Simple continued fraction

a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}}

A finite continued fraction, where

n

is a non-negative integer,

a_{0}

is an integer, and

a_{i}

is a positive integer, for

i=1,\ldots ,n

.

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.^[1] 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. In either case, all integers in the sequence, other than the first, must be positive. The integers $a_{i}$ are called the coefficients or terms of the continued fraction.^[2]

Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number ⁠ $p$ / $q$ ⁠ has two closely related expressions as a finite continued fraction, whose coefficients $a i$ can be determined by applying the Euclidean algorithm to $(p,q)$ . The numerical value of an infinite continued fraction is irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number $\alpha$ 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 incommensurable values $\alpha$ 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 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 functions, arising in their analytic theory. For this use of the term, see Padé approximation and Chebyshev rational functions.

Motivation and notation

Consider, for example, the rational number ⁠415/93⁠, which is around 4.4624. As a first approximation, start with 4, which is the integer part; $⁠ 415 / 93 ⁠ = 4 + ⁠ 43 / 93 ⁠$ . Note that the fractional part is the reciprocal of ⁠93/43⁠ which is about 2.1628. Use the integer part, 2, as an approximation for the reciprocal to obtain a second approximation of $4 + ⁠ 1 / 2 ⁠ = 4.5; ⁠ 93 / 43 ⁠ = 2 + ⁠ 7 / 43 ⁠$ . The remaining fractional part, ⁠7/43⁠, is the reciprocal of ⁠43/7⁠, and ⁠43/7⁠ is around 6.1429. Use 6 as an approximation for this to obtain $2 + ⁠ 1 / 6 ⁠$ as an approximation for ⁠93/43⁠ and $4 + ⁠ 1 / 2 + ⁠ 1 / 6 ⁠ ⁠$ , about 4.4615, as the third approximation; $⁠ 43 / 7 ⁠ = 6 + ⁠ 1 / 7 ⁠$ . Finally, the fractional part, ⁠1/7⁠, is the reciprocal of 7, so its approximation in this scheme, 7, is exact ( $⁠ 7 / 1 ⁠ = 7 + ⁠ 0 / 1 ⁠$ ) and produces the exact expression $4 + ⁠ 1 / 2 + ⁠ 1 / 6 + ⁠ 1 / 7 ⁠ ⁠ ⁠$ for ⁠415/93⁠.

The expression $4 + ⁠ 1 / 2 + ⁠ 1 / 6 + ⁠ 1 / 7 ⁠ ⁠ ⁠$ is called the continued fraction representation of ⁠415/93⁠. This can be represented by the abbreviated notation ⁠415/93⁠ = [4; 2, 6, 7]. (Note that it is customary to replace only the first comma by a semicolon.) Some older textbooks use all commas in the $(n + 1)$ -tuple, for example, [4, 2, 6, 7].^[3]^[4]

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, 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:

$\sqrt 19 = [4;2,1,3,1,2,8,2,1,3,1,2,8,\dots]$ (sequence A010124 in the OEIS). The pattern repeats indefinitely with a period of 6.
$e = [2;1,2,1,1,4,1,1,6,1,1,8,\dots]$ (sequence A003417 in the OEIS). The pattern repeats indefinitely with a period of 3 except that 2 is added to one of the terms in each cycle.
$π = [3;7,15,1,292,1,1,1,2,1,3,1,\dots]$ (sequence A001203 in the OEIS). The terms in this representation are apparently random.
$ϕ = [1;1,1,1,1,1,1,1,1,1,1,1,\dots]$ (sequence A000012 in the OEIS). The golden ratio, the most difficult irrational number to approximate rationally. See: 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 representations, 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 $⁠ 137 / 1600 ⁠ = 0.085625$ , or infinite with a repeating cycle, for example $⁠ 4 / 27 ⁠ = 0.148148148148\dots$
Every rational number has an essentially unique continued fraction representation. Each rational can be represented in exactly two ways, since $[a 0; a 1,\dots a n -1, a n] = [a 0; a 1,\dots a n -1,(a n -1),1]$ . Usually the first, shorter one is chosen as the canonical representation.
The continued fraction representation of an irrational number is unique.
The real numbers whose continued fraction eventually repeats are precisely the quadratic irrationals.^[5] For example, the repeating continued fraction [1;1,1,1,…] is the golden ratio, and the repeating continued fraction [1;2,2,2,…] is the square root of 2. In contrast, the decimal representations of quadratic irrationals are apparently 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

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

where a_i and b_i are either rational numbers, real numbers, or complex numbers. If b_i = 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.^[6]

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

Calculating continued fraction representations

Consider a real number $r$ . Let $i=\lfloor r\rfloor$ be the integer part of $r$ and let $f=r-i$ be the fractional part of $r$ . Then the continued fraction representation of $r$ is $[i;a_{1},a_{2},\ldots ]$ , where $[a_{1};a_{2},\ldots ]$ is the continued fraction representation of $1/f$ .

To calculate a continued fraction representation of a number $r$ , write down the integer part (technically the floor) of $r$ . Subtract this integer part from $r$ . If the difference is 0, stop; otherwise find the reciprocal of the difference and repeat. The procedure will halt if and only if $r$ 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; 4,12,3,1].

Step	Real Number	Integer part	Fractional part	Simplified	Reciprocal of $f$
Find the continued fraction for $3.245={\frac {649}{200}}$
$1$	$r={\frac {649}{200}}$	$i=3$	$f={\frac {649}{200}}-3$	$={\frac {49}{200}}$	${\frac {1}{f}}={\frac {200}{49}}$
$2$	$r={\frac {200}{49}}$	$i=4$	$f={\frac {200}{49}}-4$	$={\frac {4}{49}}$	${\frac {1}{f}}={\frac {49}{4}}$
$3$	$r={\frac {49}{4}}$	$i=12$	$f={\frac {49}{4}}-12$	$={\frac {1}{4}}$	${\frac {1}{f}}={\frac {4}{1}}$
$4$	$r=4$	$i=4$	$f=4-4$	$=0$	STOP
Continued fraction form for $3.245={\frac {649}{200}}$ is $[3;4,12,4]$ : $3.245=3+{\frac {1}{4+{\frac {1}{12+{\frac {1}{4}}}}}}$

Notations for continued fractions

The integers $a_{0}$ , $a_{1}$ etc., are called the coefficients or terms of the continued fraction.^[2] One can abbreviate the continued fraction

x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}}}}}}}

in the notation of Carl Friedrich Gauss

x=a_{0}+{\underset {i=1}{\overset {3}{\mathrm {K} }}}~{\frac {1}{a_{i}}}

or as

x=[a_{0};a_{1},a_{2},a_{3}]

,

or in the notation of Pringsheim as

x=a_{0}+{\frac {1\mid }{\mid a_{1}}}+{\frac {1\mid }{\mid a_{2}}}+{\frac {1\mid }{\mid a_{3}}},

or in another related notation as

x=a_{0}+{1 \over a_{1}+{}}{1 \over a_{2}+{}}{1 \over a_{3}{}}.

Sometimes angle brackets are used, like this:

x=\left\langle a_{0};a_{1},a_{2},a_{3}\right\rangle .

The semicolon in the square and angle bracket notations is sometimes replaced by a comma.^[3]^[4]

One may also define infinite simple continued fractions as limits:

[a_{0};a_{1},a_{2},a_{3},\,\ldots ]=\lim _{n\to \infty }[a_{0};a_{1},a_{2},\,\ldots ,a_{n}].

This limit exists for any choice of $a_{0}$ and positive integers $a_{1},a_{2},\ldots$ .

Finite continued fractions

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:

[a 0; a 1, a 2, \dots, a n - 1, a n, 1] = [a 0; a 1, a 2, \dots, a n - 1, a n + 1]

.

[a 0; 1] = [a 0 + 1]

.

Continued fractions of reciprocals

The continued fraction representations of a positive rational number and its 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 $[a_{0};a_{1},a_{2},\ldots ,a_{n}]$ and $[0;a_{0},a_{1},\ldots ,a_{n}]$ are reciprocals. For instance if $a$ is an integer and $x<1$ then

x=0+{\frac {1}{a+{\frac {1}{b}}}}

and

{\frac {1}{x}}=a+{\frac {1}{b}}

.

If $x>1$ then

x=a+{\frac {1}{b}}

and

{\frac {1}{x}}=0+{\frac {1}{a+{\frac {1}{b}}}}

.

The last number that generates the remainder of the continued fraction is the same for both $x$ and its reciprocal.

For example,

2.25={\frac {9}{4}}=[2;4]

and

{\frac {1}{2.25}}={\frac {4}{9}}=[0;2,4]

.

Infinite continued fractions and convergents

Every infinite continued fraction is 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.^[7]^[8] 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 $[a 0; a 1, a 2, \dots]$ , the first four convergents (numbered 0 through 3) are

⁠ a 0 / 1 ⁠, ⁠ a 1 a 0 + 1 / a 1 ⁠, ⁠ a 2 (a 1 a 0 + 1) + a 0 / a 2 a 1 + 1 ⁠, ⁠ 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 ⁠

In words, 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 polynomials called continuants.

If successive convergents are found, with numerators $h$ ₁, $h$ ₂, … and denominators $k$ ₁, $k$ ₂, … then the relevant recursive relation is:

h n = a n h n - 1 + h n - 2

,

k n = a n k n - 1 + k n - 2

.

The successive convergents are given by the formula

⁠ h n / k n ⁠ = ⁠ a n h n - 1 + h n - 2 / a n k n - 1 + k 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 ⁰⁄₁ and ¹⁄₀. For example, here are the convergents for [0;1,5,2,2].

$n$	−2	−1	0	1	2	3	4
$a n$			0	1	5	2	2
$h n$	0	1	0	1	5	11	27
$k n$	1	0	1	1	6	13	32

When using the 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, … , $2 k -1$ , ... For example, the continued fraction expansion for √3 is [1;1,2,1,2,1,2,1,2,…]. Comparing the convergents with the approximants derived from the Babylonian method:

$n$	−2	−1	0	1	2	3	4	5	6	7
$a n$			1	1	2	1	2	1	2	1
$h n$	0	1	1	2	5	7	19	26	71	97
$k n$	1	0	1	1	3	4	11	15	41	56

x 0 = 1 = ⁠ 1 / 1 ⁠

x 1 = ⁠ 1 / 2 ⁠ (1 + ⁠ 3 / 1 ⁠) = ⁠ 2 / 1 ⁠ = 2

x 2 = ⁠ 1 / 2 ⁠ (2 + ⁠ 3 / 2 ⁠) = ⁠ 7 / 4 ⁠

x 3 = ⁠ 1 / 2 ⁠ (⁠ 7 / 4 ⁠ + ⁠ 3 / ⁠ 7 / 4 ⁠ ⁠) = ⁠ 97 / 56 ⁠

Properties

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 usual topology on the reals). The infinite continued fraction also provides a map between the quadratic irrationals and the dyadic rationals, 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 Mobius transformations having integer values in the transform. Roughly speaking, continued fraction convergents can be taken to be Mobius transformations acting on the (hyperbolic) upper half-plane; this is what leads to the fractal self-symmetry.

Some useful theorems

If $a_{0}$ , $a_{1}$ , $a_{2}$ , $\ldots$ is an infinite sequence of positive integers, define the sequences $h_{n}$ and $k_{n}$ recursively:

$h_{n}=a_{n}h_{n-1}+h_{n-2}$			$h_{-1}=1$		$h_{-2}=0$
$k_{n}=a_{n}k_{n-1}+k_{n-2}$			$k_{-1}=0$		$k_{-2}=1$

Theorem 1. For any positive real number $z$
$\left[a_{0};a_{1},\,\dots ,a_{n-1},z\right]={\frac {zh_{n-1}+h_{n-2}}{zk_{n-1}+k_{n-2}}}.$

Theorem 2. The convergents of [ $a_{0}$ ; $a_{1}$ , $a_{2}$ , $\ldots$ ] are given by
$\left[a_{0};a_{1},\,\dots ,a_{n}\right]={\frac {h_{n}}{k_{n}}}.$

Theorem 3. If the $n$ th convergent to a continued fraction is $h_{n}$ / $k_{n}$ , then
$k_{n}h_{n-1}-k_{n-1}h_{n}=(-1)^{n}.$

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

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

{\frac {h_{n}}{k_{n}}}-{\frac {h_{n-1}}{k_{n-1}}}={\frac {h_{n}k_{n-1}-k_{n}h_{n-1}}{k_{n}k_{n-1}}}={\frac {(-1)^{n+1}}{k_{n}k_{n-1}}}.

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

a_{0}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{k_{n}k_{n+1}}}.

Corollary 4: The matrix

{\begin{bmatrix}h_{n}&h_{n-1}\\k_{n}&k_{n-1}\end{bmatrix}}

has determinant plus or minus one, and thus belongs to the group of $2\times 2$ unimodular matrices $\mathrm {GL} (2,Z)$ .

Theorem 4. Each ( $s$ th) convergent is nearer to a subsequent ( $n$ th) convergent than any preceding ( $r$ th) convergent is. In symbols, if the $n$ th convergent is taken to be $[a_{0};a_{1},\ldots ,a_{n}]=x_{n}$ , then
$\left|x_{r}-x_{n}\right|>\left|x_{s}-x_{n}\right|$
for all $r<s<n$ .

Corollary 1: The even convergents (before the $n$ th) continually increase, but are always less than $x_{n}$ .

Corollary 2: The odd convergents (before the $n$ th) continually decrease, but are always greater than $x_{n}$ .

Theorem 5.
${\frac {1}{k_{n}(k_{n+1}+k_{n})}}<\left|x-{\frac {h_{n}}{k_{n}}}\right|<{\frac {1}{k_{n}k_{n+1}}}.$

Corollary 1: Any convergent is nearer to the continued fraction than any other fraction whose denominator is less than that of the convergent.

Corollary 2: Any convergent which immediately precedes a large quotient is a near approximation to the continued fraction.

Semiconvergents

If

{\frac {h_{n-1}}{k_{n-1}}},{\frac {h_{n}}{k_{n}}}

are consecutive convergents, then any fractions of the form

{\frac {h_{n-1}+mh_{n}}{k_{n-1}+mk_{n}}},

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

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

Best rational approximations

One can choose to define a best rational approximation to a real number $x$ as a rational number ⁠ $n$ / $d$ ⁠, $d > 0$ , that is closer to $x$ than any approximation with a smaller or equal denominator. The simple continued fraction for $x$ generates all of the best rational approximations for $x$ according to three rules:

Truncate the continued fraction, and possibly reduce its last term.
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.

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]
Rational approximation	1	⁠3/4⁠	⁠4/5⁠	⁠5/6⁠	⁠11/13⁠	⁠16/19⁠	⁠27/32⁠
Decimal equivalent	1	0.75	0.8	~0.83333	~0.84615	~0.84211	0.84375
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 is that when $a$ _$k$ is even, the halved term $a$ _$k$/2 is admissible if and only if $| x - [a 0; a 1, \dots, a k - 1]| > | x - [a 0; a 1, \dots, a k - 1, a k /2]|$ ^[9] This is equivalent^[9] to:^[10]

[a k; a k - 1, \dots, a 1] > [a k; a k + 1, \dots]

.

The convergents to $x$ are best approximations in an even stronger sense: $n$ / $d$ is a convergent for $x$ if and only if $| dx - n |$ is the least among all approximations $m$ / $c$ with $c \leq d$ ; that is, we have $| dx - n | < | cx - m |$ so long as $c < d$ . (Note also that $| d k x - n k | \to 0$ as $k \to \infty$ .)

Best rational within an interval

A rational that falls within the interval $(x, y)$ , for $0 < x < y$ , can be found with the continued fractions for $x$ and $y$ . When both $x$ and $y$ are irrational and

x = [a 0; a 1, a 2, \dots, a k - 1, a k, a k + 1, \dots]

y = [a 0; a 1, a 2, \dots, a k - 1, b k, b k + 1, \dots]

where $x$ and $y$ have identical continued fraction expansions up through $a k -1$ , a rational that falls within the interval $(x, y)$ is given by the finite continued fraction,

z (x, y) = [a 0; a 1, a 2, \dots, a k - 1, min(a k, b k) + 1]

This rational will be best in the sense that no other rational in $(x, y)$ will have a smaller numerator or a smaller denominator.

If $x$ is rational, it will have two continued fraction representations that are finite, $x 1$ and $x 2$ , and similarly a rational $y$ will have two representations, $y 1$ and $y 2$ . The coefficients beyond the last in any of these representations should be interpreted as $+\infty$ ; and the best rational will be one of $z (x 1, y 1)$ , $z (x 1, y 2)$ , $z (x 2, y 1)$ , or $z (x 2, y 2)$ .

For example, the decimal representation 3.1416 could be rounded from any number in the interval $[3.14155, 3.14165]$ . The continued fraction representations of 3.14155 and 3.14165 are

3.14155 = [3; 7, 15, 2, 7, 1, 4, 1, 1] = [3; 7, 15, 2, 7, 1, 4, 2]

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

[3; 7, 16] = ⁠ 355 / 113 ⁠ = 3.1415929....

Thus, ⁠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,

z = [a 0; a 1, \dots, a k - 1, a k, 1] = [a 0; a 1, \dots, a k - 1, a 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

x = [a 0; a 1, \dots, a k - 1, a k, 2]

and

y = [a 0; a 1, \dots, a k - 1, a k + 2]

The numbers $x$ and $y$ are formed by incrementing the last coefficient in the two representations for $z$ . It is the case that $x < y$ when $k$ is even, and $x > y$ when $k$ is odd.

For example, the number ⁠355/113⁠ has the continued fraction representations

⁠355113⁠ = [3; 7, 15, 1] = [3; 7, 16]

and thus ⁠355/113⁠ is a convergent of any number strictly between

$[3; 7, 15, 2]$	=	$⁠ 688 / 219 ⁠ \approx 3.1415525$
$[3; 7, 17]$	=	$⁠ 377 / 120 ⁠ \approx 3.1416667$

Comparison of continued fractions

Consider $x = [a 0; a 1, \dots]$ and $y = [b 0; b 1, \dots]$ . If $k$ is the smallest index for which $a k$ is unequal to $b k$ then $x < y$ if $(-1) k (a k - b k) < 0$ and $y < x$ otherwise.

If there is no such $k$ , but one expansion is shorter than the other, say $x = [a 0; a 1, \dots, a n]$ and $y = [b 0; b 1, \dots, b n, b n + 1, \dots]$ with $a i = b i$ for $0 \leq i \leq n$ , then $x < y$ if $n$ is even and $y < x$ if $n$ is odd.

Continued fraction expansions of $π$

To calculate the convergents of $π$ we may set $a 0 = ⌊ π ⌋ = 3$ , define $u 1 = ⁠ 1 / π - 3 ⁠ \approx 7.0625$ and $a 1 = ⌊ u 1 ⌋ = 7$ , $u 2 = ⁠ 1 / u 1 - 7 ⁠ \approx 15.9966$ and $a 2 = ⌊ u 2 ⌋ = 15$ , $u 3 = ⁠ 1 / u 2 - 15 ⁠ \approx 1.0034$ . Continuing like this, one can determine the infinite continued fraction of $π$ as

[3;7,15,1,292,1,1,…] (sequence A001203 in the OEIS).

The fourth convergent of $π$ is [3;7,15,1] = ⁠355/113⁠ = 3.14159292035..., sometimes called Milü, which is fairly close to the true value of $π$ .

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, ⁠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, ⁠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 (22 × 15 = 330) + 3 = 333, and for our denominator, (7 × 15 = 105) + 1 = 106. The third convergent, therefore, is ⁠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:

⁠31⁠, ⁠227⁠, ⁠333106⁠, ⁠355113⁠, ….

These convergents are alternately smaller and larger than the true value of $π$ , and approach nearer and nearer to $π$ . The difference between a given convergent and $π$ is less than the reciprocal of the product of the denominators of that convergent and the next convergent. For example, the fraction ⁠22/7⁠ is greater than $π$ , but ⁠22/7⁠ − $π$ is less than ⁠1/7 × 106⁠ = ⁠1/742⁠ (in fact, ⁠22/7⁠ − $π$ is just more than ⁠1/791⁠ = ⁠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 ⁠22/7⁠ and ⁠3/1⁠ is ⁠1/7⁠, in excess; between ⁠333/106⁠ and ⁠22/7⁠, ⁠1/742⁠, in deficit; between ⁠355/113⁠ and ⁠333/106⁠, ⁠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:

⁠31⁠ + ⁠11 × 7⁠ − ⁠17 × 106⁠ + ⁠1106 × 113⁠ − …

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

Generalized continued fraction

A generalized continued fraction is an expression of the form

x=b_{0}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+{\cfrac {a_{4}}{b_{4}+\ddots \,}}}}}}}}

where the a_n (n > 0) are the partial numerators, the b_n are the partial denominators, and the leading term b₀ 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 $π$ does not show any obvious pattern:

\pi =[3;7,15,1,292,1,1,1,2,1,3,1,\ldots ]

or

\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 }}}}}}}}}}}}}}}}}}}}}}

However, several generalized continued fractions for $π$ have a perfectly regular structure, such as:

\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 }}}}}}}}}}

\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\cdot 2}{1+{\cfrac {2\cdot 3}{1+{\cfrac {3\cdot 4}{1+\ddots }}}}}}}}

\displaystyle \pi =2+{\cfrac {4}{3+{\cfrac {1\cdot 3}{4+{\cfrac {3\cdot 5}{4+{\cfrac {5\cdot 7}{4+\ddots }}}}}}}}

The first two of these are special cases of the arctangent function with $π$ = 4 arctan (1).

\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 }}}}}}}}}}

The above continued fraction of pi consisting of cubes uses the Nilakantha series and an exploit from Leonhard Euler.^[11]

Other continued fraction expansions

Periodic continued fractions

The numbers with periodic continued fraction expansion are precisely the irrational solutions of quadratic equations 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 √2 = [1;2,2,2,2,…], while √14 = [3;1,2,1,6,1,2,1,6…] and √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 √2) or 1,2,1 (for √14), followed by the double of the leading integer.

A property of the golden ratio φ

Because the continued fraction expansion for 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^[12] states that any real number $k$ can be approximated by infinitely many rational ⁠m/n⁠ with

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

While virtually all real numbers $k$ will eventually have infinitely many convergents ⁠m/n⁠ whose distance from $k$ is significantly smaller than this limit, the convergents for φ (i.e., the numbers ⁠5/3⁠, ⁠8/5⁠, ⁠13/8⁠, ⁠21/13⁠, etc.) consistently "toe the boundary", keeping a distance of almost exactly ${\scriptstyle {1 \over n^{2}{\sqrt {5}}}}$ away from φ, thus never producing an approximation nearly as impressive as, for example, ⁠355/113⁠ for $π$ . It can also be shown that every real number of the form ⁠a + bφ/c + dφ⁠, where $a$ , $b$ , $c$ , and $d$ are integers such that $a d - b c = \pm1$ , 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 $π$ , there is one for $e$ , the base of the natural logarithm:

e=e^{1}=[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,1,\dots ],

which is a special case of this general expression for positive integer $n$ :

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

Another, more complex pattern appears in this continued fraction expansion for positive odd $n$ :

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]\,\!,

with a special case for $n = 1$ :

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 ]\,\!.

Other continued fractions of this sort are

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

where $n$ is a positive integer; also, for integral $n$ :

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

with a special case for $n = 1$ :

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

If $I n (x)$ is the modified, or hyperbolic, Bessel function of the first kind, we may define a function on the rationals ⁠p/q⁠ by

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

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

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

with similar formulas for negative rationals; in particular we have

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

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, Khinchin proved that for almost all real numbers $x$ , the $a i$ (for $i = 1, 2, 3, \dots$ ) have an astonishing property: their geometric mean is a constant (known as Khinchin's constant, $K \approx 2.6854520010\dots$ ) independent of the value of $x$ . Paul Lévy showed that the $n$ th root of the denominator of the $n$ 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 $n$ th convergent of the continued fraction expansion of almost all real numbers determines the number to an average accuracy of just over $n$ decimal places.

Applications

Square roots

Generalized continued fractions are used in a method for computing square roots.

The identity

{\sqrt {x}}=1+{\frac {x-1}{1+{\sqrt {x}}}}

(1)

leads via recursion to the generalized continued fraction for any square root:^[13]

{\sqrt {x}}=1+{\cfrac {x-1}{2+{\cfrac {x-1}{2+{\cfrac {x-1}{2+{\ddots }}}}}}}

(2)

Pell's equation

Continued fractions play an essential role in the solution of Pell's equation. For example, for positive integers $p$ and $q$ , and non-square $n$ , it is true that $p 2 - nq 2 = \pm1$ if and only if $⁠ p / q ⁠$ is a convergent of the regular continued fraction for √ $n$ .

Dynamical systems

Continued fractions also play a role in the study of dynamical systems, where they tie together the 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 $h (x) = 1/ x - ⌊1/ x ⌋$ called the Gauss map, which lops off digits of a continued fraction expansion: $h ([0; a 1, a 2, a 3, \dots]) = [0; a 2, a 3, \dots]$ . 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.^[14]

Examples of rational and irrational numbers

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

An infinite continued fraction is defined by the sequences

\{a_{i}\},\{b_{i}\}

, for

i=0,1,2,\ldots

, with

a_{0}=0

.

A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, the continued fraction is finite or infinite.

Different fields of mathematics have different terminology and notation for continued fraction. In number theory the standard unqualified use of the term continued fraction refers to the special case where all numerators are 1, and is treated in the article Simple continued fraction. The present article treats the case where numerators and denominators are sequences $\{a_{i}\},\{b_{i}\}$ of constants or functions. From the perspective of number theory, these are called generalized continued fraction. From the perspective of complex analysis or numerical analysis, however, they are just standard, and in the present article they will simply be called "continued fraction".

Formulation

A continued fraction is an expression of the form

x=b_{0}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+{\cfrac {a_{4}}{b_{4}+\ddots \,}}}}}}}}

where the $a n$ ( $n > 0$ ) are the partial numerators, the $b n$ are the partial denominators, and the leading term $b 0$ is called the integer part of the continued fraction.

The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas:

{\begin{aligned}x_{0}&={\frac {A_{0}}{B_{0}}}=b_{0},\\[4px]x_{1}&={\frac {A_{1}}{B_{1}}}={\frac {b_{1}b_{0}+a_{1}}{b_{1}}},\\[4px]x_{2}&={\frac {A_{2}}{B_{2}}}={\frac {b_{2}(b_{1}b_{0}+a_{1})+a_{2}b_{0}}{b_{2}b_{1}+a_{2}}},\ \dots \end{aligned}}

where $A n$ is the numerator and $B n$ is the denominator, called continuants,^[15]^[16] of the $n$ th convergent. They are given by the three-term recurrence relation ^[17]

{\begin{aligned}A_{n}&=b_{n}A_{n-1}+a_{n}A_{n-2},\\B_{n}&=b_{n}B_{n-1}+a_{n}B_{n-2}\qquad {\text{for }}n\geq 1\end{aligned}}

with initial values

{\begin{aligned}A_{-1}&=1,&A_{0}&=b_{0},\\B_{-1}&=0,&B_{0}&=1.\end{aligned}}

If the sequence of convergents ${x n}$ approaches a limit, the continued fraction is convergent and has a definite value. If the sequence of convergents never approaches a limit, the continued fraction is divergent. It may diverge by oscillation (for example, the odd and even convergents may approach two different limits), or it may produce an infinite number of zero denominators $B n$ .

History

The story of continued fractions begins with the Euclidean algorithm,^[18] a procedure for finding the greatest common divisor of two natural numbers $m$ and $n$ . That algorithm introduced the idea of dividing to extract a new remainder – and then dividing by the new remainder repeatedly.

Nearly two thousand years passed before Bombelli (1579) devised a technique for approximating the roots of quadratic equations with continued fractions in the mid-sixteenth century. Now the pace of development quickened. Just 24 years later, in 1613, Pietro Cataldi introduced the first formal notation for the generalized continued fraction.^[19] Cataldi represented a continued fraction as

{a_{0}\cdot }\,\&\,{\frac {n_{1}}{d_{1}\cdot }}\,\&\,{\frac {n_{2}}{d_{2}\cdot }}\,\&\,{\frac {n_{3}}{d_{3}}}

with the dots indicating where the next fraction goes, and each $&$ representing a modern plus sign.

Late in the seventeenth century John Wallis introduced the term "continued fraction" into mathematical literature.^[20] New techniques for mathematical analysis (Newton's and Leibniz's calculus) had recently come onto the scene, and a generation of Wallis' contemporaries put the new phrase to use.

In 1748 Euler published a theorem showing that a particular kind of continued fraction is equivalent to a certain very general infinite series.^[21] Euler's continued fraction formula is still the basis of many modern proofs of convergence of continued fractions.

In 1761, Johann Heinrich Lambert gave the first proof that $π$ is irrational, by using the following continued fraction for $tan x$ :^[22]

\tan(x)={\cfrac {x}{1+{\cfrac {-x^{2}}{3+{\cfrac {-x^{2}}{5+{\cfrac {-x^{2}}{7+{}\ddots }}}}}}}}

Continued fractions can also be applied to problems in number theory, and are especially useful in the study of Diophantine equations. In the late eighteenth century Lagrange used continued fractions to construct the general solution of Pell's equation, thus answering a question that had fascinated mathematicians for more than a thousand years.^[23] Lagrange's discovery implies that the canonical continued fraction expansion of the square root of every non-square integer is periodic and that, if the period is of length $p > 1$ , it contains a palindromic string of length $p - 1$ .

In 1813 Gauss derived from complex-valued hypergeometric functions what is now called Gauss's continued fractions.^[24] They can be used to express many elementary functions and some more advanced functions (such as the Bessel functions), as continued fractions that are rapidly convergent almost everywhere in the complex plane.

Notation

The long continued fraction expression displayed in the introduction is easy for an unfamiliar reader to interpret. However, it takes up a lot of space and can be difficult to typeset. So mathematicians have devised several alternative notations. One convenient way to express a generalized continued fraction sets each nested fraction on the same line, indicating the nesting by dangling plus signs in the denominators:

x=b_{0}+{\frac {a_{1}}{b_{1}+}}\,{\frac {a_{2}}{b_{2}+}}\,{\frac {a_{3}}{b_{3}+\cdots }}

Sometimes the plus signs are typeset to vertically align with the denominators but not under the fraction bars:

x=b_{0}+{\frac {a_{1}}{b_{1}}}{{} \atop +}{\frac {a_{2}}{b_{2}}}{{} \atop +}{\frac {a_{3}}{b_{3}}}{{} \atop \!{}+\cdots }

Pringsheim wrote a generalized continued fraction this way:

x=b_{0}+{{} \atop {{\big |}\!}}\!{\frac {a_{1}}{\,b_{1}\,}}\!{{\!{\big |}} \atop {}}+{{} \atop {{\big |}\!}}\!{\frac {a_{2}}{\,b_{2}\,}}\!{{\!{\big |}} \atop {}}+{{} \atop {{\big |}\!}}\!{\frac {a_{3}}{\,b_{3}\,}}\!{{\!{\big |}} \atop {}}+\cdots

Carl Friedrich Gauss evoked the more familiar infinite product $Π$ when he devised this notation:

x=b_{0}+{\underset {i=1}{\overset {\infty }{\operatorname {K} }}}{\frac {a_{i}}{b_{i}}}.\,

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.

Some elementary considerations

Here are some elementary results that are of fundamental importance in the further development of the analytic theory of continued fractions.

Partial numerators and denominators

If one of the partial numerators $a n +1$ is zero, the infinite continued fraction

b_{0}+{\underset {i=1}{\overset {\infty }{\operatorname {K} }}}{\frac {a_{i}}{b_{i}}}\,

is really just a finite continued fraction with $n$ fractional terms, and therefore a rational function of $a 1$ to $a n$ and $b 0$ to $b n +1$ . Such an object is of little interest from the point of view adopted in mathematical analysis, so it is usually assumed that all $a i \neq 0$ . There is no need to place this restriction on the partial denominators $b i$ .

The determinant formula

When the $n$ th convergent of a continued fraction

x_{n}=b_{0}+{\underset {i=1}{\overset {n}{\operatorname {K} }}}{\frac {a_{i}}{b_{i}}}\,

is expressed as a simple fraction $x n = ⁠ A n / B n ⁠$ we can use the determinant formula

A_{n-1}B_{n}-A_{n}B_{n-1}=\left(-1\right)^{n}a_{1}a_{2}\cdots a_{n}=\prod _{i=1}^{n}(-a_{i})

(1)

to relate the numerators and denominators of successive convergents $x n$ and $x n - 1$ to one another. The proof for this can be easily seen by induction.

Proof

Base case

The case

n = 1

results from a very simple computation.

Inductive step

Assume that (1) holds for

n - 1

. Then we need to see the same relation holding true for

n

. Substituting the value of

A n

and

B n

in (1) we obtain:

{\begin{aligned}&=b_{n}A_{n-1}B_{n-1}+a_{n}A_{n-1}B_{n-2}-b_{n}A_{n-1}B_{n-1}-a_{n}A_{n-2}B_{n-1}\\&=a_{n}(A_{n-1}B_{n-2}-A_{n-2}B_{n-1})\end{aligned}}

which is true because of our induction hypothesis.

A_{n-1}B_{n}-A_{n}B_{n-1}=\left(-1\right)^{n}a_{1}a_{2}\cdots a_{n}=\prod _{i=1}^{n}(-a_{i})\,

Specifically, if neither

B n

nor

B n - 1

is zero (

n > 0

) we can express the difference between the

(n - 1)

th and

n

th convergents like this:

x_{n-1}-x_{n}={\frac {A_{n-1}}{B_{n-1}}}-{\frac {A_{n}}{B_{n}}}=\left(-1\right)^{n}{\frac {a_{1}a_{2}\cdots a_{n}}{B_{n}B_{n-1}}}={\frac {\prod _{i=1}^{n}(-a_{i})}{B_{n}B_{n-1}}}.\,

The equivalence transformation

If ${c i} = {c 1, c 2, c 3, ...}$ is any infinite sequence of non-zero complex numbers we can prove, by induction, that

b_{0}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+{\cfrac {a_{4}}{b_{4}+\ddots \,}}}}}}}}=b_{0}+{\cfrac {c_{1}a_{1}}{c_{1}b_{1}+{\cfrac {c_{1}c_{2}a_{2}}{c_{2}b_{2}+{\cfrac {c_{2}c_{3}a_{3}}{c_{3}b_{3}+{\cfrac {c_{3}c_{4}a_{4}}{c_{4}b_{4}+\ddots \,}}}}}}}}

where equality is understood as equivalence, which is to say that the successive convergents of the continued fraction on the left are exactly the same as the convergents of the fraction on the right.

The equivalence transformation is perfectly general, but two particular cases deserve special mention. First, if none of the $a i$ are zero, a sequence ${c i}$ can be chosen to make each partial numerator a 1:

b_{0}+{\underset {i=1}{\overset {\infty }{\operatorname {K} }}}{\frac {a_{i}}{b_{i}}}=b_{0}+{\underset {i=1}{\overset {\infty }{\operatorname {K} }}}{\frac {1}{c_{i}b_{i}}}\,

where $c 1 = ⁠ 1 / a 1 ⁠$ , $c 2 = ⁠ a 1 / a 2 ⁠$ , $c 3 = ⁠ a 2 / a 1 a 3 ⁠$ , and in general $c n +1 = ⁠ 1 / a n +1 c n ⁠$ .

Second, if none of the partial denominators $b i$ are zero we can use a similar procedure to choose another sequence ${d i}$ to make each partial denominator a 1:

b_{0}+{\underset {i=1}{\overset {\infty }{\operatorname {K} }}}{\frac {a_{i}}{b_{i}}}=b_{0}+{\underset {i=1}{\overset {\infty }{\operatorname {K} }}}{\frac {d_{i}a_{i}}{1}}\,

where $d 1 = ⁠ 1 / b 1 ⁠$ and otherwise $d n +1 = ⁠ 1 / b n b n +1 ⁠$ .

These two special cases of the equivalence transformation are enormously useful when the general convergence problem is analyzed.

Notions of convergence

As mentioned in the introduction, the continued fraction

x=b_{0}+{\underset {i=1}{\overset {\infty }{\operatorname {K} }}}{\frac {a_{i}}{b_{i}}}\,

converges if the sequence of convergents ${x n$ } tends to a finite limit. This notion of convergence is very natural, but it is sometimes too restrictive. It is therefore useful to introduce the notion of general convergence of a continued fraction. Roughly speaking, this consists in replacing the $\operatorname {K} _{i=n}^{\infty }{\tfrac {a_{i}}{b_{i}}}$ part of the fraction by $w n$ , instead of by 0, to compute the convergents. The convergents thus obtained are called modified convergents. We say that the continued fraction converges generally if there exists a sequence $\{w_{n}^{*}\}$ such that the sequence of modified convergents converges for all $\{w_{n}\}$ sufficiently distinct from $\{w_{n}^{*}\}$ . The sequence $\{w_{n}^{*}\}$ is then called an exceptional sequence for the continued fraction. See Chapter 2 of Lorentzen & Waadeland (1992) for a rigorous definition.

There also exists a notion of absolute convergence for continued fractions, which is based on the notion of absolute convergence of a series: a continued fraction is said to be absolutely convergent when the series

f=\sum _{n}\left(f_{n}-f_{n-1}\right),

where $f_{n}=\operatorname {K} _{i=1}^{n}{\tfrac {a_{i}}{b_{i}}}$ are the convergents of the continued fraction, converges absolutely.^[25] The Śleszyński–Pringsheim theorem provides a sufficient condition for absolute convergence.

Finally, a continued fraction of one or more complex variables is uniformly convergent in an open neighborhood $Ω$ when its convergents converge uniformly on $Ω$ ; that is, when for every $ε > 0$ there exists $M$ such that for all $n > M$ , for all $z\in \Omega$ ,

|f(z)-f_{n}(z)|<\varepsilon .

Even and odd convergents

It is sometimes necessary to separate a continued fraction into its even and odd parts. For example, if the continued fraction diverges by oscillation between two distinct limit points $p$ and $q$ , then the sequence ${x 0, x 2, x 4, ...}$ must converge to one of these, and ${x 1, x 3, x 5, ...}$ must converge to the other. In such a situation it may be convenient to express the original continued fraction as two different continued fractions, one of them converging to $p$ , and the other converging to $q$ .

The formulas for the even and odd parts of a continued fraction can be written most compactly if the fraction has already been transformed so that all its partial denominators are unity. Specifically, if

x={\underset {i=1}{\overset {\infty }{\operatorname {K} }}}{\frac {a_{i}}{1}}\,

is a continued fraction, then the even part $x even$ and the odd part $x odd$ are given by

x_{\text{even}}={\cfrac {a_{1}}{1+a_{2}-{\cfrac {a_{2}a_{3}}{1+a_{3}+a_{4}-{\cfrac {a_{4}a_{5}}{1+a_{5}+a_{6}-{\cfrac {a_{6}a_{7}}{1+a_{7}+a_{8}-\ddots }}}}}}}}\,

and

x_{\text{odd}}=a_{1}-{\cfrac {a_{1}a_{2}}{1+a_{2}+a_{3}-{\cfrac {a_{3}a_{4}}{1+a_{4}+a_{5}-{\cfrac {a_{5}a_{6}}{1+a_{6}+a_{7}-{\cfrac {a_{7}a_{8}}{1+a_{8}+a_{9}-\ddots }}}}}}}}\,

respectively. More precisely, if the successive convergents of the continued fraction $x$ are ${x 1, x 2, x 3, ...}$ , then the successive convergents of $x even$ as written above are ${x 2, x 4, x 6, ...}$ , and the successive convergents of $x odd$ are ${x 1, x 3, x 5, ...}$ .^[26]

Conditions for irrationality

If $a 1, a 2,...$ and $b 1, b 2,...$ are positive integers with $a k \leq b k$ for all sufficiently large $k$ , then

x=b_{0}+{\underset {i=1}{\overset {\infty }{\operatorname {K} }}}{\frac {a_{i}}{b_{i}}}\,

converges to an irrational limit.^[27]

Fundamental recurrence formulas

The partial numerators and denominators of the fraction's successive convergents are related by the fundamental recurrence formulas:

{\begin{aligned}A_{-1}&=1&B_{-1}&=0\\A_{0}&=b_{0}&B_{0}&=1\\A_{n+1}&=b_{n+1}A_{n}+a_{n+1}A_{n-1}&B_{n+1}&=b_{n+1}B_{n}+a_{n+1}B_{n-1}\,\end{aligned}}

The continued fraction's successive convergents are then given by

x_{n}={\frac {A_{n}}{B_{n}}}.\,

These recurrence relations are due to John Wallis (1616–1703) and Leonhard Euler (1707–1783).^[28] These recurrence relations are simply a different notation for the relations obtained by Pietro Antonio Cataldi (1548-1626).

As an example, consider the regular continued fraction in canonical form that represents the golden ratio $φ$ :

x=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots \,}}}}}}}}

Applying the fundamental recurrence formulas we find that the successive numerators $A n$ are ${1, 2, 3, 5, 8, 13, ...}$ and the successive denominators $B n$ are ${1, 1, 2, 3, 5, 8, ...}$ , the Fibonacci numbers. Since all the partial numerators in this example are equal to one, the determinant formula assures us that the absolute value of the difference between successive convergents approaches zero quite rapidly.

Linear fractional transformations

A linear fractional transformation (LFT) is a complex function of the form

w=f(z)={\frac {az+b}{cz+d}},\,

where $z$ is a complex variable, and $a, b, c, d$ are arbitrary complex constants such that $cz + d \neq 0$ . An additional restriction that $ad \neq bc$ is customarily imposed, to rule out the cases in which $w = f (z)$ is a constant. The linear fractional transformation, also known as a Möbius transformation, has many fascinating properties. Four of these are of primary importance in developing the analytic theory of continued fractions.

If $c \neq 0$ the LFT has one or two fixed points. This can be seen by considering the equation

f(z)=z\Rightarrow az+b=cz^{2}+dz\Rightarrow cz^{2}+(d-a)z-b=0,

which is clearly a quadratic equation in

z

. The roots of this equation are the fixed points of

f (z)

. If the discriminant

(d - a) 2 + 4 bc

is zero the LFT fixes a single point; otherwise it has two fixed points.

If $ad \neq bc$ the LFT is an invertible conformal mapping of the extended complex plane onto itself. In other words, this LFT has an inverse function

z=g(w)={\frac {{\phantom {+}}dw-b}{-cw+a}}\,

such that

f (g (z)) = g (f (z)) = z

for every point

z

in the extended complex plane, and both

f

and

g

preserve angles and shapes at vanishingly small scales. From the form of

z = g (w)

we see that

g

is also an LFT.

The composition of two different LFTs for which $ad \neq bc$ is itself an LFT for which $ad \neq bc$ . In other words, the set of all LFTs for which $ad \neq bc$ is closed under composition of functions. The collection of all such LFTs, together with the "group operation" composition of functions, is known as the automorphism group of the extended complex plane.
If $a = 0$ the LFT reduces to

w=f(z)={\frac {b}{cz+d}},\,

which is a very simple meromorphic function of

z

with one simple pole (at

- ⁠ d / c ⁠

) and a residue equal to

⁠ b / c ⁠

. (See also Laurent series.)

The continued fraction as a composition of LFTs

Consider a sequence of simple linear fractional transformations

{\begin{aligned}\tau _{0}(z)&=b_{0}+z,\\[4px]\tau _{1}(z)&={\frac {a_{1}}{b_{1}+z}},\\[4px]\tau _{2}(z)&={\frac {a_{2}}{b_{2}+z}},\\[4px]\tau _{3}(z)&={\frac {a_{3}}{b_{3}+z}},\\&\;\vdots \end{aligned}}

Here we use $τ$ to represent each simple LFT, and we adopt the conventional circle notation for composition of functions. We also introduce a new symbol $Τ n$ to represent the composition of $n + 1$ transformations $τ i$ ; that is,

{\begin{aligned}{\boldsymbol {\mathrm {T} }}_{\boldsymbol {1}}(z)&=\tau _{0}\circ \tau _{1}(z)=\tau _{0}{\big (}\tau _{1}(z){\big )},\\{\boldsymbol {\mathrm {T} }}_{\boldsymbol {2}}(z)&=\tau _{0}\circ \tau _{1}\circ \tau _{2}(z)=\tau _{0}{\Big (}\tau _{1}{\big (}\tau _{2}(z){\big )}{\Big )},\,\end{aligned}}

and so forth. By direct substitution from the first set of expressions into the second we see that

{\begin{aligned}{\boldsymbol {\mathrm {T} }}_{\boldsymbol {1}}(z)&=\tau _{0}\circ \tau _{1}(z)&=&\quad b_{0}+{\cfrac {a_{1}}{b_{1}+z}}\\[4px]{\boldsymbol {\mathrm {T} }}_{\boldsymbol {2}}(z)&=\tau _{0}\circ \tau _{1}\circ \tau _{2}(z)&=&\quad b_{0}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+z}}}}\,\end{aligned}}

and, in general,

{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(z)=\tau _{0}\circ \tau _{1}\circ \tau _{2}\circ \cdots \circ \tau _{n}(z)=b_{0}+{\underset {i=1}{\overset {n}{\operatorname {K} }}}{\frac {a_{i}}{b_{i}}}\,

where the last partial denominator in the finite continued fraction $K$ is understood to be $b n + z$ . And, since $b n + 0 = b n$ , the image of the point $z = 0$ under the iterated LFT $Τ n$ is indeed the value of the finite continued fraction with $n$ partial numerators:

{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(0)={\boldsymbol {\mathrm {T} }}_{\boldsymbol {n+1}}(\infty )=b_{0}+{\underset {i=1}{\overset {n}{\operatorname {K} }}}{\frac {a_{i}}{b_{i}}}.\,

A geometric interpretation

Defining a finite continued fraction as the image of a point under the iterated linear fractional transformation $Τ n (z)$ leads to an intuitively appealing geometric interpretation of infinite continued fractions.

The relationship

x_{n}=b_{0}+{\underset {i=1}{\overset {n}{\operatorname {K} }}}{\frac {a_{i}}{b_{i}}}={\frac {A_{n}}{B_{n}}}={\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(0)={\boldsymbol {\mathrm {T} }}_{\boldsymbol {n+1}}(\infty )\,

can be understood by rewriting $Τ n (z)$ and $Τ n +1 (z)$ in terms of the fundamental recurrence formulas:

{\begin{aligned}{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(z)&={\frac {(b_{n}+z)A_{n-1}+a_{n}A_{n-2}}{(b_{n}+z)B_{n-1}+a_{n}B_{n-2}}}&{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(z)&={\frac {zA_{n-1}+A_{n}}{zB_{n-1}+B_{n}}};\\[6px]{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n+1}}(z)&={\frac {(b_{n+1}+z)A_{n}+a_{n+1}A_{n-1}}{(b_{n+1}+z)B_{n}+a_{n+1}B_{n-1}}}&{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n+1}}(z)&={\frac {zA_{n}+A_{n+1}}{zB_{n}+B_{n+1}}}.\,\end{aligned}}

In the first of these equations the ratio tends toward $⁠ A n / B n ⁠$ as $z$ tends toward zero. In the second, the ratio tends toward $⁠ A n / B n ⁠$ as $z$ tends to infinity. This leads us to our first geometric interpretation. If the continued fraction converges, the successive convergents $⁠ A n / B n ⁠$ are eventually arbitrarily close together. Since the linear fractional transformation $Τ n (z)$ is a continuous mapping, there must be a neighborhood of $z = 0$ that is mapped into an arbitrarily small neighborhood of $Τ n (0) = ⁠ A n / B n ⁠$ . Similarly, there must be a neighborhood of the point at infinity which is mapped into an arbitrarily small neighborhood of $Τ n (\infty) = ⁠ A n -1 / B n -1 ⁠$ . So if the continued fraction converges the transformation $Τ n (z)$ maps both very small $z$ and very large $z$ into an arbitrarily small neighborhood of $x$ , the value of the continued fraction, as $n$ gets larger and larger.

For intermediate values of $z$ , since the successive convergents are getting closer together we must have

{\frac {A_{n-1}}{B_{n-1}}}\approx {\frac {A_{n}}{B_{n}}}\quad \Rightarrow \quad {\frac {A_{n-1}}{A_{n}}}\approx {\frac {B_{n-1}}{B_{n}}}=k\,

where $k$ is a constant, introduced for convenience. But then, by substituting in the expression for $Τ n (z)$ we obtain

{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(z)={\frac {zA_{n-1}+A_{n}}{zB_{n-1}+B_{n}}}={\frac {A_{n}}{B_{n}}}\left({\frac {z{\frac {A_{n-1}}{A_{n}}}+1}{z{\frac {B_{n-1}}{B_{n}}}+1}}\right)\approx {\frac {A_{n}}{B_{n}}}\left({\frac {zk+1}{zk+1}}\right)={\frac {A_{n}}{B_{n}}}\,

so that even the intermediate values of $z$ (except when $z \approx - k -1$ ) are mapped into an arbitrarily small neighborhood of $x$ , the value of the continued fraction, as $n$ gets larger and larger. Intuitively, it is almost as if the convergent continued fraction maps the entire extended complex plane into a single point.^[29]

Notice that the sequence ${Τ n}$ lies within the automorphism group of the extended complex plane, since each $Τ n$ is a linear fractional transformation for which $ab \neq cd$ . And every member of that automorphism group maps the extended complex plane into itself: not one of the $Τ n$ can possibly map the plane into a single point. Yet in the limit the sequence ${Τ n}$ defines an infinite continued fraction which (if it converges) represents a single point in the complex plane.

When an infinite continued fraction converges, the corresponding sequence ${Τ n}$ of LFTs "focuses" the plane in the direction of $x$ , the value of the continued fraction. At each stage of the process a larger and larger region of the plane is mapped into a neighborhood of $x$ , and the smaller and smaller region of the plane that's left over is stretched out ever more thinly to cover everything outside that neighborhood.^[30]

For divergent continued fractions, we can distinguish three cases:

The two sequences ${Τ 2 n -1}$ and ${Τ 2 n}$ might themselves define two convergent continued fractions that have two different values, $x odd$ and $x even$ . In this case the continued fraction defined by the sequence ${Τ n}$ diverges by oscillation between two distinct limit points. And in fact this idea can be generalized: sequences ${Τ n}$ can be constructed that oscillate among three, or four, or indeed any number of limit points. Interesting instances of this case arise when the sequence ${Τ n}$ constitutes a subgroup of finite order within the group of automorphisms over the extended complex plane.
The sequence ${Τ n}$ may produce an infinite number of zero denominators $B i$ while also producing a subsequence of finite convergents. These finite convergents may not repeat themselves or fall into a recognizable oscillating pattern. Or they may converge to a finite limit, or even oscillate among multiple finite limits. No matter how the finite convergents behave, the continued fraction defined by the sequence ${Τ n}$ diverges by oscillation with the point at infinity in this case.^[31]
The sequence ${Τ n}$ may produce no more than a finite number of zero denominators $B i$ . while the subsequence of finite convergents dances wildly around the plane in a pattern that never repeats itself and never approaches any finite limit either.

Interesting examples of cases 1 and 3 can be constructed by studying the simple continued fraction

x=1+{\cfrac {z}{1+{\cfrac {z}{1+{\cfrac {z}{1+{\cfrac {z}{1+\ddots }}}}}}}}\,

where $z$ is any real number such that $z < - ⁠ 1 / 4 ⁠$ .^[32]

Euler's continued fraction formula

Euler proved the following identity:^[21]

a_{0}+a_{0}a_{1}+a_{0}a_{1}a_{2}+\cdots +a_{0}a_{1}a_{2}\cdots a_{n}={\frac {a_{0}}{1-}}{\frac {a_{1}}{1+a_{1}-}}{\frac {a_{2}}{1+a_{2}-}}\cdots {\frac {a_{n}}{1+a_{n}}}.\,

From this many other results can be derived, such as

{\frac {1}{u_{1}}}+{\frac {1}{u_{2}}}+{\frac {1}{u_{3}}}+\cdots +{\frac {1}{u_{n}}}={\frac {1}{u_{1}-}}{\frac {u_{1}^{2}}{u_{1}+u_{2}-}}{\frac {u_{2}^{2}}{u_{2}+u_{3}-}}\cdots {\frac {u_{n-1}^{2}}{u_{n-1}+u_{n}}},\,

and

{\frac {1}{a_{0}}}+{\frac {x}{a_{0}a_{1}}}+{\frac {x^{2}}{a_{0}a_{1}a_{2}}}+\cdots +{\frac {x^{n}}{a_{0}a_{1}a_{2}\ldots a_{n}}}={\frac {1}{a_{0}-}}{\frac {a_{0}x}{a_{1}+x-}}{\frac {a_{1}x}{a_{2}+x-}}\cdots {\frac {a_{n-1}x}{a_{n}+x}}.\,

Euler's formula connecting continued fractions and series is the motivation for the fundamental inequalities^{[link or clarification needed]}, and also the basis of elementary approaches to the convergence problem.

Examples

Transcendental functions and numbers

Here are two continued fractions that can be built via Euler's identity.

e^{x}={\frac {x^{0}}{0!}}+{\frac {x^{1}}{1!}}+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots =1+{\cfrac {x}{1-{\cfrac {1x}{2+x-{\cfrac {2x}{3+x-{\cfrac {3x}{4+x-\ddots }}}}}}}}

\log(1+x)={\frac {x^{1}}{1}}-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots ={\cfrac {x}{1-0x+{\cfrac {1^{2}x}{2-1x+{\cfrac {2^{2}x}{3-2x+{\cfrac {3^{2}x}{4-3x+\ddots }}}}}}}}

Here are additional generalized continued fractions:

\arctan {\cfrac {x}{y}}={\cfrac {xy}{1y^{2}+{\cfrac {(1xy)^{2}}{3y^{2}-1x^{2}+{\cfrac {(3xy)^{2}}{5y^{2}-3x^{2}+{\cfrac {(5xy)^{2}}{7y^{2}-5x^{2}+\ddots }}}}}}}}={\cfrac {x}{1y+{\cfrac {(1x)^{2}}{3y+{\cfrac {(2x)^{2}}{5y+{\cfrac {(3x)^{2}}{7y+\ddots }}}}}}}}

e^{\frac {x}{y}}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+{\cfrac {x^{2}}{18y+\ddots }}}}}}}}}}\quad \Rightarrow \quad e^{2}=7+{\cfrac {2}{5+{\cfrac {1}{7+{\cfrac {1}{9+{\cfrac {1}{11+\ddots }}}}}}}}

\log \left(1+{\frac {x}{y}}\right)={\cfrac {x}{y+{\cfrac {1x}{2+{\cfrac {1x}{3y+{\cfrac {2x}{2+{\cfrac {2x}{5y+{\cfrac {3x}{2+\ddots }}}}}}}}}}}}={\cfrac {2x}{2y+x-{\cfrac {(1x)^{2}}{3(2y+x)-{\cfrac {(2x)^{2}}{5(2y+x)-{\cfrac {(3x)^{2}}{7(2y+x)-\ddots }}}}}}}}

This last is based on an algorithm derived by Aleksei Nikolaevich Khovansky in the 1970s.^[33]

Example: the natural logarithm of 2 (= $[0; 1, 2, 3, 1, 5, ⁠ 2 / 3 ⁠, 7, ⁠ 1 / 2 ⁠, 9, ⁠ 2 / 5 ⁠,..., 2 k - 1, ⁠ 2 / k ⁠,...]$ ≈ 0.693147...):^[34]

\log 2=\log(1+1)={\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{3+{\cfrac {2}{2+{\cfrac {2}{5+{\cfrac {3}{2+\ddots }}}}}}}}}}}}={\cfrac {2}{3-{\cfrac {1^{2}}{9-{\cfrac {2^{2}}{15-{\cfrac {3^{2}}{21-\ddots }}}}}}}}

$π$

Here are three of $π$ 's best-known generalized continued fractions, the first and third of which are derived from their respective arctangent formulas above by setting $x = y = 1$ and multiplying by 4. The Leibniz formula for $π$ :

\pi ={\cfrac {4}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+\ddots }}}}}}}}=\sum _{n=0}^{\infty }{\frac {4(-1)^{n}}{2n+1}}={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+-\cdots

converges too slowly, requiring roughly $3 \times 10 n$ terms to achieve $n$ correct decimal places. The series derived by Nilakantha Somayaji:

\pi =3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+\ddots }}}}}}=3-\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n(n+1)(2n+1)}}=3+{\frac {1}{1\cdot 2\cdot 3}}-{\frac {1}{2\cdot 3\cdot 5}}+{\frac {1}{3\cdot 4\cdot 7}}-+\cdots

is a much more obvious expression but still converges quite slowly, requiring nearly 50 terms for five decimals and nearly 120 for six. Both converge sublinearly to $π$ . On the other hand:

\pi ={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+\ddots }}}}}}}}=4-1+{\frac {1}{6}}-{\frac {1}{34}}+{\frac {16}{3145}}-{\frac {4}{4551}}+{\frac {1}{6601}}-{\frac {1}{38341}}+-\cdots

converges linearly to $π$ , adding at least three digits of precision per four terms, a pace slightly faster than the arcsine formula for $π$ :

\pi =6\sin ^{-1}\left({\frac {1}{2}}\right)=\sum _{n=0}^{\infty }{\frac {3\cdot {\binom {2n}{n}}}{16^{n}(2n+1)}}={\frac {3}{16^{0}\cdot 1}}+{\frac {6}{16^{1}\cdot 3}}+{\frac {18}{16^{2}\cdot 5}}+{\frac {60}{16^{3}\cdot 7}}+\cdots \!

which adds at least three decimal digits per five terms.^[35]

Note: this continued fraction's rate of convergence $μ$ tends to $3 - \sqrt 8 \approx 0.1715729$ , hence $⁠ 1 / μ ⁠$ tends to $3 + \sqrt 8 \approx 5.828427$ , whose common logarithm is $0.7655... \approx ⁠ 13 / 17 ⁠ > ⁠ 3 / 4 ⁠$ . The same $⁠ 1 / μ ⁠ = 3 + \sqrt 8$ (the silver ratio squared) also is observed in the unfolded general continued fractions of both the natural logarithm of 2 and the $n$ th root of 2 (which works for any integer $n > 1$ ) if calculated using $2 = 1 + 1$ . For the folded general continued fractions of both expressions, the rate convergence $μ = (3 - \sqrt 8) 2 = 17 - \sqrt 288 \approx 0.02943725$ , hence $⁠ 1 / μ ⁠ = (3 + \sqrt 8) 2 = 17 + \sqrt 288 \approx 33.97056$ , whose common logarithm is $1.531... \approx ⁠ 26 / 17 ⁠ > ⁠ 3 / 2 ⁠$ , thus adding at least three digits per two terms. This is because the folded GCF folds each pair of fractions from the unfolded GCF into one fraction, thus doubling the convergence pace. The Manny Sardina reference further explains "folded" continued fractions.
Note: Using the continued fraction for $arctan ⁠ x / y ⁠$ cited above with the best-known Machin-like formula provides an even more rapidly, although still linearly, converging expression:

\pi =16\tan ^{-1}{\cfrac {1}{5}}\,-\,4\tan ^{-1}{\cfrac {1}{239}}={\cfrac {16}{u+{\cfrac {1^{2}}{3u+{\cfrac {2^{2}}{5u+{\cfrac {3^{2}}{7u+\ddots }}}}}}}}\,-\,{\cfrac {4}{v+{\cfrac {1^{2}}{3v+{\cfrac {2^{2}}{5v+{\cfrac {3^{2}}{7v+\ddots }}}}}}}}.

with $u = 5$ and $v = 239$ .

Roots of positive numbers

The $n$ th root of any positive number $z m$ can be expressed by restating $z = x n + y$ , resulting in

{\sqrt[{n}]{z^{m}}}={\sqrt[{n}]{\left(x^{n}+y\right)^{m}}}=x^{m}+{\cfrac {my}{nx^{n-m}+{\cfrac {(n-m)y}{2x^{m}+{\cfrac {(n+m)y}{3nx^{n-m}+{\cfrac {(2n-m)y}{2x^{m}+{\cfrac {(2n+m)y}{5nx^{n-m}+{\cfrac {(3n-m)y}{2x^{m}+\ddots }}}}}}}}}}}}

which can be simplified, by folding each pair of fractions into one fraction, to

{\sqrt[{n}]{z^{m}}}=x^{m}+{\cfrac {2x^{m}\cdot my}{n(2x^{n}+y)-my-{\cfrac {(1^{2}n^{2}-m^{2})y^{2}}{3n(2x^{n}+y)-{\cfrac {(2^{2}n^{2}-m^{2})y^{2}}{5n(2x^{n}+y)-{\cfrac {(3^{2}n^{2}-m^{2})y^{2}}{7n(2x^{n}+y)-{\cfrac {(4^{2}n^{2}-m^{2})y^{2}}{9n(2x^{n}+y)-\ddots }}}}}}}}}}.

The square root of $z$ is a special case with $m = 1$ and $n = 2$ :

{\sqrt {z}}={\sqrt {x^{2}+y}}=x+{\cfrac {y}{2x+{\cfrac {y}{2x+{\cfrac {3y}{6x+{\cfrac {3y}{2x+\ddots }}}}}}}}=x+{\cfrac {2x\cdot y}{2(2x^{2}+y)-y-{\cfrac {1\cdot 3y^{2}}{6(2x^{2}+y)-{\cfrac {3\cdot 5y^{2}}{10(2x^{2}+y)-\ddots }}}}}}

which can be simplified by noting that $⁠ 5 / 10 ⁠ = ⁠ 3 / 6 ⁠ = ⁠ 1 / 2 ⁠$ :

{\sqrt {z}}={\sqrt {x^{2}+y}}=x+{\cfrac {y}{2x+{\cfrac {y}{2x+{\cfrac {y}{2x+{\cfrac {y}{2x+\ddots }}}}}}}}=x+{\cfrac {2x\cdot y}{2(2x^{2}+y)-y-{\cfrac {y^{2}}{2(2x^{2}+y)-{\cfrac {y^{2}}{2(2x^{2}+y)-\ddots }}}}}}.

The square root can also be expressed by a periodic continued fraction, but the above form converges more quickly with the proper $x$ and $y$ .

Example 1

The cube root of two (2^1/3 or ³√2 ≈ 1.259921...) can be calculated in two ways:

Firstly, "standard notation" of $x = 1$ , $y = 1$ , and $2 z - y = 3$ :

{\sqrt[{3}]{2}}=1+{\cfrac {1}{3+{\cfrac {2}{2+{\cfrac {4}{9+{\cfrac {5}{2+{\cfrac {7}{15+{\cfrac {8}{2+{\cfrac {10}{21+{\cfrac {11}{2+\ddots }}}}}}}}}}}}}}}}=1+{\cfrac {2\cdot 1}{9-1-{\cfrac {2\cdot 4}{27-{\cfrac {5\cdot 7}{45-{\cfrac {8\cdot 10}{63-{\cfrac {11\cdot 13}{81-\ddots }}}}}}}}}}.

Secondly, a rapid convergence with $x = 5$ , $y = 3$ and $2 z - y = 253$ :

{\sqrt[{3}]{2}}={\cfrac {5}{4}}+{\cfrac {0.5}{50+{\cfrac {2}{5+{\cfrac {4}{150+{\cfrac {5}{5+{\cfrac {7}{250+{\cfrac {8}{5+{\cfrac {10}{350+{\cfrac {11}{5+\ddots }}}}}}}}}}}}}}}}={\cfrac {5}{4}}+{\cfrac {2.5\cdot 1}{253-1-{\cfrac {2\cdot 4}{759-{\cfrac {5\cdot 7}{1265-{\cfrac {8\cdot 10}{1771-\ddots }}}}}}}}.

Example 2

Pogson's ratio (100^1/5 or ⁵√100 ≈ 2.511886...), with $x = 5$ , $y = 75$ and $2 z - y = 6325$ :

{\sqrt[{5}]{100}}={\cfrac {5}{2}}+{\cfrac {3}{250+{\cfrac {12}{5+{\cfrac {18}{750+{\cfrac {27}{5+{\cfrac {33}{1250+{\cfrac {42}{5+\ddots }}}}}}}}}}}}={\cfrac {5}{2}}+{\cfrac {5\cdot 3}{1265-3-{\cfrac {12\cdot 18}{3795-{\cfrac {27\cdot 33}{6325-{\cfrac {42\cdot 48}{8855-\ddots }}}}}}}}.

Example 3

The twelfth root of two (2^1/12 or ¹²√2 ≈ 1.059463...), using "standard notation":

{\sqrt[{12}]{2}}=1+{\cfrac {1}{12+{\cfrac {11}{2+{\cfrac {13}{36+{\cfrac {23}{2+{\cfrac {25}{60+{\cfrac {35}{2+{\cfrac {37}{84+{\cfrac {47}{2+\ddots }}}}}}}}}}}}}}}}=1+{\cfrac {2\cdot 1}{36-1-{\cfrac {11\cdot 13}{108-{\cfrac {23\cdot 25}{180-{\cfrac {35\cdot 37}{252-{\cfrac {47\cdot 49}{324-\ddots }}}}}}}}}}.

Example 4

Equal temperament's perfect fifth (2^7/12 or ¹²√2⁷ ≈ 1.498307...), with $m = 7$ :

With "standard notation":

{\sqrt[{12}]{2^{7}}}=1+{\cfrac {7}{12+{\cfrac {5}{2+{\cfrac {19}{36+{\cfrac {17}{2+{\cfrac {31}{60+{\cfrac {29}{2+{\cfrac {43}{84+{\cfrac {41}{2+\ddots }}}}}}}}}}}}}}}}=1+{\cfrac {2\cdot 7}{36-7-{\cfrac {5\cdot 19}{108-{\cfrac {17\cdot 31}{180-{\cfrac {29\cdot 43}{252-{\cfrac {41\cdot 55}{324-\ddots }}}}}}}}}}.

A rapid convergence with $x = 3$ , $y = -7153$ , and $2 z - y = 2 19 + 3 12$ :

{\sqrt[{12}]{2^{7}}}={\cfrac {1}{2}}{\sqrt[{12}]{3^{12}-7153}}={\cfrac {3}{2}}-{\cfrac {0.5\cdot 7153}{4\cdot 3^{12}-{\cfrac {11\cdot 7153}{6-{\cfrac {13\cdot 7153}{12\cdot 3^{12}-{\cfrac {23\cdot 7153}{6-{\cfrac {25\cdot 7153}{20\cdot 3^{12}-{\cfrac {35\cdot 7153}{6-{\cfrac {37\cdot 7153}{28\cdot 3^{12}-{\cfrac {47\cdot 7153}{6-\ddots }}}}}}}}}}}}}}}}

{\sqrt[{12}]{2^{7}}}={\cfrac {3}{2}}-{\cfrac {3\cdot 7153}{12(2^{19}+3^{12})+7153-{\cfrac {11\cdot 13\cdot 7153^{2}}{36(2^{19}+3^{12})-{\cfrac {23\cdot 25\cdot 7153^{2}}{60(2^{19}+3^{12})-{\cfrac {35\cdot 37\cdot 7153^{2}}{84(2^{19}+3^{12})-\ddots }}}}}}}}.

More details on this technique can be found in General Method for Extracting Roots using (Folded) Continued Fractions.

Higher dimensions

Another meaning for generalized continued fraction is a generalization to higher dimensions. For example, there is a close relationship between the simple continued fraction in canonical form for the irrational real number $α$ , and the way lattice points in two dimensions lie to either side of the line $y = αx$ . Generalizing this idea, one might ask about something related to lattice points in three or more dimensions. One reason to study this area is to quantify the mathematical coincidence idea; for example, for monomials in several real numbers, take the logarithmic form and consider how small it can be. Another reason is to find a possible solution to Hermite's problem.

There have been numerous attempts to construct a generalized theory. Notable efforts in this direction were made by Felix Klein (the Klein polyhedron), Georges Poitou and George Szekeres.

Notes

^ http://www.britannica.com/EBchecked/topic/135043/continued-fraction
^ ^a ^b Pettofrezzo & Byrkit (1970, p. 150)
^ ^a ^b Long (1972, p. 173)
^ ^a ^b Pettofrezzo & Byrkit (1970, p. 152)
^ Weisstein, Eric W. "Periodic Continued Fraction". MathWorld.
^ Collins, Darren C. "Continued Fractions" (PDF). MIT Undergraduate Journal of Mathematics. Archived from the original (PDF) on 2001-11-20. {{cite journal}}: Unknown parameter |dead-url= ignored (|url-status= suggested) (help)
^ Long (1972, p. 177)
^ Pettofrezzo & Byrkit (1970, pp. 162–163)
^ ^a ^b M. Thill (2008), "A more precise rounding algorithm for rational numbers", Computing, 82: 189–198, doi:10.1007/s00607-008-0006-7
^ Shoemake, Ken (1995), "I.4: Rational Approximation", in Paeth, Alan W. (ed.), Graphic Gems V, San Diego, California: Academic Press, pp. 25–31, ISBN 0-12-543455-3
^ Foster, Tony (June 22, 2015). "Theorem of the Day: Theorem no. 203". Robin Whitty. Retrieved June 25, 2015.
^ Theorem 193: Hardy, G.H.; Wright, E.M. (1979). An Introduction to the Theory of Numbers (Fifth ed.). Oxford.
^ Ben Thurston, "Estimating square roots, generalized continued fraction expression for every square root", The Ben Paul Thurston Blog
^ Martin, Richard M. (2004), Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press, p. 557, ISBN 9781139643658.
^ Cusick & Flahive 1989.
^ Chrystal 1999.
^ Jones & Thron 1980, p. 20.
^ Euclid (2008) - The Euclidean algorithm generates a continued fraction as a by-product.
^ Cataldi 1613.
^ Wallis 1699.
^ ^a ^b Euler 1748, Chapter 18.
^ Havil 2012, pp. 104–105.
^ Brahmagupta (598–670) was the first mathematician to make a systematic study of Pell's equation.
^ Gauss 1813.
^ Lorentzen & Waadeland 1992.
^ Oskar Perron derives even more general extension and contraction formulas for continued fractions. See Perron (1977a), Perron (1977b).
^ Angell 2021.
^ Porubský 2008.
^ This intuitive interpretation is not rigorous because an infinite continued fraction is not a mapping: it is the limit of a sequence of mappings. This construction of an infinite continued fraction is roughly analogous to the construction of an irrational number as the limit of a Cauchy sequence of rational numbers.
^ Because of analogies like this one, the theory of conformal mapping is sometimes described as "rubber sheet geometry".
^ One approach to the convergence problem is to construct positive definite continued fractions, for which the denominators $B i$ are never zero.
^ This periodic fraction of period one is discussed more fully in the article convergence problem.
^ An alternative way to calculate log(x)
^ Borwein, Crandall & Fee 2004, p. 278, 280.
^ Beckmann 1971.

References

Angell, David (2010). "A family of continued fractions" (PDF). Journal of Number Theory. 130 (4). Elsevier: 904–911. doi:10.1016/j.jnt.2009.12.003.

Angell, David (2021). Irrationality and Transcendence in Number Theory. Chapman and Hall/CRC. ISBN 9780367628376.

Beckmann, Petr (1971). A History of Pi. St. Martin's Press, Inc. pp. 131–133, 140–143. ISBN 0-88029-418-3.

Bombelli, Rafael (1579). L'algebra.

Borwein, Jonathan Michael; Crandall, Richard E.; Fee, Greg (2004). "On the Ramanujan AGM Fraction, I: The Real-Parameter Case". Experimental Mathematics. 13 (3): 275–285. doi:10.1080/10586458.2004.10504540. S2CID 17758274.

Cataldi, Pietro Antonio (1613). Trattato del modo brevissimo di trovar la radice quadra delli numeri [A treatise on a quick way to find square roots of numbers].

Chrystal, George (1999). Algebra, an Elementary Text-book for the Higher Classes of Secondary Schools and for Colleges: Pt. 1. American Mathematical Society. p. 500. ISBN 0-8218-1649-7.

Cusick, Thomas W.; Flahive, Mary E. (1989). The Markoff and Lagrange Spectra. American Mathematical Society. pp. 89. ISBN 0-8218-1531-8.

Euclid (2008) [300 BC]. "Elements". Clay Mathematics Institute.

Euler, Leonhard (1748). "E101 – Introductio in analysin infinitorum, volume 1". The Euler Archive. Retrieved 2 May 2022.

Gauss, Carl Friedrich (1813). Disquisitiones generales circa seriem infinitam.

Havil, Julian (2012). The Irrationals: A Story of the Numbers You Can't Count On. Princeton University Press. p. 280. ISBN 978-0691143422. JSTOR j.ctt7smdw.

Jones, William B.; Thron, W.J. (1980). Continued fractions. Analytic theory and applications. Encyclopedia of Mathematics and its Applications. Vol. 11. Reading, MA: Addison-Wesley. ISBN 0-201-13510-8. Zbl 0445.30003. (Covers both analytic theory and history.)

Lorentzen, Lisa; Waadeland, Haakon (1992). Continued Fractions with Applications. Reading, MA: North Holland. ISBN 978-0-444-89265-2. (Covers primarily analytic theory and some arithmetic theory.)

Perron, Oskar (1977a) [1954]. Die Lehre von den Kettenbrüchen. Vol. Band I: Elementare Kettenbrüche (3 ed.). Vieweg + Teubner Verlag. ISBN 9783519020219.

Perron, Oskar (1977b) [1954]. Die Lehre von den Kettenbrüchen. Vol. Band II: Analytisch-funktionentheoretische Kettenbrüche (3 ed.). Vieweg + Teubner Verlag. ISBN 9783519020226.

Porubský, Štefan (2008). "Basic definitions for continued fractions". Interactive Information Portal for Algorithmic Mathematics. Prague, Czech Republic: Institute of Computer Science of the Czech Academy of Sciences. Retrieved 2 May 2022.

Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 5.2. Evaluation of Continued Fractions". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8. Archived from the original on 2021-05-06. Retrieved 2011-08-08.

Sardina, Manny (2007). "General Method for Extracting Roots using (Folded) Continued Fractions" (PDF). Surrey (UK).

Szekeres, George (1970). "Multidimensional continued fractions". Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 13: 113–140.

Von Koch, Helge (1895). "Sur un théorème de Stieltjes et sur les fonctions définies par des fractions continues". Bulletin de la Société Mathématique de France. 23: 33–40. doi:10.24033/bsmf.508. JFM 26.0233.01.

Wall, Hubert Stanley (1967). Analytic Theory of Continued Fractions (Reprint ed.). Chelsea Pub Co. ISBN 0-8284-0207-8. (This reprint of the D. Van Nostrand edition of 1948 covers both history and analytic theory.)

Wallis, John (1699). Opera mathematica [Mathematical Works].

External links

The first twenty pages of Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, ISBN 0-521-81805-2, contains generalized continued fractions for √2 and the golden mean.
OEIS sequence A133593 ("Exact" continued fraction for Pi)

History of continued fractions

300 BC 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
1579 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

a_{0}

&

{\frac {n_{1}}{d_{1}\cdot }}

&

{\frac {n_{2}}{d_{2}\cdot }}

&

{\frac {n_{3}}{d_{3}\cdot }}

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 is irrational.^[1]
1748 Euler, 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.^[2]
1761 Johann Lambert – gave the first proof of the irrationality of $π$ using a continued fraction for 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 irrationals expand to periodic continued fractions.
1813 Carl Friedrich Gauss, Werke, Vol. 3, pp. 134–138 – derived a very general complex-valued continued fraction via a clever identity involving the hypergeometric function
1892 Henri Padé defined Padé approximant
1972 Bill Gosper – First exact algorithms for continued fraction arithmetic.

Notes

^ Sandifer, Ed (February 2006). "How Euler Did It: Who proved e is irrational?" (PDF). MAA Online.
^ "E101 – Introductio in analysin infinitorum, volume 1". Retrieved 2008-03-16.

References

Siebeck, H. (1846). "Ueber periodische Kettenbrüche". J. Reine Angew. Math. Vol. 33. pp. 68–70.
Heilermann, J. B. H. (1846). "Ueber die Verwandlung von Reihen in Kettenbrüche". J. Reine Angew. Math. Vol. 33. pp. 174–188.
Magnus, Arne (1962). "Continued fractions associated with the Padé Table". Math. Zeitschr. Vol. 78. pp. 361–374.
Chen, Chen-Fan; Shieh, Leang-San (1969). "Continued fraction inversion by Routh's Algorithm". IEEE Trans. Circ. Theory. Vol. 16, no. 2. pp. 197–202. doi:10.1109/TCT.1969.1082925.
Gragg, William B. (1974). "Matrix interpretations and applications of the continued fraction algorithm". Rocky Mount. J. Math. Vol. 4, no. 2. p. 213. doi:10.1216/RJM-1974-4-2-213.
Jones, William B.; Thron, W. J. (1980). Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications. Vol. 11. Reading. Massachusetts: Addison-Wesley Publishing Company. ISBN 0-201-13510-8.
Khinchin, A. Ya. (1964) [Originally published in Russian, 1935]. Continued Fractions. University of Chicago Press. ISBN 0-486-69630-8.
Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77-171950
Perron, Oskar (1950). Die Lehre von den Kettenbrüchen. New York, NY: Chelsea Publishing Company.
Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766
Rockett, Andrew M.; Szüsz, Peter (1992). Continued Fractions. 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
Cuyt, A.; Brevik Petersen, V.; Verdonk, B.; Waadeland, H.; Jones, W. B. (2008). Handbook of Continued fractions for Special functions. Springer Verlag. ISBN 978-1-4020-6948-2.
Rieger, G. J. (1982). "A new approach to the real numbers (motivated by continued fractions)". Abh. Braunschweig.Wiss. Ges. Vol. 33. pp. 205–217.

External links

"Continued fraction", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
An Introduction to the Continued Fraction
Linas Vepstas Continued Fractions and Gaps (2004) reviews chaotic structures in continued fractions.
Continued Fractions on the Stern-Brocot Tree at cut-the-knot
The Antikythera Mechanism I: Gear ratios and continued fractions
Continued fraction calculator, WIMS.
Continued Fraction Arithmetic Gosper's first continued fractions paper, unpublished. Cached on the Internet Archive's Wayback Machine
Weisstein, Eric W. "Continued Fraction". MathWorld.
Continued Fractions by Stephen Wolfram and Continued Fraction Approximations of the Tangent Function by Michael Trott, Wolfram Demonstrations Project.
OEIS: A133593 Exact Continued Fraction for Pi
A view into "fractional interpolation" of a continued fraction ${1; 1, 1, 1, ...$ }

Warning: Default sort key "Continued Fraction" overrides earlier default sort key "Generalized Continued Fraction".

[1] ttp://www.britannica.com/EBchecked/topic/135043/continued-fraction

[Pettofrezzo_1970_150-2] Pettofrezzo & Byrkit (1970, p. 150)

[Long_1972_173-3] Long (1972, p. 173)

[Pettofrezzo_1970_152-4] Pettofrezzo & Byrkit (1970, p. 152)

[5] Weisstein, Eric W. "Periodic Continued Fraction". MathWorld.

[6] Collins, Darren C. "Continued Fractions" (PDF). MIT Undergraduate Journal of Mathematics. Archived from the original (PDF) on 2001-11-20. {{cite journal}}: Unknown parameter |dead-url= ignored (|url-status= suggested) (help)

[7] Long (1972, p. 177)

[8] Pettofrezzo & Byrkit (1970, pp. 162–163)

[thill-9] M. Thill (2008), "A more precise rounding algorithm for rational numbers", Computing, 82: 189–198, doi:10.1007/s00607-008-0006-7

[10] Shoemake, Ken (1995), "I.4: Rational Approximation", in Paeth, Alan W. (ed.), Graphic Gems V, San Diego, California: Academic Press, pp. 25–31, ISBN 0-12-543455-3

[11] Foster, Tony (June 22, 2015). "Theorem of the Day: Theorem no. 203". Robin Whitty. Retrieved June 25, 2015.

[12] Theorem 193: Hardy, G.H.; Wright, E.M. (1979). An Introduction to the Theory of Numbers (Fifth ed.). Oxford.

[13] Ben Thurston, "Estimating square roots, generalized continued fraction expression for every square root", The Ben Paul Thurston Blog

[14] Martin, Richard M. (2004), Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press, p. 557, ISBN 9781139643658.

[FOOTNOTECusickFlahive1989-15] Cusick & Flahive 1989.

[FOOTNOTEChrystal1999-16] Chrystal 1999.

[FOOTNOTEJonesThron198020-17] Jones & Thron 1980, p. 20.

[18] Euclid (2008) - The Euclidean algorithm generates a continued fraction as a by-product.

[FOOTNOTECataldi1613-19] Cataldi 1613.

[FOOTNOTEWallis1699-20] Wallis 1699.

[FOOTNOTEEuler1748Chapter_18-21] Euler 1748, Chapter 18.

[FOOTNOTEHavil2012104–105-22] Havil 2012, pp. 104–105.

[23] Brahmagupta (598–670) was the first mathematician to make a systematic study of Pell's equation.

[FOOTNOTEGauss1813-24] Gauss 1813.

[FOOTNOTELorentzenWaadeland1992-25] Lorentzen & Waadeland 1992.

[26] Oskar Perron derives even more general extension and contraction formulas for continued fractions. See Perron (1977a), Perron (1977b).

[FOOTNOTEAngell2021-27] Angell 2021.

[FOOTNOTEPorubský2008-28] Porubský 2008.

[29] This intuitive interpretation is not rigorous because an infinite continued fraction is not a mapping: it is the limit of a sequence of mappings. This construction of an infinite continued fraction is roughly analogous to the construction of an irrational number as the limit of a Cauchy sequence of rational numbers.

[30] Because of analogies like this one, the theory of conformal mapping is sometimes described as "rubber sheet geometry".

[31] One approach to the convergence problem is to construct positive definite continued fractions, for which the denominators $B i$ are never zero.

[32] This periodic fraction of period one is discussed more fully in the article convergence problem.

[33] An alternative way to calculate log(x)

[FOOTNOTEBorweinCrandallFee2004278,_280-34] Borwein, Crandall & Fee 2004, p. 278, 280.

[FOOTNOTEBeckmann1971-35] Beckmann 1971.

[sandifer-36] Sandifer, Ed (February 2006). "How Euler Did It: Who proved e is irrational?" (PDF). MAA Online.

[IntroductioI-37] "E101 – Introductio in analysin infinitorum, volume 1". Retrieved 2008-03-16.

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v t e Fractions and ratios
	Division and ratio	Dividend ÷ Divisor = Quotient
	Fraction	⁠Numerator/Denominator⁠ = Quotient
	Algebraic Aspect Binary Continued Decimal Dyadic Egyptian Golden Silver Integer Irreducible Reduction Just intonation LCD Musical interval Paper size Percentage Unit

Motivation and notation

Basic formula

Calculating continued fraction representations

Notations for continued fractions

Finite continued fractions

Continued fractions of reciprocals

Infinite continued fractions and convergents

Properties

Some useful theorems

Semiconvergents

Best rational approximations

Best rational within an interval

Interval for a convergent

Comparison of continued fractions

Continued fraction expansions of π

Generalized continued fraction

Other continued fraction expansions

Periodic continued fractions

A property of the golden ratio φ

Regular patterns in continued fractions

Typical continued fractions

Applications

Square roots

Pell's equation

Dynamical systems

Eigenvalues and eigenvectors

Examples of rational and irrational numbers

Formulation

History

Notation

Some elementary considerations

Partial numerators and denominators

The determinant formula

The equivalence transformation

Notions of convergence

Even and odd convergents

Conditions for irrationality

Fundamental recurrence formulas

Linear fractional transformations

The continued fraction as a composition of LFTs

A geometric interpretation

Euler's continued fraction formula

Examples

Transcendental functions and numbers

π

Roots of positive numbers

Example 1

Example 2

Example 3

Example 4

Higher dimensions

See also

Notes

References

External links

History of continued fractions

See also

Notes

References

External links

Continued fraction expansions of $π$

$π$