Continued fractions

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The unqualified term continued fraction implies simple continued fraction (also called regular continued fraction).

Simple continued fractions

Finite simple continued fractions

A finite simple continued fraction is an expression of the form

$c=a_{0}+{\underset {k=1}{\overset {n}{\rm {K}}}}~{\frac {1}{a_{k}}}:=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots {\cfrac {\ddots }{a_{n-1}+{\cfrac {1}{a_{n}}}}}}}}}}},\,$

where

a0

is the integer part of the continued fraction, the partial quotients

ak , 1   ≤   k   ≤   n

, are positive integers, and

n

is a positive integer. (See Gauss’ Kettenbruch notation for the continued fraction operator

K

.)

Finite simple continued fractions obviously represent rational numbers, and every rational number can be represented in precisely one way as a finite simple continued fraction.

Infinite simple continued fractions

A infinite simple continued fraction is an expression of the form

$c=a_{0}+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {1}{a_{k}}}:=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{a_{4}+{\cfrac {1}{\ddots }}}}}}}}}},\,$

where

a0

is the integer part of the continued fraction and the partial denominators

ak , k   ≥   1

, are positive integers, all the partial numerators being 1. (See Gauss’ Kettenbruch notation for the continued fraction operator

K

.)

A compact representation is

$c=a_{0}+1/(a_{1}+1/(a_{2}+1/(a_{3}+1/(a_{4}+1/(a_{5}+1/(a_{6}+\ldots ))))))\,$

A compact notation is

$c=[a_{0};a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},\ldots ]\,$

A sequence representation is

$\{a_{0},a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},\ldots \}\,$

Every infinite simple continued fraction represent an irrational number, and every irrational number can be represented in precisely one way as an infinite simple continued fraction.

Eventually periodic infinite simple continued fractions

Every eventually periodic infinite simple continued fraction represent an irrational quadratic number (root of an irreducible quadratic polynomial with integer coefficients), and every irrational quadratic number can be represented in precisely one way as an eventually periodic infinite simple continued fraction, i.e.

$c=[a_{0};a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},\ldots ]\,$

and, for some integer

m

and some integer

k > 0

, we have

a n = a n  + k

for all

n   ≥   m

.

All non-quadratic irrational numbers have non-periodic infinite simple continued fractions.

Simple continued fractions convergents

An infinite simple continued fraction representation for an irrational number is mainly useful because its initial segments provide excellent rational approximations to the number. These rational numbers are called the convergents of the continued fraction. Even-numbered convergents are smaller than the original number, while odd-numbered ones are bigger.

The first few convergents (numbered from 0) are

$c_{0}={\frac {p_{0}}{q_{0}}}={\frac {a_{0}}{1}},\ c_{1}={\frac {p_{1}}{q_{1}}}={\frac {a_{1}a_{0}+1}{a_{1}}},\ c_{2}={\frac {p_{2}}{q_{2}}}={\frac {a_{2}(a_{1}a_{0}+1)+a_{0}}{a_{2}a_{1}+1}},\ c_{3}={\frac {p_{3}}{q_{3}}}={\frac {a_{3}(a_{2}(a_{1}a_{0}+1)+a_{0})+(a_{1}a_{0}+1)}{a_{3}(a_{2}a_{1}+1)+a_{1}}},\ \ldots \,$

or equivalently

$c_{0}={\frac {p_{0}}{q_{0}}}={\frac {a_{0}p_{-1}+p_{-2}}{a_{0}q_{-1}+q_{-2}}}={\frac {a_{0}}{1}},\ c_{1}={\frac {p_{1}}{q_{1}}}={\frac {a_{1}p_{0}+p_{-1}}{a_{1}q_{0}+q_{-1}}},\ c_{2}={\frac {p_{2}}{q_{2}}}={\frac {a_{2}p_{1}+p_{0}}{a_{2}q_{1}+q_{0}}},\ c_{3}={\frac {p_{3}}{q_{3}}}={\frac {a_{3}p_{2}+p_{1}}{a_{3}q_{2}+q_{1}}},\ \ldots \,$

with

$p_{-2}=0,\,q_{-2}=1,\,$ i.e. $c_{-2}={\frac {p_{-2}}{q_{-2}}}={\frac {0}{1}}=0\,$

$p_{-1}=1,\,q_{-1}=0,\,$ i.e. $c_{-1}={\frac {p_{-1}}{q_{-1}}}={\frac {1}{0}}=\infty \,$

giving

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

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

with

$p_{-2}=0,\,q_{-2}=1,\,$ i.e. $c_{-2}={\frac {p_{-2}}{q_{-2}}}={\frac {0}{1}}=0\,$

$p_{-1}=1,\,q_{-1}=0,\,$ i.e. $c_{-1}={\frac {p_{-1}}{q_{-1}}}={\frac {1}{0}}=\infty \,$

where

$c_{n}={\frac {p_{n}}{q_{n}}},\,n\geq -2.\,$

$\lim _{n\to \infty }c_{n}=c\quad {\rm {iff}}\quad \lim _{n\to \infty }\sum _{k=0}^{n}a_{k}\to \infty \,$

These recurrence relations (a special case of generalized continued fractions convergents) are due to John Wallis.

Table of nice simple continued fractions

Continued fraction	Closed form	Decimal expansion	A-number
$1+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {1}{1}}=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}\,$ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}	Phi Golden ratio $\varphi ={\frac {1+{\sqrt {5}}}{2}}\,$	1.618033988749894848204586834...	CF: A000012 Conv. nums:   pn = A000045   (n + 2), n   ≥   0 Conv. dens:   qn = A000045   (n + 1), n   ≥   0 Base 10: A001622
$0+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {1}{1+({\frac {2k}{3}}-1)\cdot 0^{(k{\bmod {3}})}}}={\textbf {0}}+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{{\textbf {2}}+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{{\textbf {4}}+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}\,$ {0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, ...}	1/(e-1) ${\frac {1}{e-1}}=\sum _{n=1}^{\infty }e^{-n}$	0.5819767068693264243850020051...	CF:   a (n) = A005131   (n + 1), n   ≥   0 Conv. nums: A?????? Conv. dens: A?????? Base 10: A073333
$1+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {1}{1+{\frac {4k}{3}}\cdot 0^{(k{\bmod {3}})}}}={\textbf {1}}+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{{\textbf {5}}+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{{\textbf {9}}+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}\,$ {1, 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17, 1, 1, 21, 1, 1, 25, 1, 1, 29, ...}	1/(sqrt(e)-1) ${\frac {1}{{\sqrt {e}}-1}}\,$	1.54149408253679828413110344447...	CF:   a (n) = A058281   (n + 1), n   ≥   0 Conv. nums: A?????? Conv. dens: A?????? Base 10: A113011
$0+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {1}{2(2k-1)}}=0+{\cfrac {1}{2+{\cfrac {1}{6+{\cfrac {1}{10+{\cfrac {1}{14+{\cfrac {1}{18+{\cfrac {1}{22+{\cfrac {1}{26+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}\,$ {0, 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, ...}	tanh(1/2) = (e-1)/(e+1) $\tanh({\tfrac {1}{2}})={\frac {e-1}{e+1}}\,$	0.46211715726000975850231848364...	CF:   a (n) = A016825   (n  −  1), n   ≥   1 Conv. nums: A?????? Conv. dens: A?????? Base 10: A160327
$0+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {1}{k}}=0+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{3+{\cfrac {1}{4+{\cfrac {1}{5+{\cfrac {1}{6+{\cfrac {1}{7+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}\,$ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...}	BesselI[1, 2]/BesselI[0, 2] ${\frac {I_{1}(2)}{I_{0}(2)}}\,$	0.697774657964007982006790592...	CF: A001477 Conv. nums: A?????? Conv. dens: A?????? Base 10: A052119
$1+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {1}{k+1}}=1+{\cfrac {1}{2+{\cfrac {1}{3+{\cfrac {1}{4+{\cfrac {1}{5+{\cfrac {1}{6+{\cfrac {1}{7+{\cfrac {1}{8+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}\,$ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...}	BesselI[0, 2]/BesselI[1, 2] ${\frac {I_{0}(2)}{I_{1}(2)}}\,$	1.433127426722311758317183455...	CF: A000027 Conv. nums: A?????? Conv. dens: A?????? Base 10: A060997
$p_{1}+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {1}{p_{k+1}}}=2+{\cfrac {1}{3+{\cfrac {1}{5+{\cfrac {1}{7+{\cfrac {1}{11+{\cfrac {1}{13+{\cfrac {1}{17+{\cfrac {1}{19+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}\,$ {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, ...}	? $?\,$	2.3130367364335829063839516...	CF: A000040 Conv. nums: A?????? Conv. dens: A?????? Base 10: A064442

Generalized continued fractions

Generalized continued fractions are also called general continued fractions.

Finite generalized continued fractions

A finite generalized continued fraction is an expression of the form

$c=a_{0}+{\underset {k=1}{\overset {n}{\rm {K}}}}~{\frac {b_{k}}{a_{k}}}:=a_{0}+{\cfrac {b_{1}}{a_{1}+{\cfrac {b_{2}}{a_{2}+{\cfrac {b_{3}}{\ddots {\cfrac {\ddots }{a_{n-1}+{\cfrac {b_{n}}{a_{n}}}}}}}}}}},\,$

where

a 0

is the integer part of the continued fraction, the

bk , 1   ≤   k   ≤   n,

are the partial numerators, the

ak , 1   ≤   k   ≤   n,

are the partial denominators, and

n

is a positive integer. (See Gauss’ Kettenbruch notation for the continued fraction operator

K

.)

Finite generalized continued fractions obviously represent rational numbers, although rational numbers can be represented in many (finitely many?) ways as a finite generalized continued fraction.

Infinite generalized continued fractions

A infinite generalized continued fraction is an expression of the form

$c=a_{0}+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {b_{k}}{a_{k}}}:=a_{0}+{\cfrac {b_{1}}{a_{1}+{\cfrac {b_{2}}{a_{2}+{\cfrac {b_{3}}{a_{3}+{\cfrac {b_{4}}{a_{4}+{\cfrac {b_{5}}{\ddots }}}}}}}}}},\,$

where

a0

is the integer part of the continued fraction, the

bk , k   ≥   1,

are the partial numerators, the

ak , k   ≥   1,

are the partial denominators. (See Gauss’ Kettenbruch notation for the continued fraction operator

K

.)

A compact representation is

$c=a_{0}+b_{1}/(a_{1}+b_{2}/(a_{2}+b_{3}/(a_{3}+b_{4}/(a_{4}+b_{5}/(a_{5}+b_{6}/(a_{6}+\ldots ))))))\,$

A compact notation could be

$c=[a_{0};b_{1}/a_{1},b_{2}/a_{2},b_{3}/a_{3},b_{4}/a_{4},b_{5}/a_{5},b_{6}/a_{6},\ldots ]\,$

A sequence representation could be

$\{a_{0},b_{1},a_{1},b_{2},a_{2},b_{3},a_{3},b_{4},a_{4},b_{5},a_{5},b_{6},a_{6},\ldots \}\,$

Every infinite generalized continued fraction represent an irrational number, although irrational numbers can be represented in many (infinitely many?) ways as an infinite generalized continued fraction.

??? Eventually periodic infinite generalized continued fractions ???

The numbers having at least one eventually periodic infinite generalized continued fraction representation are... ?????

where

$c=[a_{0};b_{1}/a_{1},b_{2}/a_{2},b_{3}/a_{3},b_{4}/a_{4},b_{5}/a_{5},b_{6}/a_{6},\ldots ]\,$

and, for some integer

m

and some integer

k > 0

, we have

bn an = bn +k an +k

for all

n   ≥   m

.

The numbers with only non-periodic infinite generalized continued fraction representations are ?????. (Are there such numbers...?)

Generalized continued fractions convergents

The first few convergents (numbered from 0) are

$c_{0}={\frac {p_{0}}{q_{0}}}={\frac {a_{0}}{1}},\ c_{1}={\frac {p_{1}}{q_{1}}}={\frac {a_{1}a_{0}+b_{1}}{a_{1}}},\ c_{2}={\frac {p_{2}}{q_{2}}}={\frac {a_{2}(a_{1}a_{0}+b_{1})+b_{2}a_{0}}{a_{2}a_{1}+b_{2}}},\ c_{3}={\frac {p_{3}}{q_{3}}}={\frac {a_{3}(a_{2}(a_{1}a_{0}+b_{1})+b_{2}a_{0})+b_{3}(a_{1}a_{0}+b_{1})}{a_{3}(a_{2}a_{1}+b_{2})+b_{3}a_{1}}},\ \ldots \,$

or equivalently

$c_{0}={\frac {p_{0}}{q_{0}}}={\frac {a_{0}p_{-1}+b_{0}p_{-2}}{a_{0}q_{-1}+b_{0}q_{-2}}}={\frac {a_{0}}{1}},\,c_{1}={\frac {p_{1}}{q_{1}}}={\frac {a_{1}p_{0}+b_{1}p_{-1}}{a_{1}q_{0}+b_{1}q_{-1}}},\,c_{2}={\frac {p_{2}}{q_{2}}}={\frac {a_{2}p_{1}+b_{2}p_{0}}{a_{2}q_{1}+b_{2}q_{0}}},\,c_{3}={\frac {p_{3}}{q_{3}}}={\frac {a_{3}p_{2}+b_{3}p_{1}}{a_{3}q_{2}+b_{3}q_{1}}},\,\ldots \,$

with

$b_{0}\equiv 1\,$

$p_{-2}\equiv ,\,q_{-2}\equiv 1,\,$ i.e. $c_{-2}\equiv {\frac {p_{-2}}{q_{-2}}}={\frac {0}{1}}=0\,$

$p_{-1}\equiv ,\,q_{-1}\equiv 0,\,$ i.e. $c_{-1}\equiv {\frac {p_{-1}}{q_{-1}}}={\frac {1}{0}}=\infty \,$

giving

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

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

with

$b_{0}\equiv 1\,$

$p_{-2}\equiv ,\,q_{-2}\equiv 1,\,$ i.e. $c_{-2}\equiv {\frac {p_{-2}}{q_{-2}}}={\frac {0}{1}}=0\,$

$p_{-1}\equiv ,\,q_{-1}\equiv 0,\,$ i.e. $c_{-1}\equiv {\frac {p_{-1}}{q_{-1}}}={\frac {1}{0}}=\infty \,$

where

$c_{n}={\frac {p_{n}}{q_{n}}},\,n\geq -2.\,$

$c=\lim _{n\to \infty }c_{n}\,$

These recurrence relations are due to John Wallis.

Table of nice generalized continued fractions

Continued fraction	Closed form	Decimal expansion	A-number
$1+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {k}{1}}=1+{\cfrac {1}{1+{\cfrac {2}{1+{\cfrac {3}{1+{\cfrac {4}{\ddots }}}}}}}}\,$ {1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, ...}	sqrt(2/(pi*e))/erfc(1/sqrt(2)) ${\sqrt {\frac {2}{\pi e}}}{\frac {1}{{\rm {~erfc}}({\tfrac {1}{\sqrt {2}}})}}\,$	1.525135276160981209089090536...	CF:   a (n) = A057979   (n + 2), n   ≥   0 Conv. nums: A?????? Conv. dens: A?????? Base 10: A111129
$0+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {k}{k}}=0+{\cfrac {1}{1+{\cfrac {2}{2+{\cfrac {3}{3+{\cfrac {4}{\ddots }}}}}}}}\,$ {0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, ...}	1/(e-1) ${\frac {1}{e-1}}=\sum _{n=1}^{\infty }e^{-n}\,$	0.5819767068693264243850020051...	CF: A110654 Conv. nums: A?????? Conv. dens: A?????? Base 10: A073333
$1+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {2k}{2k+1}}=1+{\cfrac {2}{3+{\cfrac {4}{5+{\cfrac {6}{7+{\cfrac {8}{\ddots }}}}}}}}\,$ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...}	1/(sqrt(e)-1) ${\frac {1}{{\sqrt {e}}-1}}\,$	1.54149408253679828413110344447...	CF: A000027 Conv. nums: A?????? Conv. dens: A?????? Base 10: A113011
$1+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {k^{2}}{2k+1}}=1+{\cfrac {1}{3+{\cfrac {4}{5+{\cfrac {9}{7+{\cfrac {16}{\ddots }}}}}}}}\,$ {1, 1, 3, 4, 5, 9, 7, 16, 9, 25, 11, 36, 13, 49, 15, 64, 17, 81, ...}	4/pi ${\frac {4}{\pi }}\,$	1.27323954473516268615107010698...	CF: A079097 Conv. nums: A?????? Conv. dens: A?????? Base 10: A088538
$p_{1}+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {p_{2k}}{p_{2k+1}}}=2+{\cfrac {3}{5+{\cfrac {7}{11+{\cfrac {13}{17+{\cfrac {19}{\ddots }}}}}}}}\,$ {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, ...}	Herkommer number $?\,$	2.5360270816893383923069490821...	CF: A000040 Conv. nums: A?????? Conv. dens: A?????? Base 10: A085825

Gauss’ Kettenbruch notation

Karl Friedrich Gauss evoked the more familiar product operator

Π

when he devised his notation for the continued fraction (Kettenbruch in german) operator

$x=a_{0}+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {b_{k}}{a_{k}}}:=a_{0}+{\cfrac {b_{1}}{a_{1}+{\cfrac {b_{2}}{a_{2}+{\cfrac {b_{3}}{a_{3}+{\cfrac {b_{4}}{a_{4}+{\cfrac {b_{5}}{\ddots }}}}}}}}}}.\,$

Here the

K

stands for Kettenbruch, the German word for “continued fraction.” This is probably the most compact and convenient way to express continued fractions; however, it is not widely used by English typesetters.

See also

• Continued fractions (nested fractions)
  • Category:Continued fractions for constants
    • Category:Continued fractions for rational numbers
    • Category:Continued fractions for irrational numbers
      • Category:Continued fractions for quadratic numbers
  • Category:Continued fractions for functions
• Continued radicals (nested radicals)
• Table of convergents constants

External links

• Marek Wolf, Continued fractions constructed from prime numbers, 2010.
• A Continued Fraction Calculator version 4Oct10, © 2003-2010 Dr Ron Knott, updated: 4 October 2010.