Fractals/Continued fraction

Notation

Basic formula

A continued fraction is an expression of the form

a_{0}+{\cfrac {b_{1}}{a_{1}+{\cfrac {b_{2}}{a_{2}+{\cfrac {b_{3}}{a_{3}+{_{\ddots }}}}}}}}

where :

$a_{n}$ and $b_n$ are either integers, rational numbers, real numbers, or complex numbers.
$a_0$ , $a_{1}$ etc., are called the coefficients or terms of the continued fraction

Variants or types :

If $b_{n}=1$ for all $n$ the expression is called a simple continued fraction.
If the expression contains a finite number of terms, it is called a finite continued fraction.
If the expression contains an infinite number of terms, it is called an infinite continued fraction.^[1]

Thus, all of the following illustrate valid finite simple continued fractions:

Examples of finite simple continued fractions
Formula	Numeric	Remarks
$\ a_{0}$	$\ 2$	All integers are a degenerate case
$\ a_{0}+{\cfrac {1}{a_{1}}}$	$\ 2+{\cfrac {1}{3}}$	Simplest possible fractional form
$\ a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}}}}}$	$\ -3+{\cfrac {1}{2+{\cfrac {1}{18}}}}$	First integer may be negative
$\ a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}}}}}}}$	$\ {\cfrac {1}{15+{\cfrac {1}{1+{\cfrac {1}{102}}}}}}$	First integer may be zero

Finite simple continued fractions

Notation :

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

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

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

Infinite continued fractions

Notation :

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

Every infinite continued fraction is irrational number $\alpha$ :

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

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

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

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

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

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

Key words :

the sequence of continued fraction convergents ${\frac {p_{n}}{q_{n}}}$ of irrational number $\alpha$
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

References

↑ Darren C. Collins, Continued Fractions, MIT Undergraduate Journal of Mathematics,

This article is issued from Wikibooks. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.