# 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.