and the multiplicand none. We pointed off as many places from the right of the multiplicand, as there were in the multiplier, and then used the multiplier as a whole number. This in fact makes the same number of decimal places in the product as there are in the multiplier. and 3 We may arrive at this result by another mode of reasoning. Units multiplied by tenths will produce tenths; units multiplied by hundredths will produce hundredths; units multiplied by thousandths will produce thousandths, &c. In the second operation of the above example, observe, that .375 ise, and 1, then of 3 is 10001 and of 3 is, which is and, set down the 5 thousandths in the place of thousandths, reserving the r Then of 80 is or 18, and 5 times is and (which was reserved) are 41, equal to Set down the T in the hundredth's place, &c. also, that when there are no decimals in the multiplicand. 80 8 100 40 1009 4 and 10 1 100° This shows there must be as many decimal places in the product as in the multiplier. It was observed that when a whole number is to be multiplied by 10, 100, &c. it is done by annexing as many zeros to the right of the number as there are in the multiplier, and to divide by these numbers, it is done by cutting off as many places as there are zeros in the divisor. When a number containing decimals is to be multiplied or divided by 10, 100, &c. it is done by removing the decimal point as many places to the right for multiplication, and to the left for division, as there are zeros in the multiplier or divisor. If, for example, we wish to multiply 384.785 by 10, we remove the point one place to the right, thus, 3847.85, if by 100, we remove it two places, thus, 38478.5. If we wish to divide the same number by 10, we remove the point one place to the left, thus, 38.4785; if by 100, we remove it two places, thus, 3.84785. The reason is evident, for removing the point one place towards the right, units become tens, and the the tenths become units, and each figure in the number is increased tenfold, and when removed the other way each figure is diminished tenfold, &c. How much cotton is there in 37% bales, each bale containing 4 cwt. 33.7; 43: = 4.75. In this example there are decimals in both multiplicand and multiplier. 4.75 3.7 3325 1425 Ans. 17.575 cwt. 3.7 is the same as 17, we have to find 37 of 4.75. Now to of 4.75, we have just seen, must be .475, and 37 is 37 times as much. We must therefore multiply .475 by 37, which gives 17.575 cwt. 75 475, and We shall obtain the same result if we express the whole in the form of common fractions. 4.75 3.737. Now according to Art. XVII. 100) 4700 100 10001 and 37 will be 37 times as much, that is 17575 17.575 1000 1000 17.575 as before. |