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88. A Decimal Fraction is usually expressed by placing a point (.), called the decimal point, before the numerator, and omitting the denominator. Thus,

4 0

127 100000

0

4 0

4

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Ten

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89. In the common system of notation the value of figures decreases from left to right in a tenfold ratio. If this law be continued below units, the first place at the right of units will express tenths, the second place hundredths, the third place thousandths, etc.

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From this it appears that decimals are not only closely connected with common fractions, but are also a continuation of the common system of notation. This is more clearly shown in the following

.3,

.04,

0

0

0

0.

0 0 0

0

0

0

0

0 ·

0

0

0 0.

4

4 4 4 4

Integers.

NUMERATION TABLE.

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Thousandths,

Ten-thousandths,

. Hundred-thousandths,

0

0 0 0

4

4 4

1000 .005,

τοστ = .0006,

348

1000000

0

4

Decimals.

Millionths.

=

4

.000348, etc.

4, etc.

Is read,

4 Tenths,

4 Hundredths,

4 Thousandths,

4 Ten-thousandths,

4 Hundred-thousandths,

4 Millionths.

90. A Pure Decimal is one which consists of decimal figures only; as, .4, .25, etc.

A Mixed Decimal is one which consists of an integer and a decimal; as, 4.25.

A Complex Decimal is one which has a common fraction at the right of the decimal; as, .3.

NUMERATION OF DECIMALS.

91. In reading a decimal, first read the integral part, if any; then the decimal part regarded as an integer; and, third, the name of the right-hand decimal place. To prevent ambiguity, pause after reading each part.

Thus, 5.408 is read five-and four hundred and eightthousandths.

400.004 is read four hundred-and four-thousandths; and .404 is read four hundred and four- -thousandths.

Read the following decimals:

1. .5.

2. .8.

3. .12.

4. .64.

5. .96.

6. .437.

7. .525.

8. .06.

9. .008.

10. .4267.

11. .0261.

12. .0028.

13. .0009.

14. 1.0001.

15. 300.003.
16. .303.

17. 600.005.

18. .605.

19. 42.06.

20. 125.125.

21. 1.205.

22. .000325.

23. 100.001.

24. .000035.
25. 1.2345.

26. .463.
27. .085.

28. .03.

29. Read the following, and notice the relative position of tens and tenths, hundreds and hundredths, etc.:

10.1; 100.01; 100.001; 10000.0001;

1000000.000001.

30. Read the following, and notice the effect of moving the decimal point one, two, three, or more places to the left:

516; 51.6; 5.16; .516; .0516;

.00516.

31. Read the following, and notice the effect of moving the decimal point one, two, three, or more places to the right:

.4862; 4.862; 48.62; 486.2; 4862.

32. Read the following, and notice the effect of placing one, two, three, or more zeros between the decimal point and the first decimal figure:

26.4; 26.04; 26.004;

26.0004; 26.00004.

33. Read the following, and notice the effect of placing zeros on both sides of the decimal point:

3.3; 30.03; 300.003;

3000.0003.

34. Read the following, and notice the similarity of sound and difference of value:

404000; 400.004; .404; 303000; 300.003; .303.

NOTATION OF DECIMALS.

92. In writing a decimal, write the numerator as a whole number, and then place the decimal point so that the right-hand figure expresses the denomination required.

Write in figures

1. Four tenths.

Six tenths. Nine tenths. 2. Five hundredths. Forty-five hundredths. 3. Seven thousandths. Seventy-five thousandths. 4. Eighty-five hundredths. Eight, and five tenths. 5. Sixty, and six hundredths. 125 thousandths. 6. Three hundred and four ten-thousandths.

7. Five, and fifteen thousandths. One millionth. 8. Ninety, and nine hundredths. Four millionths. 9. One hundred and twenty, and four millionths. 10. Sixty-five hundred, and sixty-five hundredths. 11. Seventy-five thousand, and seventy-five thousandths.

12. Three hundred, and three thousandths.
13. Three hundred and three thousandths.
14. Eight tens. Eight units. Eight tenths.
Four tens. Four tenths.
Six thousandths.

15. Four hundred.
16. Six thousand.
17. Twenty-five tens.
18. Twenty-five hundred.
19. 325 hundredths.

Twenty-five tenths.

325 tenths.

20. 125 ten-thousandths. 125 millionths. 21. 4211 hundredths. 4211 thousandths. 22. 16, and 25 thousandths.

23. 9, and 9 ten-millionths.

14 ten-millionths.

24. 1007 millionths.
25. Four and one-half tenths.
26. Twenty-four and two-thirds hundredths.
27. One-half tenth. One-third hundredth.

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Twenty-five hundredths.

.3 becomes .30
.3 becomes .300

=

34. 163.
35. 33%

100

36. 21610
21610039.

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Hence the value is not changed.

48. 2000.6.

49. 200.06.

50. 20.006.

51. 2.0006.

87. 1258.

1000

93. Principles.-1. Annexing ciphers to a decimal does not alter its value.

Take as an example .3 =

30 ==

by annexing one cipher.

100 by annexing two ciphers, etc.

38. 8100000.

162

39. 7000000.

52. .331.
53. .66.

54. .163.

55. .37.

2. Prefixing ciphers to a decimal diminishes its value ten times for every cipher prefixed.

Take as an example .3 = 1%.

18 by prefixing one cipher.

Too by prefixing two ciphers, etc.

Hence the value is diminished ten times for each cipher prefixed.

.3 becomes .03
.3 becomes .003

3. Moving the decimal point to the right increases the value of the decimal ten times for every place passed over.

Take as an example .333 1000

333

.333 becomes 3.33
.333 becomes 33.3

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=

333 by moving the decimal point one place. 333 by moving it two places, etc.

Hence the value is increased ten times for each place passed over.

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4. Moving the decimal point to the left diminishes the value of the decimal ten times for every place passed over.

Take as an example 33.3

333.

33.3 becomes 3.33
33.3 becomes .333

=

133 by moving the decimal point one place. 133 by moving it two places, etc.

Hence the value is diminished ten times for each place passed over.

REDUCTION OF DECIMALS.

CASE I.

94. To reduce a decimal to a common fraction.

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RULE.

Express the decimal as a common fraction, and reduce it to its lowest terms.

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