Continued Fraction Expansion of Irrational Square Root/Example

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Examples of Continued Fraction Expansion of Irrational Square Root

\(\displaystyle \sqrt 2\) \(=\) \(\displaystyle \left[{1, \left \langle{2}\right \rangle}\right]\)
\(\displaystyle \sqrt 3\) \(=\) \(\displaystyle \left[{1, \left \langle{1, 2}\right \rangle}\right]\)
\(\displaystyle \sqrt 5\) \(=\) \(\displaystyle \left[{2, \left \langle{4}\right \rangle}\right]\)
\(\displaystyle \sqrt 6\) \(=\) \(\displaystyle \left[{2, \left \langle{2, 4}\right \rangle}\right]\)
\(\displaystyle \sqrt 7\) \(=\) \(\displaystyle \left[{2, \left \langle{1, 1, 1, 4}\right \rangle}\right]\)
\(\displaystyle \sqrt {13}\) \(=\) \(\displaystyle \left[{3, \left \langle{1, 1, 1, 1, 6}\right \rangle}\right]\)
\(\displaystyle \sqrt {19}\) \(=\) \(\displaystyle \left[{4, \left \langle{2, 1, 3, 1, 2, 8}\right \rangle}\right]\)
\(\displaystyle \sqrt {28}\) \(=\) \(\displaystyle \left[{5, \left \langle{3, 2, 3, 10}\right \rangle}\right]\)
\(\displaystyle \sqrt {31}\) \(=\) \(\displaystyle \left[{5, \left \langle{1, 1, 3, 5, 3, 1, 1, 10}\right \rangle}\right]\)


Continued Fraction Expansion of $\sqrt 2$

The continued fraction expansion of the square root of $2$ is given by:

$\sqrt 2 = \sqbrk {1, \sequence 2}$


Continued Fraction Expansion of $\sqrt {13}$

The continued fraction expansion of the square root of $13$ is given by:

$\sqrt {13} = \sqbrk {3, \sequence {1, 1, 1, 1, 6} }$


Continued Fraction Expansion of $\sqrt {29}$

The continued fraction expansion of the square root of $29$ is given by:

$\sqrt {29} = \sqbrk {5, \sequence {2, 1, 1, 2, 10} }$


Continued Fraction Expansion of $\sqrt {61}$

The continued fraction expansion of the square root of $61$ is given by:

$\sqrt {61} = \sqbrk {7, \sequence {1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14} }$