Finite

Notation :

 ${\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}}}}}}}=[a_{0};a_{1},a_{2},a_{3}]}$

Every finite continued fraction represents a rational number ${\displaystyle {\frac {p}{q}}}$:

  ${\displaystyle {\frac {p}{q}}=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}}}}}}}=[a_{0};a_{1},a_{2},a_{3}]}$

Infinite

Notation :

 ${\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\ddots }}}}}}}=[a_{0};a_{1},a_{2},a_{3},\ldots ]}$

Every infinite continued fraction is irrational number ${\displaystyle \alpha }$ :

 ${\displaystyle \alpha =a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\ddots }}}}}}}=[a_{0};a_{1},a_{2},a_{3},\ldots ]}$

The rational number ${\displaystyle {\frac {p_{n}}{q_{n}}}}$ obtained by limited number of terms in a continued fraction is called a n-th convergent

  ${\displaystyle {\frac {p_{n}}{q_{n}}}=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{{\ddots }+{\cfrac {1}{a_{n}}}}}}}}}}}=[a_{0};a_{1},a_{2},a_{3},\ldots ,a_{n}]}$

because sequence of rational numbers ${\displaystyle {\frac {p_{n}}{q_{n}}}}$ converges to irrational number ${\displaystyle \alpha }$

 ${\displaystyle \lim _{n\rightarrow \infty }{\frac {p_{n}}{q_{n}}}=\alpha }$

In other words irrational number ${\displaystyle \alpha }$ is the limit of convergent sequence.

Nominator p and denominator q can be found using the relevant recursive relation:

${\displaystyle p_{n}=a_{n}p_{n-1}+p_{n-2}}$

${\displaystyle q_{n}=a_{n}q_{n-1}+q_{n-2}}$

so

${\displaystyle {\frac {p_{n}}{q_{n}}}={\frac {a_{n}p_{n-1}+p_{n-2}}{a_{n}q_{n-1}+q_{n-2}}}}$

Key words :

• the sequence of continued fraction convergents ${\displaystyle {\frac {p_{n}}{q_{n}}}}$ of irrational number ${\displaystyle \alpha }$
• sequence of the convergents
• continued fraction expansion
• rational aproximation of irrational number
• a best rational approximation to a real number r by rational number p/q