Continued Fraction Expansion/Examples

Examples of Continued Fraction Expansions

Continued Fraction Expansion of Golden Mean

The golden mean has the simplest possible continued fraction expansion, namely $\sqbrk {1, 1, 1, 1, \ldots}$:

$\phi = 1 + \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots} } }$

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 $\pi$

The constant $\pi$ (pi) has the continued fraction expansion:

$\pi = \sqbrk {3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, \ldots}$

Continued Fraction Expansion of $e$

The constant Euler's number $e$ has the continued fraction expansion:

\(\ds e\)

\(=\)

\(\ds \sqbrk {2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \ldots }\)

\(\ds \)

\(=\)

\(\ds \sqbrk {1, 0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \ldots }\)