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2. Decimal Fractions.

A DECIMAL FRACTION is one whose numerator only is expressed. Were the denominator to be written, it would, in all cases, be 10, 100, or 1, with as many ciphers annexed as there are figures in the decimal.

When there are whole numbers and decimals in the same sum, it is called a mixed number as, 16.44, which is read sixteen and forty-four hundredths.

In decimals, unity is considered a fixed point, each way from which the value of the numbers varies in a ten-fold proportion, increasing towards the left hand, and decreasing towards the right, as in the following

TABLE.

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Ciphers in the right hand of decimals do not alter their value, but on the left hand, decrease their value in a ten-fold proportion. Thus, .5 .50 and .500 express the same value, viz. 2, and are read 5 tenths, 50 hundredths, and 500 thousandths; but .5 .05 and .005 decrease in value in a ten fold proportion, and are read 5 tenths, 5 hundredths, and 5 thousandths.

In order to make the scholar familiar with the notation of decimals, he is requested to write out the following

Examples.

1. Express the decimal .36 in. words.

2. Express .03 in words.
3. Express .1002 in words.
4. Express 27.27 in words.
5. Express 34.14 in words.

1. Write forty-three thousandths in characters.

2. Write 60 and nine hundred thousandths in characters. 3. Write one hundred and four thousandths.

1. ADDITION OF DECIMALS.

Rule.

Place the numbers under each other according to the value of their places, and add as in whole numbers. Point off as many decimal places from the sum as are equal to the greatest number of decimal places in either of the given numbers.*

When the numbers are all written properly, and the amount properly pointed, the separatrices, or decimal points, will all stand in a column, or directly

Examples.

1. What is the sum of 25.4 rods, 16.05 rods, 8.842 rods and 46.004, when added together?

25.4

16.05

8.842 46.004

96.296 Ans.

Here it will be seen that the decimal points all fall in the same column, that the decimals are arranged towards the right hand from this column, and the whole numbers towards the left, and that the number of decimals in the sum is equal to the greatest given number of decimals.

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2. SUBTRACTION OF DECIMALS.

RULE. Place the less number under the greater, with one of the decimal points directly under the other; then subtract as in whole numbers, and point off as in addition.

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over one another. All the difficulty in Decimal Fractions is in placing the numbers, and pointing off the decimals. In other respects, they are managed precisely as whole numbers. The scholar should endeavor to become familiar with the management of Decimals, as to us they form one of the most useful parts of Arithmetick. Our lawful mode of reckoning money is purely decimal.

3. MULTIPLICATION OF DECIMALS.

RULE. Write the multiplier under the multiplicand; then multiply as in whole numbers, and from the product point off as many places for decimals, as there are decimal places in both the factors. If there be not so many figures in the product as there ought to be decimals, supply the deficiency by prefixing ciphers.

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RULE.-Divide as in whole numbers, and point off so many places for decimals in the quotient as the decimal places in the dividend exceed those in the divisor. If there are not so many figures in the quotient as the number of decimals required, supply the defect by prefixing ciphers. If the decimal places in the divisor exceed those in the dividend, make them equal by annexing ciphers to the latter. When there is a remainder, by annexing ciphers, more decimal places may be obtained in the quotient.

26

1000

26

78

* The truth of this rule will appear by considering that .026 and .003 are equivalent to and whence 1880X1800-1000000 .000078 by the nature of notation; that is, the decimal consists of as many places as there are ciphers in the denominator, and when the product falls short of this number, the deficiency must be made up by ciphers on the left hand. There are usually given several methods of contraction under this rule; but they are of no essential service, and might perplex the young scholar. It may not be amiss, however, to observe that in dividing by 10, 100, or 1 with any number of ciphers, we have only to remove the separatrix as many places towards the right hand as there are ciphers in the multiplier; thus 2.71 multiplied by 10 is 27.1; by 100, it is 271, &c.

†The reason of the rule for pointing off the decimal places in the quotient will appear obvious by considering that the divisor and quotient are two factors whose product is the dividend, and that the decimal places in both the factors are equal to the decimal places in their product, as was shown in Multiplication of decimals.

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The Scholar is requested to point the two following examples.

4. Divide 4263 by 2.5

Ans. 17052

5. Divide 4.2 by 36.

Ans. 116+

RECIPROCALS.

The Reciprocal of a given number is one, which, multiplied by the given number, gives a unit for the product; thus .2 is reciprocal of 5, because 5 ×.2=1.

If the given number be a multiplier, its reciprocal may be employed as a divisor of the multiplicand, and the quotient will be equal to the product of the multiplicand by the multiplier; but if the given number be a divisor, its reciprocal may be employed as a multiplier of the dividend, and the product will be equal to the quotient of the dividend by the divisor. Examples.-1. Multiply 7 by 5. 7×5-35, and 7.235. 2. Divide by 5. 7-5-1.4, and 7X.2=1.4.

Problem.

To find the reciprocal of any number.

RULE.-Divide a unit by the given number, and the quotient will be its reciprocal.

Examples.

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