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n both; the only peculiarity in decimals relates to the placing of the decimal mark, which will easily be underfrom the rules and explanations, which follow.

In addition to the foregoing observations, it has been at necessary to subjoin the following table, whereby the system of notation in whole numbers and decimals will be torily shewn.

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ppears, by inspection of this table, that the figure on each

f the units place, and equally distant from it, is of like deation, one in parts, and the other in wholes; but as in g large whole numbers we make use of an abbreviated of expression, the same is found convenient in reading de. The above table is thus read; Nine hundred and sixtymillions, four hundred and seventy-two thousands, five ed and thirty-one, and thirty-five millions, two hundred venty-four thousands, eight hundred and sixty-nine, hundred nths: this, as far as it relates to the whole numbers, is sufly obvious. With respect to the decimals, this latter mode pressing them, and that in the table, may at first sight r to be different; we will shew that they are exactly of the import.

Thus, 3 tenths........ are equal to

5 hundredths

307

millions

5

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Hundred Millionths.

is, the decimals as expressed in the table are together equal

to 35 millions, 274 thousands, 869 hundred millionths, as we expressed their value above; which was to be shewn.

220. There is another method of reading decimals, which is very convenient for practice, as follows:

The first place of decimals next to unity is called the place of primes; the next place is called the place of seconds, &c. and the figures occupying those places are called respectively primes, seconds, thirds, fourths, &c.

Thus the number 5.27 is read five, two primes, and seven seconds: and nine thirds, eight fourths, and six sevenths, is thus expressed in figures, .0098006.

221. EXAMPLES IN DECIMAL NOTATION AND NUMERATION. Write in words the following decimals.

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Write in figures the following decimals.

Six tenths. Thirty-nine hundredths. Four hundred and fifty-six thousandths. One millionth. Five, and twenty-three hundredths. Three thousand three hundred and thirty-three ten thousandths. One, and twenty-four ten thousandths. Seven, and eight primes. Two, and two seconds. Nine thirds, eight fourths, seven fifths, and six sixths. Seven sixths, eight ninths, and nine tenths.

222. ADDITION OF DECIMALS.

RULE. Place the numbers so that the decimal marks may stand in a line under each other; then will units stand under units, tens under tens, tenths under tenths, hundredths under hundredths, &c. then, beginning at the right hand, add the numbers together like whole numbers; and from the right hand of the sum cut off as many figures by the decimal mark as are equal to the greatest number of decimal places in any of the given numbers.

The rule for placing the numbers to be added is extremely obvious; for, since figures of different denominations cannot be added together, it is plain,

EXAMPLES.

1. Add 2.34 + 35.2 + .7831 + 1.2481 +8.0379 together.

OPERATION.

2.34

35.2 .7831 1.2481 8.0379 Sum 47.6091

45.2691

Proof 47.6091

Explanation.

Having placed the numbers so that like places may stand under like, I add up exactly as whole numbers are added; then I count four (equal to the greatest number of decimal places) from the right hand of the sum, and place the decimal mark to the left of the fourth figure. The proof is the same as in simple Addition.

2. Add 12.9 + 7.38 + 8.2 + .945 +1.805 together, Sum 31.23.

3. Add 7.239 + 10.0046 + 3.27.89 + .0073 together. Sum 21.4109.

4. Add 942.64 + 2.301 + 71.5 + 8.457 + 3091.9 together. Sum 4116.798.

5. Add .4937 + .008 + .37042+.89139 +1.290037 together. Sum 3.053547.

6. Add 3748.2 + 9.8073 + 120.965 + 1374.7 +48. together.

223. SUBTRACTION OF DECIMALS.

RULE. Place the less number below the greater, with the decimal marks under each other, so that units may stand under units, tenths under tenths, &c. as in Addition; then subtract as in whole numbers, and cut off from the right hand of the remainder as many figures for decimals as there are decimal places in either of the two given numbers".

that those of the same denomination must be placed under each other, as they alone are capable of being added.

With respect to the operation, it is plain that 10 hundredths make I tenth, 10 tenths make I unit, 10 units 1 ten, and so on in every denomination, whether it be above or below unity; wherefore, since the same law obtains in both decimal parts and whole numbers, both must evidently be added by the same rule, namely, by simple Addition.

The observations contained in the preceding note apply equally to this rule, which is obvious from the nature of simple Subtraction,

When ciphers occur in one or more of the left hand decimal places, they

PS

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6. From 12.3456 take .78095. Diff. 11.56465.
7. From .081059 take .0003741. Diff. .0806849.
8. From 987.6 take .05432. Diff. 987.54568.
9. From .638176 take .03749. Diff. 600686.

224. MULTIPLICATION OF DECIMALS.

RULE I. Place the factors so that the right hand figure of the multiplier may stand under the right hand figure of the multiplicand, and multiply as in whole numbers.

II. Count the decimal places in both factors, and from the right hand of the product mark off as many figures for decimals as there are decimals in both factors together.

To prove the operation, multiply the multiplier by the multiplicand, and proceed as before; or cast out the nines, as in simple multiplication.

III. When the number of decimals in both factors exceeds the number of figures in the product, prefix as many ciphers to the left of the product as will make up the number, and to the left of them place the decimal mark *.

must always be put down, but ciphers occupying the right hand decimal places may be omitted; thus in ex. 2, the two ciphers at the left of the difference are put down, and in ex. 3, the two that arise at the right hand of the proof are omitted: likewise when any of the right hand places of decimals are wanting, as in ex. 2 and 3, the operation is to be performed as though there were ciphers in those vacant places.

s To make the truth of this rule plain, recourse must be had to an easy example; thus, let .5 be multiplied by .3; these numbers are equivalent to

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as appears from the nature of decimal notation; now

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prove the operation, multiply the multiplier by the multi

d, and mark off decimals in the product as before; or cast e nines, as in simple multiplication.

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re are 5 decimals in one factor, and 4 in the other, that is, 9 in both; I ore count 9 places from the right of the product, and put the decimal to the left of the ninth place.

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15

15

by Vulgar Fractions; also .5 X 3.15 by the rule: but

00

= .15 100

Decimal Notation; that is, the result obtained by this rule, and that obed by Vulgar Fractions, are the same: the rule is therefore true.

f any doubt should remain respecting the truth of the rule when there are le numbers concerned, let the factors in ex. 1. be turned into vulgar frac

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