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6. What vulgar fraction is equivalent to 0.0123456789 ?

CASE III.
When the decimal is a compound repetend.
In this case we obviously have the following

RULE.
First, find the vulgar fraction, which is equivalent to the decis
mal figures which precede those that circulate, by rule under Case
I, of this Article.

Secondly, find the vulgar fraction, which is equivalent to the circulating part of the decimal, by rule under Case II, of this Article; to the denominator of this fraction annex as many ciphers as there are decimals which precede the circulating part of the repetend; then add these two fractions together.

Examples. 1. What vulgar fraction is equivalent to the compound repetend 0.343 ?

. Ans. So todo=8=!%. 2. What vulgar fraction is equivalent to the compound repetend 0.087837 ?

Ans. 3. What vulgar, fraction is equivalent to 0.083?

Ans. th 4. What vulgar fraction is equivalent to the compound repetend 0.0357142857 ?

Ans. zł. 5. What vulgar fraction is equivalent to the compound repetend 0.0714285?

6. What vulgar fraction is equivalent to 0.123456 ?

CHAPTER IV.

CONTINUED FRACTIONS. 55. If we divide both numerator and denominator of the fraction it by the numerator, we obtain,

351 1
965=2+263 .

351. Again, performing the same operation upon the fraction 2013, we find 263 1

' 3511 +88

263; this value of Isi substituted in I. we get,

35 13

II. 351

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III. 351 1

965=2+1

1+1
. . 2+87

87 1.
88. Again, 38=17 I . .

87; this value substituted in III. we finally obtain

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Such fractions as the above are called continued fractions.

In the last example, the parts 1, 1, 2, &c. are called the first, second, third, &c. partial fractions.

If we seek for the greatest common measure of the numerator and denominator of the first fraction 35}, by the rule un. der Art. 9, we shall obtain,

OPERATION.
351)965(2

702
263)351(1

263
83)263(2

176
87)88(1
87

1)87(87
- i

87 .. .

0 Here we discover that the successive quotients are the same as the successive denominators of the partial fractions, which compose the continued fraction already drawn from

Hence, to convert a vulgar fraction into a continued frac. tion, we have this

RULE. Seek, by rule under Art. 9, the greatest common measure of the numerator and denominator of the given fraction; the re. ciprocals of the successive quotients will form the partial fractions which constitute the continued fraction required.

Examples.
1. Convert 451 into a continued fraction.

OPERATION.
251)764)3

753
11)251(22

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The partial fractions are g, as, 1, 2, , therefore we shall have 251_1 764 3+ 1

22+1 - 1+1

4+1

2. What continued fraction is equivalent to 3754?

Ans.

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3. What continued fraction is equivalent to 111?

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56. Let us now endeavor to reverse the foregoing process, that is, let us seek the vulgar fraction which is equivalent to a continued fraction.

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