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I know that any rational number can be expressed as a continued fraction, but what about irrational numbers? For example, what is the continued fractional representation of Pi, or e for that matter? Can all real numbers be expressed as continued fractions? For example, can the following number be expressed as a continued fraction?



marked as duplicate by MJD, Amzoti, Start wearing purple, Lord_Farin, Davide Giraudo Jun 27 '13 at 17:35

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    $\begingroup$ From the second paragraph of the Wikipedia article on continued fractions: "every irrational number $\alpha$ is the value of a unique infinite continued fraction..." (which, combined with the fact that every rational number can be expressed as a continued fraction, means that all real numbers can). $\endgroup$ – Zev Chonoles Jun 24 '13 at 4:04
  • $\begingroup$ In fact, $\pi$ has a very nice continued fraction decomposition. $\endgroup$ – Cameron Williams Jun 24 '13 at 4:06
  • $\begingroup$ Every real number has a representation as a simple continued fraction. For irrationals the expression is unique. $\endgroup$ – André Nicolas Jun 24 '13 at 4:15

All real numbers can be expressed (not necessarily uniquely) as continued fractions. Check out the Wikipedia article on the subject, in particular from this point on down for the special examples you requested.


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