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On Wolfram Alpha, I see continued fractions being listed in the results. Although I understand continued fractions, and how they can be used for approximations, what is a better approximation than a decimal representation of the number?

For example,

31.999999999664

tells me so much more than

enter image description here

So, I do not get why continued fractions are important. Are they supposed to tell me something which I am missing all these days?

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    Have you read [the Wikipedia article](https://en.wikipedia.org/wiki/Continued_fraction#Motivation_and_notation)? It lists "several desirable properties".2012-05-27
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    Decimals appear to tell you so much more because you don't know what to look for in continued fractions. You could just as well say English tells you so much more than French if you barely know French. Anyway, the theory of continued fractions becomes much more interesting for irrational numbers, and they have important applications in number theory (efficiently solving Pell's equation and $ax+by=1$ if $a$ and $b$ are relatively prime). On an historical note, the first proof that $\pi$ is irrational used continued fraction expansions of functions, not numbers.2012-05-27
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    Yes, read that. But how are they better? That what I dont see!2012-05-27
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    @Lazer KCd just gave you a list of applications in which they are easier to work with than decimal expansions. Are you asking what makes them easier to work with in these applications, or why the applications are significant?2012-05-27
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    @Alex: See, I am not into Maths much, and am not solving Pell's equation here :) I wanted to confirm if these numbers should mean something to a layman. Guess, they are not supposed to.2012-05-27
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    @Lazer It is difficult to respond to a question about what continued fractions tell you if you are only interested in the things a decimal expansion tell you.2012-05-27
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    The answer to this question is, "Where do I begin?"2012-05-27
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    Lazer, your judgment is sound. They are not interesting or useful.2012-05-27
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    Continued fractions are useful to get very good rational approximations of irrational real numbers. Lambert proved that $\pi$ is irrational using the continued fraction of the tangens-function. It is rather opinion-based whether continued fractions have any use for a layman. For practical everyday problems probably not, but this is true for most interesting mathematical topics. But I would definitely say that continued fractions are interesting (even for laymen).2017-04-05
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    Large entries indicate particular good approximations. The reason why $\pi\approx\frac{355}{113}$ is good to $6$ decimal digits is that the continued fraction starts with $[3,7,15,1,292,1,\cdots]$. A matter of taste how useful or interesting such stuff is ...2017-04-05

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