Continued Fraction Expansion of Irrational Square Root/Examples

Examples of Continued Fraction Expansion of Irrational Square Root

\(\ds \sqrt 2\)

\(=\)

\(\ds \sqbrk {1, \sequence 2}\)

\(\ds \sqrt 3\)

\(=\)

\(\ds \sqbrk {1, \sequence {1, 2} }\)

\(\ds \sqrt 5\)

\(=\)

\(\ds \sqbrk {2, \sequence 4}\)

\(\ds \sqrt 6\)

\(=\)

\(\ds \sqbrk {2, \sequence {2, 4} }\)

\(\ds \sqrt 7\)

\(=\)

\(\ds \sqbrk {2, \sequence {1, 1, 1, 4} }\)

\(\ds \sqrt {13}\)

\(=\)

\(\ds \sqbrk {3, \sequence {1, 1, 1, 1, 6} }\)

\(\ds \sqrt {19}\)

\(=\)

\(\ds \sqbrk {4, \sequence {2, 1, 3, 1, 2, 8} }\)

\(\ds \sqrt {28}\)

\(=\)

\(\ds \sqbrk {5, \sequence {3, 2, 3, 10} }\)

\(\ds \sqrt {31}\)

\(=\)

\(\ds \sqbrk {5, \sequence {1, 1, 3, 5, 3, 1, 1, 10}\ }\)

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 $\sqrt 5$

The continued fraction expansion of the square root of $5$ is given by:

$\sqrt 5 = \sqbrk {2, \sequence 4}$

Continued Fraction Expansion of $\sqrt 8$

The continued fraction expansion of the square root of $8$ is given by:

$\sqrt 8 = \sqbrk {2, \sequence {1, 4} }$

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:

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