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Thus, is 5, and is 25, and is 075, and is 00124; 35

100

124 100000

where ciphers are prefixed to make up as many places as are in the numerator, when there is a deficiency of figures.

A mixed number is made up of a whole number with some decimal fraction, the one being separated from the other by a point. Thus 3.25 is the same as 31, or 225

Ciphers on the right hand of decimals make no alteration in their value; for .5 or 50 or 500, are decimals having all the same value, being each = or 4. But if they are placed on the left hand, they decrease the value in a tenfold proportion. Thus 5 is or 5 tenths, but 05 is only or 5 hundredths, and 005 is but 10 or 5 thousandths.

The first place of decimals, counted from the left hand towards the right, is called the place of primes, or 12ths; the second is the place of seconds, or 100ths; the third is the place of thirds, 1000ths; and so on. For, in decimals, as well as in whole numbers, the values of the places increase towards the left hand, and decrease towards the right, both in the same tenfold proportion; as in the following Scale or Table of Notation.

ADDITION OF DECIMALS.

RULE.-Set the numbers under each other according to the value of their places, like as in whole numbers; in which state the decimal separating points will stand all exactly under each other. Then, beginning at the right hand, add up all the columns of numbers as in integers; and point off as many places, for decimals, as are in the greatest number of decimal places in any of the lines that are added; or place the point directly below all the other points.

EXAMPLES.

1. To add together 29-0146, and 31465, and 2109, and 62417 and 14·16.

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2. To find the sum of 376-25 + 86·125 + 637-4725 + 6·5 + 41·02 + 358.865.

Ans. 1506-2325.

3. Required the sum of 35 + 47·25 + 20073 + 92701 + 1·5.

Ans. 981 2673.

4 Required the sum of 276 + 54321 + 112 + 0·65 + 12·5 + ·0463.

Ans. 455 5173.

SUBTRACTION OF DECIMALS.

RULE. Place the numbers under each other according to the value of their places, as in the last rule. Then, beginning at the right hand, subtract as in whole numbers, and point off the decimals as in Addition.

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RULE.*—Place the factors, and multiply them together the same as if they were whole numbers.-Then point off in the product just as many places of decimals as there are decimals in both the factors. But if there be not so many figures in the product, then supply the defect by prefixing ciphers.

EXAMPLES.

1. Multiply 321096

by •2465

1605480

1926576

1284384

642192

Ans. 0791501640 the product.

* The rule will be evident from this example: Let it be required to multiply ·12 by ·361; these numbers are equivalent to and the product of which is 133804332, by the nature of Notation, which consists of as many places as there are ciphers, that is, of as many places as there are in both numbers. And in like manner for any other numbers.

2. Multiply 79-347 by 23·15.

Ans. 1836.88305

3. Multiply ·63478 by ·8204.
4. Multiply 385746 by 00464.

CONTRACTION I.

Ans. 520773512. Ans. 00178986144.

To multiply decimals by 1 with any number of ciphers, as 10, or 100, or 1000, &c. THIS is done by only removing the decimal point so many places farther to the right hand as there are ciphers in the multiplier; and subjoining ciphers if need be.

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To contract the operation, so us to retain only as many decimals in the product as may be thought necessary, when the product would naturally contain several more places.

SET the units' place of the multiplier under that figure of the multiplicand whose place is the same as is to be retained for the last in the product; and dispose of the rest of the figures in the inverted or contrary order to what they are usually placed in.—Then, in multiplying, reject all the figures that are more to the right than each multiplying figure; and set down the products, so that their right hand figures may fall in a column straight below each other; but observing to increase the first figure of every line with what would arise from the figures omitted, in this manner, namely 1 from 5 to 14, 2 from 15 to 24, 3 froin 25 to 34, &c.; and the sum of all the lines will be the product as required, commonly to the nearest unit in the last figure.

EXAMPLES.

1. To multiply 27-14986 by 92-41035, so as to retain only four places of decimals in the product.

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2. Multiply 480-14936 by 2·72416, retaining only four decimals in the product.

3. Multiply 2490-3048 by ·573286, retaining only five decimals in the product. 4. Multiply 325-701428 by 7218333, retaining only three decimals in the product.

DIVISION OF DECIMALS.

RULE.—Divide as in whole numbers; and point off in the quotient as many places for decimals, as the decimal places in the dividend exceed those in the divisor.*

When the places of the quotient are not so many as the rule requires, let the defect be supplied by prefixing ciphers.

When there happens to be a remainder after the division; or when the decimal places in the divisor are more than those in the dividend; then ciphers may be annexed to the dividend, and the quotient carried on as far as required.

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WHEN the divisor is an integer, with any number of ciphers annexed; cut off those ciphers, and remove the decimal point in the dividend as many places farther to the left as there are ciphers cut off, prefixing ciphers if need be; then proceed as before.†

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* The reason of this rule is evident; for, since the divisor multiplied by the quotient gives the dividend, therefore the number of decimal places in the dividend is equal to those in the divisor and quotient taken together, by the nature of Multiplication; and consequently the quotient itself must contain as many as the dividend exceeds the divisor.

†This is no more than dividing both divisor and dividend by the same number, either IC, or 100, or 1000, &c., according to the number of ciphers cut off; which, leaving them in the same proportion, does not affect the quotient. And, in the same way, the decimal point may be moved the same num. ber of places in both the divisor and dividend, either to the right or left, whether they have ciphers or not,

CONTRACTION II.

HENCE, if the divisor be 1 with ciphers, as 10, or 100, or 1000, &c.; then the quotient will be found by merely moving the decimal point in the dividend so many places farther to the left as the divisor has ciphers; prefixing ciphers if need be.

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WHEN there are many figures in the divisor; or only a certain number of decimals are necessary to be retained in the quotient, then take only as many figures of the divisor as will be equal to the number of figures, both integers and decimals, to be in the quotient, and find how many times they may be contained in the first figures of the dividend, as usual.

Let each remainder be a new dividend; and for every such dividend, leave out one figure more on the right hand side of the divisor; remembering to carry for the increase of the figures cut off, as in the 2d contraction in Multiplication. Note. When there are not so many figures in the divisor as are required to be in the quotient, begin the operation with all the figures, and continue it as usual till the number of figures in the divisor be equal to those remaining to be found in the quotient, after which begin the contraction.

EXAMPLES.

1. Divide 2508-92806 by 92-41035, so as to have only four decimals in the quotient, in which case the quotient will contain six figures.

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2. Divide 4109-2351 by 230-409, so that the quotient may contain only four decimals.

3. Divide 37-10438 by 5713-96, that the quotient may contain only five decimals.

4. Divide 913-08 by 2137-2, that the quotient may contain only three decimals.

REDUCTION OF DECIMALS.

CASE I.

To reduce a vulgar fraction to its equivalent decimal.

RULE. Divide the numerator by the denominator as in Division of Decimals, annexing ciphers to the numerator as far as necessary; so shall the quotient be the decimal required.

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