8.3: Continued fractions
- Page ID
- 7621
This page is a draft and is under active development.
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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
\[ \frac{2520}{154}= 16 + \frac{1}{2+ \frac{1}{1+ \frac{1}{3}}}.\]