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A Question about continued fraction representations

  1. Mar 24, 2016 #1
    How powerful are continued fraction representations?

    From what I understand, they could be used to exactly represent some irrational numbers

    So, could they represent any root of an nth degree polynomial equation?

    Specially where n>4, since 5th degree roots are not guaranteed to have an algebraic representation.
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    And if they do, will there be a possibility to have a quintic formula which isn't an algebraic solution, but rather an continued fraction solution
     
  2. jcsd
  3. Mar 25, 2016 #2
    An irrational [itex]\xi[/itex] admits a finite or periodic continued fraction representation if and only if there exist integers [itex]a, b, c, d[/itex] with [itex]c \geq 1[/itex] and [itex]d[/itex] nonzero such that

    [itex]\displaystyle \xi = \frac{a + b \sqrt{c}}{d}[/itex]

    So in short the answer to your question is no; the only classes of numbers which admit finite or periodic continued fraction representations are integers, rationals, and roots to quadratic polynomials (also known as "quadratic surds"). Any higher degree algebraic number does not admit a periodic continued fraction representation.

    Of course, it is possible to have infinite continued fractions with a finitely representable series of coefficients (e.g. some aperiodic pattern, for instance the CF for [itex]e[/itex]) however this does not involve finitely many operations and so cannot be considered a closed-form solution.
     
  4. Mar 26, 2016 #3

    FactChecker

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    They are very similar to power series.

    No finite length continued fraction with rational numbers can represent an irrational number. The term "continued fractions" includes infinitely long continued fractions. They can represent irrational numbers.
     
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