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Are continued fractions just an abstract ruse, or a more natural way to describe a ratio?

Is there any paractical use? when sucha fraction is really necessary?

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    Its the Deep Mathematical State. It is a conspiracy theorem. – copper.hat Apr 08 '17 at 05:27
  • Mathematicians are able to find something interesting even if it is not useful. In fact, for some, finding a use for what they study might make it less interesting.

    An importance I see to continued fractions is to demonstrate that there are quite different ways to write real numbers than just the familiar decimal form or slight variations such as binary. It helps counters the common belief that real numbers are actually defined by their decimal representation.

     – badjohn Apr 08 '17 at 07:15
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    Yes, there are so many uses that it's not possible to list them all. Among the simplest, fo example, how do you imagine that excellent rational approximations of $\pi$ such as $22/7$ and $355/113$ have been found ? Through the continued fraction development of $\pi$, etc. etc. etc. – Jean Marie Apr 08 '17 at 07:16
  • All math is just an abstract ruse. – Ivan Neretin Apr 08 '17 at 07:49
  • https://en.wikipedia.org/wiki/Pad%C3%A9_approximant – Chappers Apr 08 '17 at 17:57

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Continued fractions provide another representation of real numbers, offering insights that are not revealed by the decimal representation. For example, the golden ratio has the continued fraction [1; 1, 1, ...], and e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, ...]. Continued fractions can be used to—

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