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8.3: Continued fractions

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    Definition: Continued fractions

    A simple continued fraction is of the form, denoted by \([a_0,a_1,\ldots]\), \[a_0 + \frac{1}{a_1+\frac{\displaystyle 1}{\displaystyle a_2+ \ldots}},\] where \(a_0\), \(\ a_1\), \(\ a_2\), \(\ldots\) \(\in \mathbb{Z}\). Continued fraction has been studied extensively, but we will only explore some of them in this class.

    Example \(\PageIndex{1}\):

    A simple finite continued fraction \[ \frac{1}{2}=[1,1]=0+\frac{1}{1+\frac{1}{1}}\]

    A simple infinite continued fraction: Golden Ratio \[ \phi =\frac{1+\sqrt{5}}{2}=[1,1,\ldots]=1+\frac{1}{1+\frac{1}{1+\ldots}},\]which can be found using \(x=1+\dfrac{1}{1+x}\).

     

    Using the Euclidean algorithm to find a simple finite continued fraction

    Let's explore the following example:

    Consider \(\dfrac{2520}{154}\).

    By Euclidean algorithm we have,

    \(2520=({16})(154)+56\)

    \(154=(2)(56)+42\)

    \(56=(1)(42)+14\)

    \(42=(3)(14)+0\).

    The quotients give us the simple finite continued fraction \([16, 2, 1, 3]\). That is

    \[ \frac{2520}{154}= 16 + \frac{1}{2+ \frac{1}{1+ \frac{1}{3}}}.\]


    8.3: Continued fractions is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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