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{{short description|Representation of a number by a (generally infinite) sequence of additions and inversions}}
{{redirect-distinguish|Recurring fraction|Repeating decimal}}
{{thumb|width=220
|content=<math>a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{ \ddots + \cfrac{1}{a_n} }}}</math>
|caption=A finite continued fraction, where <math>n</math> is a non-negative integer, <math>a_0</math> is an integer, and <math>a_i</math> is a positive integer, for <math>i=1,\ldots,n</math>.
}}

In [[mathematics]], a '''continued fraction''' is an [[expression (mathematics)|expression]] obtained through an [[iterative]] process of representing a number as the sum of its [[integer part]] and the [[multiplicative inverse|reciprocal]] of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.<ref>{{cite web|url=http://www.britannica.com/EBchecked/topic/135043/continued-fraction|title=Continued fraction - mathematics|publisher=}}</ref> In a '''finite continued fraction''' (or '''terminated continued fraction'''), the iteration/[[recursion]] is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an '''infinite continued fraction''' is an [[infinite expression (mathematics)|infinite expression]]. In either case, all integers in the sequence, other than the first, must be [[positive number|positive]]. The integers <math>a_i</math> are called the [[coefficient]]s or [[term (mathematics)|terms]] of the continued fraction.<ref name="Pettofrezzo 1970 150">{{harvtxt|Pettofrezzo|Byrkit|1970|p=150}}</ref>

Continued fractions have a number of remarkable properties related to the [[Euclidean algorithm]] for integers or [[real number]]s. Every [[rational number]] {{sfrac|<math>p</math>|<math>q</math>}} has two closely related expressions as a finite continued fraction, whose coefficients {{mvar|a<sub>i</sub>}} can be determined by applying the Euclidean algorithm to <math>(p,q)</math>. The numerical value of an infinite continued fraction is [[irrational number|irrational]]; it is defined from its infinite sequence of integers as the [[limit (mathematics)|limit]] of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite [[Prefix (computer science)|prefix]] of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number <math>\alpha</math> is the value of a ''unique'' infinite continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the [[Commensurability (mathematics)|incommensurable]] values <math>\alpha</math> and 1. This way of expressing real numbers (rational and irrational) is called their ''continued fraction representation''.

It is generally assumed that the [[numerator]] of all of the fractions is 1. If arbitrary values and/or [[function (mathematics)|functions]] are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a '''[[generalized continued fraction]]'''. When it is necessary to distinguish the first form from generalized continued fractions, the former may be called a '''simple''' or '''regular continued fraction''', or said to be in '''canonical form'''.

The term ''continued fraction'' may also refer to representations of [[rational function]]s, arising in their [[analytic function|analytic theory]]. For this use of the term, see [[Padé approximation]] and [[Chebyshev rational functions]].

==Motivation and notation==
Consider, for example, the [[rational number]] {{sfrac|415|93}}, which is around 4.4624. As a first [[approximation]], start with 4, which is the [[Floor and ceiling functions|integer part]]; {{nowrap|1={{sfrac|415|93}} = 4 + {{sfrac|43|93}}}}. The fractional part is the [[Multiplicative inverse|reciprocal]] of {{sfrac|93|43}} which is about 2.1628. Use the integer part, 2, as an approximation for the reciprocal to obtain a second approximation of {{nowrap |1=4 + {{sfrac|1|2}} = 4.5; {{sfrac|93|43}} = 2 + {{sfrac|7|43}}}}.
The remaining fractional part, {{sfrac|7|43}}, is the reciprocal of {{sfrac|43|7}}, and {{sfrac|43|7}} is around 6.1429. Use 6 as an approximation for this to obtain {{nowrap|2 + {{sfrac|1|6}}}} as an approximation for {{sfrac|93|43}} and {{nowrap|4 + {{sfrac|1|2 + {{sfrac|1|6}}}}}}, about 4.4615, as the third approximation; {{nowrap|1={{sfrac|43|7}} = 6 + {{sfrac|1|7}}}}. Finally, the fractional part, {{sfrac|1|7}}, is the reciprocal of 7, so its approximation in this scheme, 7, is exact ({{nowrap|1={{sfrac|7|1}} = 7 + {{sfrac|0|1}}}}) and produces the exact expression {{nowrap|4 + {{sfrac|1|2 + {{sfrac|1|6 + {{sfrac|1|7}}}}}}}} for {{sfrac|415|93}}.

The expression {{nowrap|4 + {{sfrac|1|2 + {{sfrac|1|6 + {{sfrac|1|7}}}}}}}} is called the continued fraction representation of {{sfrac|415|93}}. This can be represented by the abbreviated notation {{sfrac|415|93}} = [4; 2, 6, 7]. (It is customary to replace only the ''first'' comma by a semicolon.) Some older textbooks use all commas in the {{math|(''n'' + 1)}}-tuple, for example, [4, 2, 6, 7].<ref name="Long 1972 173">{{harvtxt|Long|1972|p=173}}</ref><ref name="Pettofrezzo 1970 152">{{harvtxt|Pettofrezzo|Byrkit|1970|p=152}}</ref>

If the starting number is rational, then this process exactly parallels the [[Euclidean algorithm]]. In particular, it must terminate and produce a finite continued fraction representation of the number. If the starting number is [[Irrational number|irrational]], then the process continues indefinitely. This produces a sequence of approximations, all of which are rational numbers, and these converge to the starting number as a limit. This is the (infinite) continued fraction representation of the number. Examples of continued fraction representations of irrational numbers are:
* {{math|1={{sqrt|19}} = [4;2,1,3,1,2,8,2,1,3,1,2,8,...]}} {{OEIS|A010124}}. The pattern repeats indefinitely with a period of 6.
* {{math|1=[[e (mathematical constant)|''e'']] = [2;1,2,1,1,4,1,1,6,1,1,8,...]}} {{OEIS|A003417}}. The pattern repeats indefinitely with a period of 3 except that 2 is added to one of the terms in each cycle.
* {{math|1=[[Pi|π]] = [3;7,15,1,292,1,1,1,2,1,3,1,...]}} {{OEIS|A001203}}. No pattern has ever been found in this representation.
* {{math|1=[[golden ratio|ϕ]] = [1;1,1,1,1,1,1,1,1,1,1,1,...]}} {{OEIS|A000012}}. The [[golden ratio]], the irrational number that is the "most difficult" to approximate rationally. See: [[#A property of the golden ratio φ|A property of the golden ratio φ]].

Continued fractions are, in some ways, more "mathematically natural" representations of a [[real number]] than other representations such as [[decimal representation]]s, and they have several desirable properties:
* The continued fraction representation for a rational number is finite and only rational numbers have finite representations. In contrast, the decimal representation of a rational number may be finite, for example {{nowrap|1={{sfrac|137|1600}} = 0.085625}}, or infinite with a repeating cycle, for example {{nowrap|1={{sfrac|4|27}} = 0.148148148148...}}
* Every rational number has an essentially unique continued fraction representation. Each rational can be represented in exactly two ways, since {{math|1=[''a''<sub>0</sub>;''a''<sub>1</sub>,... ''a''<sub>''n''−1</sub>,''a''<sub>''n''</sub>] = [''a''<sub>0</sub>;''a''<sub>1</sub>,... ''a''<sub>''n''−1</sub>,(''a''<sub>''n''</sub>−1),1]}}. Usually the first, shorter one is chosen as the [[canonical form|canonical representation]].
* The continued fraction representation of an irrational number is unique.
* The real numbers whose continued fraction eventually repeats are precisely the [[quadratic irrational]]s.<ref>{{MathWorld|title=Periodic Continued Fraction|urlname=PeriodicContinuedFraction}}</ref> For example, the repeating continued fraction {{nowrap|[1;1,1,1,...]}} is the [[golden ratio]], and the repeating continued fraction {{nowrap|[1;2,2,2,...]}} is the [[square root of 2]]. In contrast, the decimal representations of quadratic irrationals are apparently [[normal number|random]]. The square roots of all (positive) integers, that are not perfect squares, are quadratic irrationals, hence are unique periodic continued fractions.
* The successive approximations generated in finding the continued fraction representation of a number, that is, by truncating the continued fraction representation, are in a certain sense (described below) the "best possible".

== Basic formula ==
A continued fraction is an expression of the form

:<math>a_0 + \cfrac{b_1}{a_1 + \cfrac{b_2}{a_2 + \cfrac{b_3}{a_3 + {_\ddots}}}}</math>

where a<sub>i</sub> and b<sub>i</sub> can be any complex numbers. Usually they are required to be integers.
If b<sub>i</sub> = 1 for all ''i'' 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.<ref>{{cite journal |first=Darren C. |last=Collins |title=Continued Fractions |journal=MIT Undergraduate Journal of Mathematics |url=http://www-math.mit.edu/phase2/UJM/vol1/COLLIN~1.PDF |archive-url=https://web.archive.org/web/20011120064343/http://www-math.mit.edu/phase2/UJM/vol1/COLLIN~1.PDF |url-status=dead |archive-date=2001-11-20 }}</ref>

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

{| class="wikitable"
|+Examples of finite simple continued fractions
!Formula
!Numeric
!Remarks
|-
|<math>\ a_0</math>
|<math>\ 2</math>
|All integers are a [[degenerate case]]
|-
|<math>\ a_0 + \cfrac{1}{a_1}</math>
|<math>\ 2 + \cfrac{1}{3}</math>
|Simplest possible fractional form
|-
|<math>\ a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2}}</math>
|<math>\ -3 + \cfrac{1}{2 + \cfrac{1}{18}}</math>
|First integer may be negative
|-
|<math>\ a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3}}}</math>
|<math>\ \cfrac{1}{15 + \cfrac{1}{1 + \cfrac{1}{102}}}</math>
|First integer may be zero
|}

==Calculating continued fraction representations==
Consider a real number {{mvar|r}}.
Let <math>i=\lfloor r \rfloor </math> be the [[integer part]] of {{mvar|r}} and let
<math>f = r - i</math> be the [[fractional part]] of {{mvar|r}}.
Then the continued fraction representation of {{mvar|r}} is
<math>[i;a_1,a_2,\ldots]</math>, where <math>[a_1;a_2,\ldots]</math> is the continued fraction representation of <math>1/f</math>.

To calculate a continued fraction representation of a number {{mvar|r}}, write down the integer part (technically the [[Floor function|floor]]) of {{mvar|r}}. Subtract this integer part from {{mvar|r}}. If the difference is 0, stop; otherwise find the [[multiplicative inverse|reciprocal]] of the difference and repeat. The procedure will halt if and only if {{mvar|r}} is rational. This process can be efficiently implemented using the [[Euclidean algorithm]] when the number is rational.  The table below shows an implementation of this procedure for the number 3.245, resulting in the continued fraction expansion [3; 4,12,4].

:{| class="wikitable"
|-
|+ Find the continued fraction for <math>3.245 = \frac{649}{200} </math>
|-
!Step
!Real<br/>Number
!Integer<br/>part
!Fractional<br/>part
!Simplified
!Reciprocal<br/>of {{mvar|f}}
|-
!1
|<math>r = \frac{649}{200}</math>
|<math>i = 3</math>
|<math>f = \frac{649}{200} - 3 </math>
|<math>= \frac{49}{200}</math>
|<math>\frac{1}{f} = \frac{200}{49} </math>
|-
!2
|<math>r = \frac{200}{49} </math>
|<math>i = 4</math>
|<math>f = \frac{200}{49} - 4 </math>
|<math> = \frac{4}{49} </math>
|<math> \frac{1}{f} = \frac{49}{4} </math>
|-
!3
|<math>r = \frac{49}{4} </math>
|<math>i = 12</math>
|<math>f = \frac{49}{4} - 12 </math>
|<math>= \frac{1}{4} </math>
|<math>\frac{1}{f} = \frac{4}{1} </math>
|-
!4
|<math>r = 4</math>
|<math>i = 4 </math>
|<math>f = 4 - 4 </math>
|<math>= 0 </math>
|colspan="2"|'''STOP'''
|-
| colspan="7" style="text-align:right;" | Continued fraction form for <math> 3.245 = \frac{649}{200} = [3; 4, 12, 4]</math>
<nowiki>=</nowiki> {{math|3 + {{sfrac|1|4 + {{sfrac|1|12 + {{sfrac|1|4}}}}}}}}
|}

==Notations==
The integers <math>a_0</math>, <math>a_1</math> etc., are called the ''coefficients'' or ''terms'' of the continued fraction.<ref name="Pettofrezzo 1970 150"/> One can abbreviate the continued fraction

:<math>x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3}}}</math>

in the notation of [[Carl Friedrich Gauss]]
:<math>x = a_0 + \underset{i=1}{\overset{3}{\mathrm K}} ~ \frac{1}{a_i} </math>

or as

:<math>x = [a_0; a_1, a_2, a_3] </math>,

or in the notation of [[Alfred Pringsheim|Pringsheim]] as

:<math>x = a_0 + \frac{1 \mid}{\mid a_1} + \frac{1 \mid}{\mid a_2} + \frac{1 \mid}{\mid a_3},</math>

or in another related notation as

:<math>x = a_0 + {1 \over a_1 + {}} {1 \over a_2 + {}} {1 \over a_3 {}}.</math>

Sometimes angle brackets are used, like this:

:<math>x = \left \langle a_0; a_1, a_2, a_3 \right \rangle.</math>

The semicolon in the square and angle bracket notations is sometimes replaced by a comma.<ref name="Long 1972 173"/><ref name="Pettofrezzo 1970 152"/>

One may also define ''infinite simple continued fractions'' as [[limit of a sequence|limits]]:

:<math>[a_0; a_1, a_2, a_3, \,\ldots ] = \lim_{n \to \infty} [a_0; a_1, a_2, \,\ldots, a_n]. </math>

This limit exists for any choice of <math>a_0</math> and positive integers <math>a_1,a_2,\ldots </math><ref name="Long 1972 183">{{harvtxt|Long|1972|p=183}}</ref><ref name="Pettofrezzo 1970 158">{{harvtxt|Pettofrezzo|Byrkit|1970|p=158}}</ref>

==Finite continued fractions==<!-- This section is linked from [[Continued fraction]] -->
Every finite continued fraction represents a [[rational number]], and every rational number can be represented in precisely two different ways as a finite continued fraction, with the conditions that the first coefficient is an integer and other coefficients being positive integers. These two representations agree except in their final terms. In the longer representation the final term in the continued fraction is 1; the shorter representation drops the final 1, but increases the new final term by 1. The final element in the short representation is therefore always greater than 1, if present. In symbols:

:{{math|[''a''{{sub|0}}; ''a''{{sub|1}}, ''a''{{sub|2}}, ..., ''a''{{sub|''n'' − 1}}, ''a''{{sub|''n''}}, 1] {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ''a''{{sub|2}}, ..., ''a''{{sub|''n'' − 1}}, ''a''{{sub|''n''}} + 1]}}.
:{{math|[''a''{{sub|0}}; 1] {{=}} [''a''{{sub|0}} + 1]}}.

==Of reciprocals==
The continued fraction representations of a positive rational number and its [[multiplicative inverse|reciprocal]] are identical except for a shift one place left or right depending on whether the number is less than or greater than one respectively. In other words, the numbers represented by
<math>[a_0;a_1,a_2,\ldots,a_n]</math> and <math>[0;a_0,a_1,\ldots,a_n]</math> are reciprocals. 

For instance if <math>a</math> is an integer and <math>x < 1</math> then

:<math>x=0 + \frac{1}{a + \frac{1}{b}}</math> and <math>\frac{1}{x} = a + \frac{1}{b}</math>.
If <math>x>1</math> then
:<math>x = a + \frac{1}{b}</math> and <math>\frac{1}{x} = 0 + \frac{1}{a + \frac{1}{b}}</math>.

The last number that generates the remainder of the continued fraction is the same for both <math>x</math> and its reciprocal.

For example,
:<math>2.25 = \frac{9}{4} = [2;4]</math> and <math>\frac{1}{2.25} = \frac{4}{9} = [0;2,4]</math>.

==Infinite continued fractions and convergents==<!-- [[Convergent (continued fraction)]] links here.  Please do not change. -->
Every infinite continued fraction is [[irrational number|irrational]], and every irrational number can be represented in precisely one way as an infinite continued fraction.

An infinite continued fraction representation for an irrational number is useful because its initial segments provide rational approximations to the number. These rational numbers are called the '''convergents''' of the continued fraction.<ref>{{harvtxt|Long|1972|p=177}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|pp=162–163}}</ref> The larger a term is in the continued fraction, the closer the corresponding convergent is to the irrational number being approximated. Numbers like π have occasional large terms in their continued fraction, which makes them easy to approximate with rational numbers. Other numbers like ''e'' have only small terms early in their continued fraction, which makes them more difficult to approximate rationally. The [[golden ratio]] ϕ has terms equal to 1 everywhere—the smallest values possible—which makes ϕ the most difficult number to approximate rationally. In this sense, therefore, it is the "most irrational" of all irrational numbers. Even-numbered convergents are smaller than the original number, while odd-numbered ones are larger.

For a continued fraction {{math|[''a''{{sub|0}}; ''a''{{sub|1}}, ''a''{{sub|2}}, ...]}}, the first four convergents (numbered 0 through 3) are

:{{bigmath|{{sfrac|''a''{{sub|0}}|1}}, {{sfrac|''a''{{sub|1}}''a''{{sub|0}} + 1|''a''{{sub|1}}}}, {{sfrac|''a''{{sub|2}}(''a''{{sub|1}}''a''{{sub|0}} + 1) + ''a''{{sub|0}}|''a''{{sub|2}}''a''{{sub|1}} + 1}}, {{sfrac|''a''{{sub|3}}(''a''{{sub|2}}(''a''{{sub|1}}''a''{{sub|0}} + 1) + ''a''{{sub|0}}) + (''a''{{sub|1}}''a''{{sub|0}} + 1) | ''a''{{sub|3}}(''a''{{sub|2}}''a''{{sub|1}} + 1) + ''a''{{sub|1}}}}}}

The numerator of the third convergent is formed by multiplying the numerator of the second convergent by the third quotient, and adding the numerator of the first convergent. The denominators are formed similarly. Therefore, each convergent can be expressed explicitly in terms of the continued fraction as the ratio of certain [[multivariate polynomial]]s called ''[[Continuant (mathematics)|continuants]]''.

If successive convergents are found, with numerators {{mvar|h}}{{sub|1}}, {{mvar|h}}{{sub|2}}, ... and denominators {{mvar|k}}{{sub|1}}, {{mvar|k}}{{sub|2}}, ... then the relevant recursive relation is:

:{{math|''h''{{sub|''n''}} {{=}} ''a''{{sub|''n''}}''h''{{sub|''n'' − 1}} + ''h''{{sub|''n'' − 2}}}},
:{{math|''k''{{sub|''n''}} {{=}} ''a''{{sub|''n''}}''k''{{sub|''n'' − 1}} + ''k''{{sub|''n'' − 2}}}}.

The successive convergents are given by the formula

:{{bigmath|{{sfrac|''h''{{sub|''n''}}|''k''{{sub|''n''}}}} {{=}} {{sfrac|''a''{{sub|''n''}}''h''{{sub|''n'' − 1}} + ''h''{{sub|''n'' − 2}}|''a''{{sub|''n''}}''k''{{sub|''n'' − 1}} + ''k''{{sub|''n'' − 2}}}}}}

Thus to incorporate a new term into a rational approximation, only the two previous convergents are necessary. The initial "convergents" (required for the first two terms) are <sup>0</sup>⁄<sub>1</sub> and <sup>1</sup>⁄<sub>0</sub>. For example, here are the convergents for [0;1,5,2,2].

:{| class="wikitable"
|- align="right"
! {{mvar|n}}
| −2|| −1|| 0 || 1 || 2 || 3 || 4
|- align="right"
! {{math|''a''{{sub|''n''}}}}
| &nbsp; || &nbsp; || 0 || 1 || 5 || 2 || 2
|- align="right"
! {{math|''h''{{sub|''n''}}}}
| 0 || 1 || 0 || 1 || 5 || 11 || 27
|- align="right"
! {{math|''k''{{sub|''n''}}}}
| 1 || 0 || 1 || 1 || 6 || 13 || 32
|}

When using the [[Methods of computing square roots#Babylonian method|Babylonian method]] to generate successive approximations to the square root of an integer, if one starts with the lowest integer as first approximant, the rationals generated all appear in the list of convergents for the continued fraction. Specifically, the approximants will appear on the convergents list in positions 0, 1, 3, 7, 15, ... , {{math|2<sup>''k''</sup>−1}}, ... For example, the continued fraction expansion for [[square root of 3|{{sqrt|3}}]] is [1;1,2,1,2,1,2,1,2,...]. Comparing the convergents with the approximants derived from the Babylonian method:

:{| class="wikitable"
|- align="right"
! {{mvar|n}}
| −2|| −1|| '''0''' || '''1''' || 2 || '''3''' || 4 || 5 || 6 || '''7'''
|- align="right"
! {{math|''a''{{sub|''n''}}}}
| &nbsp; || &nbsp; || 1 || 1 || 2 || 1 || 2 || 1 || 2 || 1
|- align="right"
! {{math|''h''{{sub|''n''}}}}
| 0 || 1 || '''1''' || '''2''' || 5 || '''7''' || 19 || 26 || 71 || '''97'''
|- align="right"
! {{math|''k''{{sub|''n''}}}}
| 1 || 0 || '''1''' || '''1''' || 3 || '''4''' || 11 || 15 || 41 || '''56'''
|}

:{{math|''x''{{sub|0}} {{=}} 1 {{=}} {{sfrac|1|1}}}}
:{{math|''x''{{sub|1}} {{=}} {{sfrac|1|2}}(1 + {{sfrac|3|1}}) {{=}} {{sfrac|2|1}} {{=}} 2}}
:{{math|''x''{{sub|2}} {{=}} {{sfrac|1|2}}(2 + {{sfrac|3|2}}) {{=}} {{sfrac|7|4}}}}
:{{math|''x''{{sub|3}} {{=}} {{sfrac|1|2}}({{sfrac|7|4}} + {{sfrac|3|{{sfrac|7|4}}}}) {{=}} {{sfrac|97|56}}}}

===Properties===
[[Baire space (set theory)|Baire space]] is a topological space on infinite sequences of natural numbers. The infinite continued fraction provides a [[homeomorphism]] from Baire space to the space of irrational real numbers (with the subspace topology inherited from the [[Euclidean topology|usual topology]] on the reals). The infinite continued fraction also provides a map between the [[quadratic irrational]]s and the [[dyadic rational]]s, and from other irrationals to the set of infinite strings of binary numbers (i.e. the [[Cantor set]]); this map is called the [[Minkowski question mark]] function. The mapping has interesting self-similar [[fractal]] properties; these are given by the [[modular group]], which is the subgroup of [[Möbius transformation]]s having integer values in the transform. Roughly speaking, continued fraction convergents can be taken to be Möbius transformations acting on the (hyperbolic) [[upper half-plane]]; this is what leads to the fractal self-symmetry.

===Some useful theorems===
If <math>a_0</math>, <math>a_1</math>, <math>a_2</math>, <math>\ldots</math> is an infinite sequence of positive integers, define the sequences <math>h_n</math> and <math>k_n</math> recursively:
{| border="0" cellpadding="5" cellspacing="10" align="none"
|-
|<math>h_{n}=a_nh_{n-1}+h_{n-2}</math>
|
|
|<math>h_{-1}=1</math>
|
|<math>h_{-2}=0</math>
|-
|<math>k_{n}=a_nk_{n-1}+k_{n-2}</math>
|
|
|<math>k_{-1}=0</math>
|
|<math>k_{-2}=1</math>
|}

<blockquote>'''Theorem 1.''' For any positive real number <math>z</math>

:<math> \left[a_0; a_1, \,\dots, a_{n-1}, z \right]=\frac{z h_{n-1}+h_{n-2}}{z k_{n-1}+k_{n-2}}.</math>
</blockquote>

<blockquote>'''Theorem 2.''' The convergents of [<math>a_0</math>; <math>a_1</math>, <math>a_2</math>, <math>\ldots</math>] are given by

:<math>\left[a_0; a_1, \,\dots, a_n\right]=\frac{h_n}{k_n}.</math>
</blockquote>

<blockquote>'''Theorem 3.''' If the <math>n</math>th convergent to a continued fraction is <math>h_n</math>/<math>k_n</math>, then

:<math>k_nh_{n-1}-k_{n-1}h_n=(-1)^n.</math>
</blockquote>

'''Corollary 1:''' Each convergent is in its lowest terms (for if <math>h_n</math> and <math>k_n</math> had a nontrivial common divisor it would divide <math>k_n h_{n-1} - k_{n-1} h_n</math>, which is impossible).

'''Corollary 2:''' The difference between successive convergents is a fraction whose numerator is unity:

:<math>\frac{h_n}{k_n}-\frac{h_{n-1}}{k_{n-1}} = \frac{h_nk_{n-1}-k_nh_{n-1}}{k_nk_{n-1}}= \frac{(-1)^{n+1}}{k_nk_{n-1}}.</math>

'''Corollary 3:''' The continued fraction is equivalent to a series of alternating terms:

:<math>a_0 + \sum_{n=0}^\infty \frac{(-1)^{n}}{k_{n}k_{n+1}}.</math>

'''Corollary 4:''' The matrix
:<math>\begin{bmatrix}
h_n & h_{n-1} \\
k_n & k_{n-1}
\end{bmatrix}</math>
has [[determinant]] plus or minus one, and thus belongs to the group of
<math>2\times 2 </math> [[unimodular matrix|unimodular matrices]] <math>\mathrm{GL}(2,\mathbb{Z})</math>.

<blockquote>'''Theorem 4.''' Each (<math>s</math>th) convergent is nearer to a subsequent (<math>n</math>th) convergent than any preceding (<math>r</math>th) convergent is. In symbols, if the <math>n</math>th convergent is taken to be <math>[a_0;a_1,\ldots,a_n] = x_n</math>, then
:<math>\left| x_r - x_n \right| > \left| x_s - x_n \right|</math>
for all <math>r < s < n</math>.</blockquote>

'''Corollary 1:''' The even convergents (before the <math>n</math>th) continually increase, but are always less than <math>x_n</math>.

'''Corollary 2:''' The odd convergents (before the <math>n</math>th) continually decrease, but are always greater than <math>x_n</math>.

<blockquote>'''Theorem 5.'''

:<math>\frac{1}{k_n(k_{n+1}+k_n)}< \left|x-\frac{h_n}{k_n}\right|< \frac{1}{k_nk_{n+1}}. </math>
</blockquote>

'''Corollary 1:''' A convergent is nearer to the limit of the continued fraction than any fraction whose denominator is less than that of the convergent.

'''Corollary 2:''' A convergent obtained by terminating the continued fraction just before a large term is a close approximation to the limit of the continued fraction.

==Semiconvergents==<!-- This section is linked from [[Complete quotient]] -->

If

:<math> \frac{h_{n-1}}{k_{n-1}},\frac{h_n}{k_n} </math>

are consecutive convergents, then any fractions of the form

: <math> \frac{h_{n-1} + mh_n}{k_{n-1} + mk_n},</math>

where <math>m</math> is an integer such that <math>0\leq m\leq a_{n+1}</math>, are called ''semiconvergents'', ''secondary convergents'', or ''intermediate fractions''. The <math>(m+1)</math>-st semiconvergent equals the [[Mediant (mathematics)|mediant]] of the <math>m</math>-th one and the convergent <math>\tfrac{h_n}{k_n}</math>. It follows that semiconvergents represent a [[monotonic sequence]] of fractions between the convergents <math>\tfrac{h_{n-1}}{k_{n-1}}</math> (corresponding to <math>m=0</math>) and <math>\tfrac{h_{n+1}}{k_{n+1}}</math> (corresponding to <math>m=a_{n+1}</math>). Sometimes the term is taken to mean that being a semiconvergent excludes the possibility of being a convergent (i.e., <math>0<m<a_{n+1}</math>), rather than that a convergent is a kind of semiconvergent.

The semiconvergents to the continued fraction expansion of a real number <math>x</math> include all the rational approximations that are better than any approximation with a smaller denominator. Another useful property is that consecutive semiconvergents <math>\tfrac{a}{b}</math> and <math>\tfrac{c}{d}</math> are such that
<math>ad - bc = \pm 1</math>.

==Best rational approximations==
{{See also|Diophantine approximation|Padé approximant}}

One can choose to define a ''best rational approximation'' to a real number {{mvar|x}} as a rational number {{sfrac|{{mvar|n}}|{{mvar|d}}}}, {{math|''d'' > 0}}, that is closer to {{mvar|x}} than any approximation with a smaller or equal denominator. The simple continued fraction for {{mvar|x}} can be used to generate ''all'' of the best rational approximations for {{mvar|x}} by applying these three rules:

#Truncate the continued fraction, and reduce its last term by a chosen amount (possibly zero).
#The reduced term cannot have less than half its original value.
#If the final term is even, half its value is admissible only if the corresponding semiconvergent is better than the previous convergent. (See below.)

For example, 0.84375 has continued fraction [0;1,5,2,2]. Here are all of its best rational approximations.

:{| class="wikitable"
|- align="center"
! Continued fraction
|  [0;1]  ||  [0;1,3]  ||  [0;1,4]  ||  [0;1,5]  ||  [0;1,5,2]  ||  [0;1,5,2,1]  ||  [0;1,5,2,2] 
|- align="center"
! Rational approximation
| 1 || {{sfrac|3|4}} || {{sfrac|4|5}} || {{sfrac|5|6}} || {{sfrac|11|13}} || {{sfrac|16|19}} || {{sfrac|27|32}}
|- align="center"
! Decimal equivalent
| 1 || 0.75 || 0.8 || ~0.83333 || ~0.84615 || ~0.84211 || 0.84375
|- align="center"
! Error
| +18.519% || −11.111% || −5.1852% || −1.2346% || +0.28490% || −0.19493% || 0%
|}

The strictly monotonic increase in the denominators as additional terms are included permits an algorithm to impose a limit, either on size of denominator or closeness of approximation.

The "half rule" mentioned above requires that when {{mvar|a}}{{sub|{{mvar|k}}}} is even, the halved term {{mvar|a}}{{sub|{{mvar|k}}}}/2 is admissible if and only if {{math|{{!}}''x'' − [''a''{{sub|0}} ; ''a''{{sub|1}}, ..., ''a''{{sub|''k'' − 1}}]{{!}} > {{!}}''x'' − [''a''{{sub|0}} ; ''a''{{sub|1}}, ..., ''a''{{sub|''k'' − 1}}, ''a''{{sub|''k''}}/2]{{!}}}}<ref name=thill>{{citation |author=M. Thill |title=A more precise rounding algorithm for rational numbers |year=2008 |journal=Computing |volume=82 |pages=189–198 |doi=10.1007/s00607-008-0006-7}}</ref> This is equivalent<ref name=thill/> to:<ref>{{Citation |editor-last=Paeth |editor-first=Alan W. |first=Ken |last=Shoemake |title=Graphic Gems V |chapter=I.4: Rational Approximation |chapter-url=https://books.google.com/books?id=8CGj9_ZlFKoC&pg=PA25 |pages=25–31 |publisher=Academic Press |year=1995 |location=San Diego, California |isbn=0-12-543455-3}}</ref>

:{{math|[''a''{{sub|''k''}}; ''a''{{sub|''k'' − 1}}, ..., ''a''{{sub|1}}] > [''a''{{sub|''k''}}; ''a''{{sub|''k'' + 1}}, ...]}}.

The convergents to {{mvar|x}} are "best approximations" in a much stronger sense than the one defined above. Namely, {{mvar|n}}/{{mvar|d}} is a convergent for {{mvar|x}} if and only if {{math|{{!}}''dx'' − ''n''{{!}}}} has the smallest value among the analogous expressions for all rational approximations {{mvar|m}}/{{mvar|c}} with {{math|''c'' ≤ ''d''}}; that is, we have {{math|{{!}}''dx'' − ''n''{{!}} < {{!}}''cx'' − ''m''{{!}}}} so long as {{math|''c'' < ''d''}}. (Note also that {{math|{{!}}''d<sub>k</sub>x'' − ''n<sub>k</sub>''{{!}} → 0}} as {{math|''k'' → ∞}}.)

=== Best rational within an interval ===
A rational that falls within the interval {{open-open|''x'', ''y''}}, for {{math|0 < {{mvar|x}} < {{mvar|y}}}}, can be found with the continued fractions for {{mvar|x}} and {{mvar|y}}. When both {{mvar|x}} and {{mvar|y}} are irrational and
:{{math|''x'' {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ''a''{{sub|2}}, ..., ''a''{{sub|''k'' − 1}}, ''a''{{sub|''k''}}, ''a''{{sub|''k'' + 1}}, ...]}}
:{{math|''y'' {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ''a''{{sub|2}}, ..., ''a''{{sub|''k'' − 1}}, ''b''{{sub|''k''}}, ''b''{{sub|''k'' + 1}}, ...]}}
where {{mvar|x}} and {{mvar|y}} have identical continued fraction expansions up through {{math|''a''<sub>''k''−1</sub>}}, a rational that falls within the interval {{open-open|''x'', ''y''}} is given by the finite continued fraction,
:{{math|''z''(''x'',''y'') {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ''a''{{sub|2}}, ..., ''a''{{sub|''k'' − 1}}, min(''a''{{sub|''k''}}, ''b''{{sub|''k''}}) + 1]}}
This rational will be best in the sense that no other rational in {{open-open|''x'', ''y''}} will have a smaller numerator or a smaller denominator.{{Citation needed|date=June 2019}}

If {{mvar|x}} is rational, it will have ''two'' continued fraction representations that are ''finite'', {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}}, and similarly a rational&nbsp;{{mvar|y}} will have two representations, {{math|''y''<sub>1</sub>}} and {{math|''y''<sub>2</sub>}}. The coefficients beyond the last in any of these representations should be interpreted as {{math|+∞}}; and the best rational will be one of {{math|''z''(''x''<sub>1</sub>, ''y''<sub>1</sub>)}}, {{math|''z''(''x''<sub>1</sub>, ''y''<sub>2</sub>)}}, {{math|''z''(''x''<sub>2</sub>, ''y''<sub>1</sub>)}}, or {{math|''z''(''x''<sub>2</sub>, ''y''<sub>2</sub>)}}.

For example, the decimal representation 3.1416 could be rounded from any number in the interval {{closed-closed|3.14155, 3.14165}}. The continued fraction representations of 3.14155 and 3.14165 are
:{{math|3.14155 {{=}} [3; 7, 15, 2, 7, 1, 4, 1, 1] {{=}} [3; 7, 15, 2, 7, 1, 4, 2]}}
:{{math|3.14165 {{=}} [3; 7, 16, 1, 3, 4, 2, 3, 1] {{=}} [3; 7, 16, 1, 3, 4, 2, 4]}}
and the best rational between these two is
:{{math|[3; 7, 16] {{=}} {{sfrac|355|113}} {{=}} 3.1415929....}}
Thus, {{sfrac|355|113}} is the best rational number corresponding to the rounded decimal number 3.1416, in the sense that no other rational number that would be rounded to 3.1416 will have a smaller numerator or a smaller denominator.

=== Interval for a convergent ===
A rational number, which can be expressed as finite continued fraction in two ways,
:{{math|''z'' {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ..., ''a''{{sub|''k'' − 1}}, ''a''{{sub|''k''}}, 1] {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ..., ''a''{{sub|''k'' − 1}}, ''a''{{sub|''k''}} + 1]}}
will be one of the convergents for the continued fraction expansion of a number, if and only if the number is strictly between
:{{math|''x'' {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ..., ''a''{{sub|''k'' − 1}}, ''a''{{sub|''k''}}, 2]}} and
:{{math|''y'' {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ..., ''a''{{sub|''k'' − 1}}, ''a''{{sub|''k''}} + 2]}}
The numbers {{mvar|x}} and {{mvar|y}} are formed by incrementing the last coefficient in the two representations for {{mvar|z}}. It is the case that {{math|''x'' < ''y''}} when {{mvar|k}} is even, and {{math|''x'' > ''y''}} when {{mvar|k}} is odd.

For example, the number {{sfrac|355|113}} has the continued fraction representations
:{{sfrac|355|113}} = [3; 7, 15, 1] = [3; 7, 16]
and thus {{sfrac|355|113}} is a convergent of any number strictly between
:{| cellpadding="2" cellspacing="0"
| align="right" | {{math|[3; 7, 15, 2]}} ||{{=}}|| {{math|{{sfrac|688|219}} ≈ 3.1415525}}
|-
| align="right" | {{math|[3; 7, 17]}} ||{{=}}|| {{math|{{sfrac|377|120}} ≈ 3.1416667}}
|}

==Comparison==
Consider {{math|''x'' {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ...]}} and {{math|''y'' {{=}} [''b''{{sub|0}}; ''b''{{sub|1}}, ...]}}. If {{mvar|k}} is the smallest index for which {{math|''a''{{sub|''k''}}}} is unequal to {{math|''b''{{sub|''k''}}}} then {{math|''x'' < ''y''}} if {{math|(−1){{sup|''k''}}(''a''{{sub|''k''}} − ''b''{{sub|''k''}}) < 0}} and {{math|''y'' < ''x''}} otherwise.

If there is no such {{mvar|k}}, but one expansion is shorter than the other, say {{math|''x'' {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ..., ''a''{{sub|''n''}}]}} and {{math|''y'' {{=}} [''b''{{sub|0}}; ''b''{{sub|1}}, ..., ''b''{{sub|''n''}}, ''b''{{sub|''n'' + 1}}, ...]}} with {{math|''a''{{sub|''i''}} {{=}} ''b''{{sub|''i''}}}} for {{math|0 ≤ ''i'' ≤ ''n''}}, then {{math|''x'' < ''y''}} if {{mvar|n}} is even and {{math|''y'' < ''x''}} if {{mvar|n}} is odd.

==Continued fraction expansions of {{pi}}==
To calculate the convergents of [[pi|{{pi}}]] we may set {{math|''a''{{sub|0}} {{=}} ⌊{{pi}}⌋ {{=}} 3}}, define {{math|''u''{{sub|1}} {{=}} {{sfrac|1|{{pi}} − 3}} ≈ 7.0625}} and {{math|''a''{{sub|1}} {{=}} ⌊''u''{{sub|1}}⌋ {{=}} 7}}, {{math|''u''{{sub|2}} {{=}} {{sfrac|1|''u''{{sub|1}} − 7}} ≈ 15.9966}} and {{math|''a''{{sub|2}} {{=}} ⌊''u''{{sub|2}}⌋ {{=}} 15}}, {{math|''u''{{sub|3}} {{=}} {{sfrac|1|''u''{{sub|2}} − 15}} ≈ 1.0034}}. Continuing like this, one can determine the infinite continued fraction of {{pi}} as
:[3;7,15,1,292,1,1,...] {{OEIS|A001203}}.
The fourth convergent of {{pi}} is [3;7,15,1] = {{sfrac|355|113}} = 3.14159292035..., sometimes called [[Milü]], which is fairly close to the true value of {{pi}}.

Let us suppose that the quotients found are, as above, [3;7,15,1]. The following is a rule by which we can write down at once the convergent fractions which result from these quotients without developing the continued fraction.

The first quotient, supposed divided by unity, will give the first fraction, which will be too small, namely, {{sfrac|3|1}}. Then, multiplying the numerator and denominator of this fraction by the second quotient and adding unity to the numerator, we shall have the second fraction, {{sfrac|22|7}}, which will be too large. Multiplying in like manner the numerator and denominator of this fraction by the third quotient, and adding to the numerator the numerator of the preceding fraction, and to the denominator the denominator of the preceding fraction, we shall have the third fraction, which will be too small. Thus, the third quotient being 15, we have for our numerator {{nowrap|(22 × 15 {{=}} 330) + 3 {{=}} 333}}, and for our denominator, {{nowrap|(7 × 15 {{=}} 105) + 1 {{=}} 106}}. The third convergent, therefore, is {{sfrac|333|106}}. We proceed in the same manner for the fourth convergent. The fourth quotient being 1, we say 333 times 1 is 333, and this plus 22, the numerator of the fraction preceding, is 355; similarly, 106 times 1 is 106, and this plus 7 is 113.

In this manner, by employing the four quotients [3;7,15,1], we obtain the four fractions:

:{{sfrac|3|1}}, {{sfrac|22|7}}, {{sfrac|333|106}}, {{sfrac|355|113}}, ....

These convergents are alternately smaller and larger than the true value of {{pi}}, and approach nearer and nearer to {{pi}}. The difference between a given convergent and {{pi}} is less than the reciprocal of the product of the denominators of that convergent and the next convergent. For example, the fraction {{sfrac|22|7}} is greater than {{pi}}, but {{sfrac|22|7}} − {{pi}} is less than {{sfrac|1|7 × 106}}&nbsp;=&nbsp;{{sfrac|1|742}} (in fact, {{sfrac|22|7}} − {{pi}} is just more than {{sfrac|1|791}} = {{sfrac|1|7 × 113}}).

The demonstration of the foregoing properties is deduced from the fact that if we seek the difference between one of the convergent fractions and the next adjacent to it we shall obtain a fraction of which the numerator is always unity and the denominator the product of the two denominators. Thus the difference between {{sfrac|22|7}} and {{sfrac|3|1}} is {{sfrac|1|7}}, in excess; between {{sfrac|333|106}} and {{sfrac|22|7}}, {{sfrac|1|742}}, in deficit; between {{sfrac|355|113}} and {{sfrac|333|106}}, {{sfrac|1|11978}}, in excess; and so on. The result being, that by employing this series of differences we can express in another and very simple manner the fractions with which we are here concerned, by means of a second series of fractions of which the numerators are all unity and the denominators successively be the product of every two adjacent denominators. Instead of the fractions written above, we have thus the series:

:{{sfrac|3|1}} + {{sfrac|1|1 × 7}} − {{sfrac|1|7 × 106}} + {{sfrac|1|106 × 113}} − ...

The first term, as we see, is the first fraction; the first and second together give the second fraction, {{sfrac|22|7}}; the first, the second and the third give the third fraction {{sfrac|333|106}}, and so on with the rest; the result being that the series entire is equivalent to the original value.

==Generalized continued fraction==
{{main|Generalized continued fraction}}

A generalized continued fraction is an expression of the form

:<math>x = b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots\,}}}}</math>

where the ''a''<sub>''n''</sub> (''n'' &gt; 0) are the partial numerators, the ''b''<sub>''n''</sub> are the partial denominators, and the leading term ''b''<sub>0</sub> is called the ''integer'' part of the continued fraction.

To illustrate the use of generalized continued fractions, consider the following example. The sequence of partial denominators of the simple continued fraction of {{pi}} does not show any obvious pattern:

:<math>\pi=[3;7,15,1,292,1,1,1,2,1,3,1,\ldots]</math>

or

:<math>\pi=3+\cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{292+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{3+\cfrac{1}{1+\ddots}}}}}}}}}}}</math>

However, several generalized continued fractions for {{pi}} have a perfectly regular structure, such as:

:<math>
\pi=\cfrac{4}{1+\cfrac{1^2}{2+\cfrac{3^2}{2+\cfrac{5^2}{2+\cfrac{7^2}{2+\cfrac{9^2}{2+\ddots}}}}}}
=\cfrac{4}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\cfrac{4^2}{9+\ddots}}}}}
=3+\cfrac{1^2}{6+\cfrac{3^2}{6+\cfrac{5^2}{6+\cfrac{7^2}{6+\cfrac{9^2}{6+\ddots}}}}}
</math>

:<math>\displaystyle \pi=2+\cfrac{2}{1+\cfrac{1}{1/2+\cfrac{1}{1/3+\cfrac{1}{1/4+\ddots}}}}=2+\cfrac{2}{1+\cfrac{1\cdot2}{1+\cfrac{2\cdot3}{1+\cfrac{3\cdot4}{1+\ddots}}}}</math>

:<math> \displaystyle \pi=2+\cfrac{4}{3+\cfrac{1\cdot3}{4+\cfrac{3\cdot5}{4+\cfrac{5\cdot7}{4+\ddots}}}}</math>

The first two of these are special cases of the [[Inverse trigonometric functions#Variant: Continued fractions for arctangent|arctangent]] function with {{pi}} = 4 arctan (1).

:<math>
\pi=3+\cfrac{1^3}{6+\cfrac{1^3+2^3}{6+\cfrac{1^3+2^3+3^3+4^3}{6+\cfrac{1^3+2^3 +3^3+4^3+5^3+6^3}{6+\cfrac{1^3+2^3+3^3+4^3+5^3+6^3+7^3+8^3}{6+\ddots}}}}}
</math>

The continued fraction of <math>\pi</math> above consisting of cubes uses the Nilakantha series and an exploit from Leonhard Euler.<ref>{{Cite web|url=https://www.theoremoftheday.org/NumberTheory/EulerCF/TotDEulerCF.pdf|title=Theorem of the Day: Theorem no. 203|last=Foster|first=Tony|date=June 22, 2015|website=|publisher=Robin Whitty|access-date=June 25, 2015}}</ref>

==Other continued fraction expansions==

===Periodic continued fractions===
{{main|Periodic continued fraction}}
The numbers with periodic continued fraction expansion are precisely the [[quadratic irrational|irrational solutions]] of [[quadratic equation]]s with rational coefficients; rational solutions have finite continued fraction expansions as previously stated. The simplest examples are the [[golden ratio]] φ = [1;1,1,1,1,1,...] and {{sqrt|2}} = [1;2,2,2,2,...], while {{sqrt|14}} = [3;1,2,1,6,1,2,1,6...] and {{sqrt|42}} = [6;2,12,2,12,2,12...]. All irrational square roots of integers have a special form for the period; a symmetrical string, like the empty string (for {{sqrt|2}}) or 1,2,1 (for {{sqrt|14}}), followed by the double of the leading integer.

===A property of the golden ratio φ===
Because the continued fraction expansion for [[golden ratio|φ]] doesn't use any integers greater than 1, φ is one of the most "difficult" real numbers to approximate with rational numbers. [[Hurwitz's theorem (number theory)|Hurwitz's theorem]]<ref>Theorem 193: {{Cite book | last = Hardy | first = G.H. | last2 = Wright | first2 = E.M. | title = An Introduction to the Theory of Numbers | publisher = Oxford | year = 1979 | edition = Fifth}}</ref> states that any irrational number {{mvar|k}} can be approximated by infinitely many rational {{sfrac|''m''|''n''}} with

:<math>\left| k - {m \over n}\right| < {1 \over n^2 \sqrt 5}.</math>

While virtually all real numbers {{mvar|k}} will eventually have infinitely many convergents {{sfrac|''m''|''n''}} whose distance from {{mvar|k}} is significantly smaller than this limit, the convergents for φ (i.e., the numbers {{sfrac|5|3}}, {{sfrac|8|5}}, {{sfrac|13|8}}, {{sfrac|21|13}}, etc.) consistently "toe the boundary", keeping a distance of almost exactly <math>{\scriptstyle{1 \over n^2 \sqrt 5}}</math> away from φ, thus never producing an approximation nearly as impressive as, for example, [[Milü|{{sfrac|355|113}}]] for [[pi|{{pi}}]]. It can also be shown that every real number of the form {{sfrac|''a'' + ''b''φ|''c'' + ''d''φ}}, where {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, and {{mvar|d}} are integers such that {{math|1=''a'' ''d'' − ''b'' ''c'' = ±1}}, shares this property with the golden ratio φ; and that all other real numbers can be more closely approximated.

===Regular patterns in continued fractions===
While there is no discernable pattern in the simple continued fraction expansion of {{pi}}, there is one for {{math|''e''}}, the [[e (mathematical constant)|base of the natural logarithm]]:

:<math>e = e^1 = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, \dots],</math>

which is a special case of this general expression for positive integer {{mvar|n}}:

:<math>e^{1/n} = [1; n-1, 1, 1, 3n-1, 1, 1, 5n-1, 1, 1, 7n-1, 1, 1, \dots] \,\!.</math>

Another, more complex pattern appears in this continued fraction expansion for positive odd {{mvar|n}}:

:<math>e^{2/n} = \left[1; \frac{n-1}{2}, 6n, \frac{5n-1}{2}, 1, 1, \frac{7n-1}{2}, 18n, \frac{11n-1}{2}, 1, 1, \frac{13n-1}{2}, 30n, \frac{17n-1}{2}, 1, 1, \dots \right] \,\!,</math>

with a special case for {{math|1=''n'' = 1}}:

:<math>e^2 = [7; 2, 1, 1, 3, 18, 5, 1, 1, 6, 30, 8, 1, 1, 9, 42, 11, 1, 1, 12, 54, 14, 1, 1 \dots, 3k, 12k+6, 3k+2, 1, 1 \dots] \,\!.</math>

Other continued fractions of this sort are

:<math>\tanh(1/n) = [0; n, 3n, 5n, 7n, 9n, 11n, 13n, 15n, 17n, 19n, \dots] </math>

where {{mvar|n}} is a positive integer; also, for integer {{mvar|n}}:

:<math>\tan(1/n) = [0; n-1, 1, 3n-2, 1, 5n-2, 1, 7n-2, 1, 9n-2, 1, \dots]\,\!,</math>

with a special case for {{math|1=''n'' = 1}}:

:<math>\tan(1) = [1; 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 1, 15, 1, 17, 1, 19, 1, \dots]\,\!.</math>

If {{math|''I''<sub>''n''</sub>(''x'')}} is the modified, or hyperbolic, [[Bessel function]] of the first kind, we may define a function on the rationals {{sfrac|''p''|''q''}} by

:<math>S(p/q) = \frac{I_{p/q}(2/q)}{I_{1+p/q}(2/q)},</math>

which is defined for all rational numbers, with {{mvar|p}} and {{mvar|q}} in lowest terms. Then for all nonnegative rationals, we have

:<math>S(p/q) = [p+q; p+2q, p+3q, p+4q, \dots],</math>

with similar formulas for negative rationals; in particular we have

:<math>S(0) = S(0/1) = [1; 2, 3, 4, 5, 6, 7, \dots].</math>

Many of the formulas can be proved using [[Gauss's continued fraction]].

===Typical continued fractions===
Most irrational numbers do not have any periodic or regular behavior in their continued fraction expansion. Nevertheless, [[Aleksandr Khinchin|Khinchin]] proved that for [[almost all]] real numbers {{mvar|x}}, the {{math|''a''<sub>''i''</sub>}} (for {{math|1=''i'' = 1, 2, 3, ...}}) have an astonishing property: their [[geometric mean]] tends to a constant (known as [[Khinchin's constant]], {{math|''K'' ≈ 2.6854520010...}}) independent of the value of {{mvar|x}}. [[Paul Lévy (mathematician)|Paul Lévy]] showed that the {{mvar|n}}th root of the denominator of the {{mvar|n}}th convergent of the continued fraction expansion of almost all real numbers approaches an asymptotic limit, approximately 3.27582, which is known as [[Lévy's constant]]. [[Lochs' theorem]] states that {{mvar|n}}th convergent of the continued fraction expansion of almost all real numbers determines the number to an average accuracy of just over {{mvar|n}} decimal places.

==Applications==
===Square roots===
Generalized continued fractions are used in a [[methods of computing square roots#Continued fraction expansion|method for computing square roots]].

The identity
{{NumBlk|:|<math>\sqrt{x} = 1+\frac{x-1}{1+\sqrt{x}}</math>|{{EquationRef|1}}}}
leads via recursion to the generalized continued fraction for any square root:<ref>Ben Thurston, [http://benpaulthurstonblog.blogspot.com/2012/05/estimating-square-roots.html "Estimating square roots, generalized continued fraction expression for every square root"], ''The Ben Paul Thurston Blog''</ref>
{{NumBlk|:|<math>\sqrt{x}=1+\cfrac{x-1}{2 + \cfrac{x-1}{2 + \cfrac{x-1}{2+{\ddots}}}}</math>|{{EquationRef|2}}}}'''

===Pell's equation===
Continued fractions play an essential role in the solution of [[Pell's equation]]. For example, for positive integers {{mvar|p}} and {{mvar|q}}, and non-square {{mvar|n}}, it is true that if {{math|1=''p''<sup>2</sup> − ''nq''<sup>2</sup> = ±1}}, then {{math|{{sfrac|''p''|''q''}}}} is a convergent of the regular continued fraction for {{sqrt|{{mvar|n}}}}.  The converse holds if the period of the regular continued fraction for {{sqrt|{{mvar|n}}}} is 1, and in general the period describes which convergents give solutions to Pell's equation.<ref>{{cite book|last1=Niven|first1=Ivan|last2=Zuckerman|first2=Herbert S.|last3=Montgomery|first3=Hugh L.|title=An introduction to the theory of numbers|url=https://archive.org/details/introductiontoth0000nive|url-access=registration|edition=Fifth|location=New York|publisher=[[John Wiley & Sons|Wiley]]|date=1991|isbn=0-471-62546-9}}</ref>

===Dynamical systems===
Continued fractions also play a role in the study of [[dynamical system]]s, where they tie together the [[Farey sequence|Farey fractions]] which are seen in the [[Mandelbrot set]] with [[Minkowski's question mark function]] and the [[modular group]] Gamma.

The backwards [[shift operator]] for continued fractions is the map {{math|''h''(''x'') {{=}} 1/{{mvar|x}} − ⌊1/{{mvar|x}}⌋}} called the '''Gauss map''', which lops off digits of a continued fraction expansion: {{math|''h''([0; ''a''{{sub|1}}, ''a''{{sub|2}}, ''a''{{sub|3}}, ...]) {{=}} [0; ''a''{{sub|2}}, ''a''{{sub|3}}, ...]}}. The [[transfer operator]] of this map is called the [[Gauss–Kuzmin–Wirsing operator]]. The distribution of the digits in continued fractions is given by the zero'th [[eigenvector]] of this operator, and is called the [[Gauss–Kuzmin distribution]].

===Eigenvalues and eigenvectors===
The [[Lanczos algorithm]] uses a continued fraction expansion to iteratively approximate the eigenvalues and eigenvectors of a large sparse matrix.<ref>{{citation|title=Electronic Structure: Basic Theory and Practical Methods|first=Richard M.|last=Martin|publisher=Cambridge University Press|year=2004|isbn=9781139643658|page=557|url=https://books.google.com/books?id=v1YhAwAAQBAJ&pg=PA557}}.</ref>

=== Networking applications ===
Continued fractions have also been used in modelling optimization problems for wireless [[network virtualization]] to find a route between a source and a destination.<ref>{{Cite journal|last=Afifi|first=Haitham|display-authors=et al|date=April 2018|title=MARVELO: Wireless Virtual Network Embedding for Overlay Graphs with Loops|url=|journal=2018 IEEE Wireless Communications and Networking Conference (WCNC)|volume=|pages=|via=}}</ref>

==Examples of rational and irrational numbers==
{{Continued fraction examples}}

==History==
* 300 BCE ''[[Euclid's Elements]]'' contains an algorithm for the [[greatest common divisor]] which generates a continued fraction as a by-product
* 499 The ''[[Aryabhatiya]]'' contains the solution of indeterminate equations using continued fractions
* 1572 [[Rafael Bombelli]], ''L'Algebra Opera'' – method for the extraction of square roots which is related to continued fractions
* 1613 [[Pietro Cataldi]], ''Trattato del modo brevissimo di trovar la radice quadra delli numeri'' – first notation for continued fractions
:Cataldi represented a continued fraction as <math>a_0</math> & <math>\frac{n_1}{d_1 \cdot}</math> & <math>\frac{n_2}{d_2 \cdot}</math> & <math>\frac{n_3}{d_3 \cdot}</math> with the dots indicating where the following fractions went.
* 1695 [[John Wallis]], ''Opera Mathematica'' – introduction of the term "continued fraction"
* 1737 [[Leonhard Euler]], ''De fractionibus continuis dissertatio'' – Provided the first then-comprehensive account of the properties of continued fractions, and included the first proof that the number [[e (mathematical constant)|e]] is irrational.<ref name=sandifer>{{cite journal | last = Sandifer | first = Ed | title = How Euler Did It: Who proved e is irrational? | journal = MAA Online |date=February 2006 | url = http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2028%20e%20is%20irrational.pdf|format=PDF}}</ref>
* 1748 Euler, ''[[List of important publications in mathematics#Introductio in analysin infinitorum|Introductio in analysin infinitorum]]''. Vol. I, Chapter 18 – proved the equivalence of a certain form of continued fraction and a generalized [[infinite series]], proved that every rational number can be written as a finite continued fraction, and proved that the continued fraction of an irrational number is infinite.<ref name=IntroductioI>{{cite web | title=E101 – Introductio in analysin infinitorum, volume 1|url=http://math.dartmouth.edu/~euler/pages/E101.html| accessdate=2008-03-16}}</ref>
* 1761 [[Johann Lambert]] – gave the first proof of the irrationality of [[Pi|{{pi}}]] using a continued fraction for [[Trigonometric functions|tan(x)]].
* 1768 [[Joseph-Louis Lagrange]] – provided the general solution to Pell's equation using continued fractions similar to Bombelli's
* 1770 Lagrange – proved that [[quadratic irrational number|quadratic irrationals]] expand to [[periodic continued fraction]]s.
* 1813 [[Carl Friedrich Gauss]], ''Werke'', Vol. 3, pp.&nbsp;134–138 – derived a very general [[Gauss's continued fraction|complex-valued continued fraction]] via a clever identity involving the [[hypergeometric function]]
* 1828 [[Évariste Galois]] proved the periodicity of continued fractions for quadratic irrationals.<ref>{{cite book|last=Wolfram|first=Stephen|title=A New Kind of Science|publisher=Wolfram Media, Inc.|year=2002|page=915|isbn=1-57955-008-8|url-access=registration|url=https://archive.org/details/newkindofscience00wolf}}</ref>  
* 1892 [[Henri Padé]] defined [[Padé approximant]]
* 1972 [[Bill Gosper]] – First exact algorithms for continued fraction arithmetic.

==See also==
*[[Complete quotient]]
*[[Methods of computing square roots#Example.2C square root of 114 as a continued fraction|Computing continued fractions of square roots]]
*[[Egyptian fraction]]
*[[Engel expansion]]
*[[Euler's continued fraction formula]]
*[[Generalized continued fraction]]
*[[Infinite compositions of analytic functions]]
*[[Infinite product]]
*[[Infinite series]]
*[[Iterated binary operation]]
*[[Mathematical constants (sorted by continued fraction representation)]]
*[[Restricted partial quotients]]
*[[Stern–Brocot tree]]
*[[Śleszyński–Pringsheim theorem]]

==Notes==
{{Reflist}}

==References==
* {{cite news|first1=H. | last1=Siebeck | year=1846 | title= Ueber periodische Kettenbrüche
 |journal= J. Reine Angew. Math. | volume=33 | pages=68–70 | url=http://www.digizeitschriften.de/dms/resolveppn/?PID=PPN243919689_0033}}
* {{cite news|first1=J. B. H. | last1=Heilermann | title=Ueber die Verwandlung von Reihen in Kettenbrüche
 |journal =J. Reine Angew. Math. | volume=33 | pages=174–188 | url=http://www.digizeitschriften.de/dms/img/?PID=PPN243919689_0033| year=1846}}
* {{cite news| first1=Arne | last1=Magnus | title=Continued fractions associated with the Padé Table
 |journal=Math. Z. | volume=78 | pages=361–374 | year=1962}}
* {{cite news | first1=Chen-Fan | last1=Chen |first2=Leang-San | last2=Shieh
 |title=Continued fraction inversion by Routh's Algorithm | journal = IEEE Trans. Circuit Theory
 | volume=16 | number=2 | pages=197–202 | doi=10.1109/TCT.1969.1082925 | year=1969}}
* {{cite news|first1=William B. | last1=Gragg | title=Matrix interpretations and applications of the continued fraction algorithm
 |journal= Rocky Mount. J. Math. | volume=4 | number=2 | doi=10.1216/RJM-1974-4-2-213 | page=213 |year=1974}}
*{{Cite book | last = Jones | first = William B. | last2 = Thron | first2 = W. J. | title = Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications. | place= | publisher = Addison-Wesley Publishing Company | year = 1980 | location = Reading. Massachusetts | volume = 11 | edition = | isbn = 0-201-13510-8}}
*{{cite book |title = Continued Fractions |url = https://archive.org/details/continuedfractio00khin_0 |url-access = registration | year = 1964 | last1 = Khinchin | first1 = A. Ya. | authorlink = Aleksandr Khinchin | origyear = Originally published in Russian, 1935 | publisher = [[University of Chicago Press]] | ISBN= 0-486-69630-8 }}
* {{citation | first1 = Calvin T. | last1 = Long | year = 1972 | title = Elementary Introduction to Number Theory | edition = 2nd | publisher = [[D. C. Heath and Company]] | location = Lexington | lccn = 77-171950 }}
* {{cite book | first=Oskar | last= Perron | authorlink=Oskar Perron | title =Die Lehre von den Kettenbrüchen | publisher=Chelsea Publishing Company | place=New York, NY | year= 1950}}
* {{citation | first1 = Anthony J. | last1 = Pettofrezzo | first2 = Donald R. | last2 = Byrkit | year = 1970 | title = Elements of Number Theory | publisher = [[Prentice Hall]] | location = Englewood Cliffs | lccn = 77-81766 }}
*{{cite book | title = Continued Fractions | url = https://archive.org/details/continuedfractio0000rock | url-access = registration | last1 = Rockett | first1 = Andrew M. | last2 = Szüsz | first2 = Peter| year = 1992 | publisher = World Scientific Press | ISBN = 981-02-1047-7 }}
*H. S. Wall, ''Analytic Theory of Continued Fractions'', D. Van Nostrand Company, Inc., 1948 {{isbn|0-8284-0207-8}}
*{{cite book|first1=A. |last1= Cuyt |first2= V. | last2= Brevik Petersen |first3= B. |last3= Verdonk
 |first4= H. |last4= Waadeland |first5 = W. B. |last5=Jones |title =Handbook of Continued fractions for Special functions
 |publisher= Springer Verlag |year=2008 |isbn=978-1-4020-6948-2}}
* {{cite news | last1=Rieger |first1=G. J. | title =A new approach to the real numbers (motivated by continued fractions)
 |journal= Abh. Braunschweig.Wiss. Ges. | volume= 33 |year=1982 |pages=205–217}}

==External links==
* {{springer|title=Continued fraction|id=p/c025540}}
* [http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html An Introduction to the Continued Fraction]
* Linas Vepstas [http://www.linas.org/math/chap-gap/chap-gap.html Continued Fractions and Gaps] (2004) reviews chaotic structures in continued fractions.
* [http://www.cut-the-knot.org/blue/ContinuedFractions.shtml Continued Fractions on the Stern-Brocot Tree] at [[cut-the-knot]]
* [http://www.math.sunysb.edu/~tony/whatsnew/column/antikytheraI-0400/kyth3.html The Antikythera Mechanism I: Gear ratios and continued fractions]
* [http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html Continued fraction calculator], WIMS.
* [https://web.archive.org/web/20030202011209/http://www.tweedledum.com/rwg/cfup.htm Continued Fraction Arithmetic] Gosper's first continued fractions paper, unpublished. Cached on the [[Internet Archive]]'s [[Internet Archive#Wayback Machine|Wayback Machine]]
* {{MathWorld |title=Continued Fraction |urlname=ContinuedFraction}}
* [http://demonstrations.wolfram.com/ContinuedFractions/ Continued Fractions] by [[Stephen Wolfram]] and [http://demonstrations.wolfram.com/ContinuedFractionApproximationsOfTheTangentFunction/ Continued Fraction Approximations of the Tangent Function] by Michael Trott, [[Wolfram Demonstrations Project]].
* {{OEIS el|1=A133593|2="Exact" continued fraction for Pi}}
* [http://go.helms-net.de/math/tetdocs/FracIterAltGeom.htm A view into "fractional interpolation" of a continued fraction {{math|{1; 1, 1, 1, ...}}} ]
* [http://www.ams.org/publicoutreach/feature-column/fcarc-irrational3 Best rational approximation through continued fractions]

{{Wiktionary}}
{{Fractions and ratios}}

{{Authority control}}

[[Category:Continued fractions| ]]
[[Category:Mathematical analysis]]