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6. What vulgar fraction is equivalent to 0.0123456789 ?
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 ?
CONTINUED FRACTIONS. 55. If we divide both numerator and denominator of the fraction it by the numerator, we obtain,
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,
III. 351 1
87; this value substituted in III. we finally obtain
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,
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.
The partial fractions are g, as, 1, 2, , therefore we shall have 251_1 764 3+ 1
22+1 - 1+1