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Let the pupil compare in a similar manner 1. 42 and 4200.

2. 42,000 and 42. 3. 420 and 4200.

4. 347 and 3470. 5. *520 and 5200.

6. 86,430 and 86,430,000. Compare the values expressed by the — 7. 42 in 42 and 427.

8. 39 in 39 and 4398. 9. 28 in 128 and 432,876. 10. 497 in 1497 and 638,497. 11. 574 in 15,748 and 35,742,631. Compare the values expressed by each of the 4's — 12. In 44; in 440; in 444. 13. In 4400; in 404; in 4444. Apply the same principles in comparing each of the 4's — 14. In 4.4; in 4.04; in 40.4; in .404. 15. In 44.4; in 4.44; in .444. 16. In 400.4; in 40.04; in 4.004; in .4004.

I. It thus appears that 4, written in the first place at the right of the point, represents one-tenth of 4 units, which equals 4 tenths, while written in the second it represents ado of 4 units, or rdo; in the third, it represents Toto of 4 units, or Too; &c., with the other places.

As a similar thing would be true of any other digit, we name the places at the right of the point in the manner indicated below.

Hundredths.
Thousandths.
Ten Thousandths.
Hundred-Thousandths.
Tenths.
Ten-Millionths.
Millionths.
Hundred-Millionths.
Point.
Billionths.

000000000 1. Which place from the point is occupied by the millionths' figure ?

Ans. The sixth place at the right. Which place from the point is occupied 2. By the tenths' figure ? by the thousandths'? 3. By the hundredths'? by the billionths'? 4. By the ten-thousandths'? by the hundred-thousandths'? 5. By the millionths' ? by the hundred-millionths'? 6. What would be the denomination of a figure in the second

86
100

2. .086 = 1000:

3. .0086 = 10000*

86

3 49 1000

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place at the right of the point ? in the fifth ? in the first ? in the sixth ? in the ninth ? in the fourth ? in the eighth ? in the seventh ?

J. Numbers expressed by figures written at the right of the point are read on exactly the same principles as those expressed by figures at the left. (See letters D and F.) Thus, if we wish to read .37, we observe that the right-hand figure is in the hundredths' place. We read it, then, as though written do: The following will furnish further illustrations of the principle involved : 1. .86

1867

4. .349
5. .0349
18487.

6. .000349 = 103880. 7. 3.49 188=34%. 8. 34.9=30=343. 9. 6.07 = 18% = 6170 10. 1.01 = 181=lido. 11. 10.1 101= 1016 12. 300.3= 3483 = 30036. 13. 30.03 = 3,9=3018. 14. 3.003= 18:=310%. 15. .3003=38%. 16. 7.008706=1888788=7100206

Let the pupil now read the following, expressing the value, in all cases where it is greater than unity, both as an improper fraction and a mixed number :

17. 73 18. 50.05
19. .08 20. 0.0764
21. .798 22. 37.9427
23. 6.4 24. 876.5874
25. 7.37 26. 3299.4856328
27. 4.28 28. 1000000.0000001
29. .94 30. 87.203794
31. 3.06 32. 87.000000079
33. 40.7 34. .000000007
35. 21.3 36. 2006.000002006

K. Any fraction whose denominator is a power of ten may be expressed by writing the numerator so that its right-hand figure shall occupy the place of the same name with its denominator, and omitting the denominator. Fractions thus written are called Decimal Fractions to distinguish them froin Vulgar

Fractions, or those whose numerator and denominator are both written.

Write the following in the form of decimal fractions :
1. 7. 6. 7. 10. 18. To.
2. 1877 1887 TO 18370: Toto
3. Toulot. 17. 370. 18. 418
4. 176. 8766 90 106. 8167.
5. 80767. 4970. 231. 2 100. 137.
6. 47681573877. 57810000000 300106
7. 188:888. 410000
8. 1000000 TOOOOOO.
9. 78467320071006000000 :

L. The preceding illustrations and explanations show that removing the figures representing a number one place further towards the left, or, which is the same thing, removing the point one place to the right, multiplies the number represented by 10, while removing the figures one place to the right, or the point one place to the left, divides it by 10. A change of two places would in like manner multiply or divide by 100, of three places by 1000, &c.

These principles generalized would be stated thus :

To express the product of any number multiplied by any power of 10, remove the decimal point as many places to the right as there are zeros used in writing the given multiplier.

To express the quotient of a number divided by any power of 10, remove the decimal point as many places to the left as there are zeros used in writing the given divisor.

When, by such change, any places between the number and point are left vacant, they must be filled by zeros.

How will you express in figures the results of the following indicated operations ?

1. 87 X 10.* 143 X 10. 8.7 x 10. 2. .87 x 10. 1.43 X 10. .087 X 10. 3. .0087 X 10. .0143 X 10. 87 • 10. 4. 14.3 • 10. 8.7 • 10. .87 10. 5. .087 ; 10. .143 ; 10. .0087 • 10. * The pupil should remember that when the decimal point is not marked, it is always understood to belong at the right of the given figures.

6. 870 ; 10. 1430 ; 10. 57 • 100. 7. 5.7 • 100. 327.8 ; 100. .57 • 100. 8. .057 • 100. 3.278 : 100. .3278 • 100. 9. .3278 X 100. 4.267 X 1000. 37.9 X 100. 10. 4.5786 x 1000. 643.7 • 1000. 7; 100. 11. .4 ; 10. .08 = 100. 63 • 1,000,000. 12. 479.643 X 1000. 479.643 = 1000. .607 X 100.000.

13. Let the pupil now tell by inspection, without changing the place of the point or re-writing the numbers, the result of the above indicated operations. Thus, 87 multiplied by 10 equals 870, &c.

He should be able to do this without the slightest hesitation.

14. Where will you place the point in order that the figures 573 may express 573 tenths ? hundredths ? units ? tens? hundreds ? thousands ? thousandths ? millions? millionths ?

15. Where will you place the point in order that the figures 87064 may express 87064 hundreds ? hundredths ? ten-thousands? ten-thousandths ? hundred-millions ? hundred-mil. lionths ? tens? tenths ?

16. Where will you place the point in order that the figures 497837 may express 497837 units? millions ? billionths ? hundredths ? millionths ? hundreds ? hundred-thousandths ?

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= 875.

M. 1. Reduce to a decimal fraction.
Solution. }

f of 70 tenths =.8 with a remainder of .6; but .6 .60, and of .60 .07 with a remainder of .04; but .04 = .040, and f of .040 = .005. Therefore, s

=.8+.07 +.005 2. Reduce to a decimal; ;$; $; ii$.

Note. In the following examples, the pupil need not reduce to a lower denomination than millionths, even if the division does not terminate there.

3. Reduce to a decimal fraction ; ; ; ; ; 13; 4. What is the value of 4574 expressed decimally?

Ans. 457.5714284. But 4 of .000001 is so small that it may be omitted. We should then have for an answer 457.571428.

5. What is the value of 831 expressed decimally ? of 57 ? of 38171 ? of 2717 ? of 3271 ? of 1413 ? of 28713?

SECTION XX.

A. Addition is the process by which, having several numbers given, we find a number equal in value to all of them.

The number thus obtained is called the sum or amount. Thus, 7+3+5 would be a problem in addition, and the answer, 15, would be the sum of 7, 3, and 5.

In order that numbers may be added, it is necessary that the things they represent shall be of the same name or denomination. 2 locks and 3 keys would be neither 5 locks nor 5 keys, but since locks and keys are both things, we can change the denomination of both by calling them things; when we shall have, 2 things and 3 things are 5 things.

In like manner 2 tens and 3 units would be neither 5 tens nor 5 units; but by changing the tens to units, calling them 20 units, we shall have 2 tens † 3 units = 20 units + 3 units= 23 units. 2 shillings + 3 pence are neither 5 shillings nor 5 pence, but since 2 shillings 24 pence, 2 shillings + 3 pence must equal 24 pence + 3 pence, or 27 pence.

For such reasons it will be found convenient, in writing large numbers for addition, to write those of the same denomination near each other, which can best be done by writing them in vertical columns, so that units shall come under units, tens under tens, &c., and pounds under pounds, shillings under shillings, pence under pence, &c. In adding, we can begin with any denomination we choose, but it will usually be more convenient to begin with the lowest, or the one at the right hand.

B. Example 1. Mr. Jones owns the following named property; a farm worth 3798 dollars, a house and garden worth 7427 dollars, three horses worth 427 dollars, one yoke of oxen worth 125 dollars, twelve cows worth 339 dollars, and 1564 dollars in money.

How
many

dollars is he worth? It is evident he will be worth as many dollars as there are in 3798 + 7427 + 427 + 125 + 339 + 1564.

Placing these numbers, for the sake of convenience, in the following manner,

$3793.
7427.
427.
125.
339.
1564.

$13680,

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