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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 decimal 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 ? 34 Ans. 388=188. 2. What vulgar fraction is equivalent to the compound repetend 0.087837 1

3. What vulgar fraction is equivalent to 0.083?

Ans. 1.

Ans.

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

Ans.

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 by the numerator, we obtain,

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Such fractions as the above are called continued fractions. In the last example, the parts, 1, 1, &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 351, by the rule under Art. 9, we shall obtain,

OPERATION.

351)965(2
702

263)351(1

263

88)263(2
176

87)88(1

87

1)87(87

87

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

96

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

RULE.

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

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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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