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are called decimal fractions; and when these fractions are written without the denominator, they are usually called decimals; and to denote the value of a decimal, a point is prefixed to as many figures of the numerator as there are ciphers in the denominator.
Thus, the decimal fraction to is written .2, the decimal fraction or is written .25, the decimal fraction 37 is written .375, and so on. The expressions .2, .25, .375, &c. are called decimals. Hence, it is evident, that the figures next the decimal point indicate tenths, the next figure hundredths, the next thousandths, and so on.
The decimal .2 is read two-tenths ; the decimal .25 is read twenty-five hundredths; the decimal .875 is read eight hundred and seventy-five thousandths; and so on. Since * = 10
and so on; .2=.20.200=.2000, &c. therefore the value of a decimal not changed by annexing a cipher to the end of it, nor by taking one away
If there be not as many figures in the numerator as there are ciphers in the denominator, ciphers must be put in the place of tenths, hundredths, &c.; thus, Tég is written .02, foco is written .025; and tooboo is written .00003; and
Hence, the value of figures in decimals are diminished in the same ratio from the decimal point towards the right, as whole numbers are increased from the right towards the left.
When the fractional part of a mixed number is reduced to a decimal, the decimal part is separated from the whole number by a decimal point.
Thus, 30 is written 3.75; 4700 is written 4.005; and
From what has been already observed, it is plain, that any fraction may b; reduced to a decimal by adding ci phers to the numerator and dividing by the denominator.
Thus, the fraction is reduced to .05, by adding two ciphers to 35, and dividing the expression 35.00 by 700 ; as there are no tenths, a cipher is put in the place of tenths ; so that the decimal equivalent to the fraction 76 is five hundredths.
Again, the fraction on, reduced to a decimal, is equiva lent to .030303, and so on; here there would still be a remainder, and it is also evident that the decimal would never
terminate ; in which case, it is only necessary in most calculations to use six or seven figures of the decimals.
A quantity of one denomination may be reduced to the decimal of another qaantity of the same kind, but of a difjerent denomination, by first expressing the ratio of the jormer to the latter hy a common fraction, and then reducing the fraction thus formed to a decimal.
For example, 2 nails is the lo of a yard, or šof a yard, which reduced to a decimal is equivalent to .125; hence, 2 nails is the 125 thousandths of a yard.
The reduction of a fraction to a decimal, or of one quantity to the decimal of another, is usually called reduction of decimals.
Examples in Reduction of Decimals. Example 1. What decimal of a foot is 9 inches ?
Here, 9 inches is the id or å of a foot; which, reduced to a decimal, is equivalent to .75, or 75 hundredths.
Ex. 2. What decimal of a yard is 2 feet 6 inches ?
Here, 2 feet 6 inches is the or of a yard, which, reduced to a decimal, is .833333, &c. This is a repeating or circulating decimal, never terminating.
Ex. 3. What decimal of an acre or 160 square poles, is 2 roods and 16 square poles ?
Here, 2 roods and 16 square poles is the 1% or of an acre, which, reduced to a decimal, is .6 or 6 tenths.
Ex. 4. What decimal of a cubic foot is 144 cubic inches?
Here, 144 cubic inches is the 1940s or of a cubic foot ; and ik, reduced to a decimal, is equivalent to .083333, &c. being a repeating decimal. Ex. 5. Reduce 8 feet 6 inches to the decimal of a mile.
Answer, .0016098. Ex. 6. Reduce 2 feet 5 inches to the decimal of a yard.
Ans. .805555. Ex. 7. Reduce 51 yards to the decimal of a mile or 1760 yards.
Ans. .003125 Ex. 8. Reduce 4ă miles to the decimal of 40 miles.
Ans. .1125. Ex. 9. Reducé 3 roods 11 poles to the decimal of an acre.
Ans. .81875. The decimal of one denomination inay be reduced to whole numbers o "lower denominations, as in the reduction of quantities of a higher denomination to a lower, observing, after each multiplication, to point off for decimals as many figures towards the right as there were figures in the given decimal. The figures on the left hand of the decimal points will be the whole numbers required. For example, .3945 of a day is equal to the fraction
of a day, which, expressed as the fraction of an hour, T0000 is X24=#8888, or 9 hours and 1888 of an hour; but 10000 4680 of an hour is 268.0 X 60 of a minute, or
which is equal 28 minutes, of a minute; again, this fraction of a minute is equal to 18060X60 of a second, or which is equal to 4 seconds, and % of a second; so that the decimal .3945 of a day is equal to 9 hours, 28 minutes, 46 seconds. From this it appears that pointing off the decimals serves the same purpose as dividing by the deno minator: thus,
28.0800 minutes. Examples in finding the values of Decimals Ex. 1. Required the value of .375 of a yard.
Ans. 1 qr. 2 nails. Ex. 2. Required the value of .625 of an acre.
Ans. 2 roods, 20 poles. Ex. 3. Required the value of .875 of a mile.
Ans. 7 furlongs. Ex. 4 Required the value of .2385 of a degree.
Ans. 14' 18" 36 thirds.
ADDITION OF DECIMALS. The addition of decimals is performed like that of whole numbers, observing, however, to arrange the numbers so that the separating points may be in the same column; that is, the tenths under tenths, the hundredths under hundredths, and so on.
For instance, the decimals .571, .672, .3, ,003, 0075, being arranged as follows:
.571 672 .3 .003 .0075
1.5535 heir sum is found to be 1.5535; the reason of the arrange ment is evident, since those figures are added together which are of the same local value.
Again, the sum of the numbers 3.5, 7.005, 4.325, .0003, and 1.000007, which contain whole units, is found in like manner, thus:
3.5 7.005 4.325
Examples in Addition of Decimals. Ex. 1. Required the sum of 5.714, 3.456, .543, and 17.4957.
Ans. 27.2087. Ex. 2. Required the sum of 3.754, 47.5, .00857, and 37.5.
Ans. 88.76257 Ex. 3. Required the sum of 54.34 .375, 14.795, and 1.5.
Ans. 71.01. Ex. 4. Required the sum of 37.5, 43.75, 56.25, and 87.5.
Ans. 225. Ex. 5. Required the sum of .375, .625, .0625, .1875, .3125, 4375, .005, .9475, and .0075.
Ans. 2.96. The Subtraction of Decimals is performed in the same manner as that of whole numbers ; observing to place each figure of the less below a figure of the same local value in the greater
For instance, let the difference of .3765 and 1236 be required: the decimals being arranged thus :
.2529 the difference will be .2529.
Again, let .7562 be taken from .82; by annexing ciphers to the greater and arranging the numbers thus :
.0638 we shall find the difference to be .0638: it must be observed, that the value of a decimal is not increased nor decreased by annexing ciphers to it; for a fraction does not alter its value by annexing ciphers to its numerator and denominator, thus; p= %=1600, and so on.
This is also evident from the decimal notation, which is similar to that of whole numbers ; that is, the value of the decimal .82 is 8 tenths and 2 hundredths, the value of the decimal .820 is also the same, being 8 tenths 2 hundredths and o thousandths; and so on.
Examples in Subtraction of Decimals. Ex. 1. Required the difference between 57.49 and 5.768.
Ans. 51.722. Ex. 2. Required the difference between .0076 and 00075.
Ans. .00685. Ex. 3. Required the difference between 3.468 and 1.2591.
Ans. 2.2089. Ex. 4. Required the difference between 3.1416 and .5236.
Ans. 2.6180. From the multiplication of fractions, or even the decimal notation, it appears evident, that the multiplication of decimals is performed as in whole numbers, but if there be not as many decimals in the product as there are in both factors, ciphers must be prefixed to supply the deficiency.
For instance, the product of .06 x .004 is equal to .00024; since .06 is equal 167, and .004 is equal va a: hence, 167 XTo6o=1o do, which, expressed according mo the decimal notation, is equal to .00024.