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I know that a continued fraction can be converted to a decimal fraction by condensing it to a simple fraction and then performing long division to produce a decimal fraction.

I am wondering how the decimal fraction can be discovered "natively," meaning without condensing it first.

It seems to me the whole benefit of the continued fraction is that the value has been reduced to a series of terms which provide utility and that potential utility is thrown away if the continued fraction is condensed and terms disposed of. I would think that there probably is some way to convert the terms directly into decimal place values.

Tyler Durden
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  • I doubt that there is such a "direct" way. The main merit of the continued fractions is to get very good or with some luck even extraordinary good approximations by rational numbers , in fact even best approximations if the denominator is limited. Sometimes they help to prove a number to be irrational. – Peter Sep 26 '20 at 13:23
  • Continued fractions are like X-rays for numbers. They can reveal many secrets hidden inside the number, but they are pretty crappy for telling you the normal things about the number. if you want to calculate - add, substract, multiply, divide - you stick with decimal expansion. But if you want to know what the bones of the number looks like - how it is constructed - continued fractions will tell you that. Figuring out what a person looks like from their x-rays is not easy, and the same is true for numbers. – Paul Sinclair Sep 26 '20 at 20:21

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