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178. If ciphers are placed on the left of decimal figures, between them and the decimal point, those figures change their places, each cipher removing them one place to the right; thus, .3 1, but .03 T8ʊ, and .003

=

3

Tooo. Hence,

Every cipher placed on the left of decimal figures, between them and the decimal point, decreases the value represented by them the same as dividing by ten.

=

179. If ciphers are placed on the right of decimal figures, their places are not changed; thus, .3 f, and .30 .3. Hence,

-=

= 10

To

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~ Millions.

Ciphers placed on the right of decimals do not alter the value represented by them.

Hence, decimals may be reduced to a common denominator, by making their decimal places equal by annexing ciphers.

NUMERATION.

180. The relation of decimals to whole numbers and to each may be learned from the following

other

Hunds. of thousands.

Tens of thousands.
Thousands.

=

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7th order or place,

6th order or place,
5th order or place,
4th order or place,
3d order or place,
2d order or place,

1st order or place,

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Whole Numbers.

Decimals.

178. What effect have ciphers placed at the left hand of decimals? Why?

179. What effect if placed at the right hand? Why?-180. What may be learned from the table?

A Mixed Number is a whole number and decimal in a single expression.

The preceding table consists of a whole number and decimal forming a mixed number. The part on the left of the decimal point is the whole number, and that on the right the decimal. The decimal part is numerated from the left to the right, and its value is expressed in words thus: Two hundred thirty-four millions five hundred sixty-seven thousand eight hundred ninetythree billionths. And the mixed number thus: Seven millions six hundred fifty-four thousand three hundred twenty-one, and two hundred thirty-four millions five hundred sixty-seven thousand eight hundred ninety-three billionths. Hence the

RULE. Read the decimal as though it were a whole number, giving it the name of the right-hand order.

NOTE. A decimal with a common fraction annexed constitutes a complex decimal; as, .61, read 61 tenth.

1.

2.

3.

4.

Write in words, or read orally, the following figures: —

.72859 12.02003 121.000386 2.3058217

.5

.42

.01

.908

5.

6.

7.

8.

.3001

.0984

.00013

.82007

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NOTATION.

9.

10.

11.
12.

181. Tenths occupy the first place at the right of the decimal point, hundredths the second, &c., and each figure takes its value by its distance from the place of units; therefore, to write decimals, we have the following

RULE. Write the decimal as though it were a whole number, supplying with ciphers such places as have no significant figures.

Write in figures the following numbers:

1. Three hundred seven, and twenty-five hundredths. 2. Forty-seven, and seven tenths.

180. Of what does it consist? What is the number called, when taken together? What is the part on the left of the decimal point? The part on the right? What is the value of the decimal? The value of the mixed number? The rule for reading decimals?—181. Upon what does the value of a decimal figure depend? The rule for writing decimals?

3. Eighteen, and five hundredths.

4. Twenty-nine, and three thousandths. 5. Forty-nine ten thousandths.

6. Eight, and eight millionths.

7. Seventy-five, and nine tenths.

8. Two thou-and, and two thousandths.

9. Eighteen, and eighteen thousandths.

10. Five hundred five, and one thousand and six millionths. 11. Three hundred, and forty-two ten millionths. 12. Twenty-five hundred, and thirty-seven billionths.

182. Decimals, since they increase from right to left, and decrease from left to right, by the scale of ten, as do simple whole numbers, may be added, subtracted, multiplied, and divided, in like manner.

ADDITION.

183. Ex. 1. Add together 5.018, 171.16, 88.133, 1113.6, .00456, and 14.178. Ans. 1392.09356.

OPERATION.

5.0 18

17 1.1 6
8 8.1 3 3

1113.6

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.00 4 5 6 1 4.1 7 8

1 3 9 2.0 9 3 5 6

RULE. Write the numbers so that figures of the same decimal place shall stand in the same column.

Add as in whole numbers, and point off, in the sum, from the right hand as many places for decimals as equal the greatest number of decimal places in any of the numbers added.

Proof. The proof is the same as in addition of simple numbers.

EXAMPLES FOR PRACTICE.

2. Add together 171.61111, 16.7101, .00007, 71.0006, and 1.167895. Ans. 260.489775.

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We write the numbers so that figures of the same decimal place shall stand in the same column, and then, beginning at the right hand, add them as whole numbers, and place the decimal point in the result directly under those above.

3. Add together .16711, 1.766, 76111.1, 167.1, .000007, and 1476.1. Ans. 77756 233117.

182. How do decimals increase and decrease? How may they be added, subtracted, multiplied, and divided?-183. How are decimals arranged for addition? The rule for addition of decimals? What is the proof?

4. Add together 151.01, 611111.01, 16.5, 6.7, 46.1, and .67896. Ans. 611331.99896.

5. Add fifty-six thousand, and fourteen thousandths; nineteen, and nineteen hundredths; fifty-seven, and forty-eight ten thousandths; twenty-three thousand five, and four tenths; and fourteen millionths. Ans. 79081.608814.

6. What is the sum of forty-nine, and one hundred and five ten thousandths; eighty-nine, and one hundred seven thousandths; one hundred twenty-seven millionths; forty-eight ten thousandths? Ans. 138.122427.

7. What is the sum of three, and eighteen ten thousandths; one thousand five, and twenty-three thousand forty-three millionths; eighty-seven, and one hundred seven thousandths; fortynine ten thousandths; forty-seven thousand, and three hundred nine hundred thousandths? Ans. 48095.139833.

SUBTRACTION.

184. Ex. 1. From 74.806 take 49.054.

OPERATION.

7 4.8 0 6 4 9.0 5 4

2 5.7 5.2

RULE. Write the less number under the greater, so that figures of the

same decimal place shall stand in the same column.

Subtract as in whole numbers, and point off the remainder as in addition of decimals.

Ans. 25.752.

Having written the less number under the greater, so that figures of the same decimal place stand in the same column, we subtract as in whole numbers, and place the decimal point in the result, as in addition of decimals.

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Proof. The proof is the same as in subtraction of simple

numbers.

EXAMPLES FOR PRACTICE.

4.

4 6.1 3
7.89 15

3 8.23 8 5

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2.

1 1.0 7 8

9.8 1

1.2 68

6. From 81.35 take 11.678956.

7. From 1 take .876543.

8. From 100 take 99.111176.

9. From 87.1 take 5.6789.

47.1 17
8.78 195

3 8.3 3 5 0 5

5. 87.1 07 1.1 1 9 8 6

8 5.9 8 7 1 4 Ans. 69.671044. Ans. .123457. Ans. .888824. Ans. 81.4211.

184. What is the rule for subtraction of decimals? What is the proof?

10. From 100 take .001.

Ans. 99.999.

11. From seventy-three, take seventy-three thousandths. Ans. 72.927.

12. From three hundred sixty-five take forty-seven ten thousandths. Ans. 364.9953. 13. From three hundred fifty-seven thousand take twentyeight, and four thousand nine ten millionths.

14. From .875 take .4.
15. From .3125 take .125.
16. From .95 take .44.

17. From 3.7 take 1.8.
18. From 8.125 take 2.6875.

19. From 9.375 take 1.5.

20. From .666 take .041.

OPERATION.

1 8.7 2
7.1

MULTIPLICATION.

185. Ex. 1. Multiply 18.72 by 7.1.

OPERATION.

5.1 2

.012

1872 13104

1 3 2.9 1 2

=

=13. Then 1872 × 77 = 182912 same as in the operation.

100

Ex. 2. Multiply 5.12 by .012.

Ans. 356971.9995991.

Ans. .475. Ans. .1875.

1024

512

.0 6 1 4 4 Ans.

Ans. 132.912.

We multiply as in whole numbers, and point off on the right of the product as many figures for decimals as there are decimal figures in the multiplicand and multiplier.

The reason for pointing off decimals in the product as above will be seen, if we convert the multiplicand and multiplier into, common fractions, and multiply them together. Thus, 18.72 18726: 1872; and 7.1 770 132,912 = 132.912, Ans., the

=

100

100
1000

Ans. .51.

Ans. 1.9.

Ans. 5.4375.

Ans. 7.875.

Ans. .625.

Since the number of figures in the product is not equal to the number of decimals in the multiplicand and multiplier, we supply the deficiency by placing a cipher on the left hand.

The reason of this process will appear, if we perform the question thus: 5.12-512-413, and .012. Then 50×1080 = .06144, Ans., the same as before. Hence we deduce the following

=

6144 100000

185 In multiplication of decimals how do you point off the product? The reason for it? When the number of figures in the product is not equal to the number of decimals in the multiplicand and multiplier, what must be done?

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