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Continued fractions
The unqualified term continued fraction implies simple continued fraction (also called regular continued fraction).
Contents
Simple continued fractions
Finite simple continued fractions
A finite simple continued fraction is an expression of the form
a0 
ak , 1 ≤ k ≤ n 
n 
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
a0 
ak , k ≥ 1 
K 
A compact representation is
A compact notation is
A sequence representation is
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.
m 
k > 0 
a n = a n + k 
n ≥ m 
All nonquadratic irrational numbers have nonperiodic 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. Evennumbered convergents are smaller than the original number, while oddnumbered ones are bigger.
The first few convergents (numbered from 0) are
or equivalently
with
 i.e.
 i.e.
giving
with
 i.e.
 i.e.
where
These recurrence relations (a special case of generalized continued fractions convergents) are due to John Wallis.
Continued fraction  Closed form  Decimal expansion  Anumber  

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}

1.618033988749894848204586834... 
CF:
Base 10:  
{0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, ...}

0.5819767068693264243850020051... 
CF:
Conv. nums: Conv. dens: Base 10:  
{1, 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17, 1, 1, 21, 1, 1, 25, 1, 1, 29, ...}

1.54149408253679828413110344447... 
CF:
Conv. nums: Conv. dens: Base 10:  
{0, 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, ...}

0.46211715726000975850231848364... 
CF:
Conv. nums: Conv. dens: Base 10:  
{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] 
0.697774657964007982006790592... 
CF: Conv. nums: Conv. dens: Base 10:  
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...}

BesselI[0, 2]/BesselI[1, 2] 
1.433127426722311758317183455... 
CF: Conv. nums: Conv. dens: Base 10:  
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, ...}

2.3130367364335829063839516... 
CF: Conv. nums: Conv. dens: Base 10: 
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
a 0 
bk , 1 ≤ k ≤ n, 
ak , 1 ≤ k ≤ n, 
n 
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
a0 
bk , k ≥ 1, 
ak , k ≥ 1, 
K 
A compact representation is
A compact notation could be
A sequence representation could be
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
m 
k > 0 

n ≥ m 
The numbers with only nonperiodic infinite generalized continued fraction representations are ?????. (Are there such numbers...?)
Generalized continued fractions convergents
The first few convergents (numbered from 0) are
or equivalently
with
 i.e.
 i.e.
giving
with
 i.e.
 i.e.
where
These recurrence relations are due to John Wallis.
Continued fraction  Closed form  Decimal expansion  Anumber  

{1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, ...}

1.525135276160981209089090536... 
CF:
Conv. nums: Conv. dens: Base 10:  
{0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, ...}

0.5819767068693264243850020051... 
CF: Conv. nums: Conv. dens: Base 10:  
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...}

1.54149408253679828413110344447... 
CF: Conv. nums: Conv. dens: Base 10:  
{1, 1, 3, 4, 5, 9, 7, 16, 9, 25, 11, 36, 13, 49, 15, 64, 17, 81, ...}

1.27323954473516268615107010698... 
CF: Conv. nums: Conv. dens: Base 10:  
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, ...}

2.5360270816893383923069490821... 
CF: Conv. nums: Conv. dens: Base 10: 
Gauss’ Kettenbruch notation
Karl Friedrich Gauss evoked the more familiar product operatorΠ 
K 
See also
 Continued fractions (nested fractions)
 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, © 20032010 Dr Ron Knott, updated: 4 October 2010.