Continued fraction

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_i} are called the coefficients or terms of the continued fraction.^[2]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{ \ddots + \cfrac{1}{a_n} }}}}

A finite regular continued fraction, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is a non-negative integer, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0} is an integer, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_i} is a positive integer, for

i=1,\ldots ,n

.

It is generally assumed that the numerator of all of the fractions is 1. If arbitrary values 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.

Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} /Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} ⁠ has two closely related expressions as a finite continued fraction, whose coefficients $a i$ can be determined by applying the Euclidean algorithm to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is the value of a unique infinite regular continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} and 1. This way of expressing real numbers (rational and irrational) is called their continued fraction representation.

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. 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. Now, ⁠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. Further, ⁠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]. (It is customary to replace only the first comma by a semicolon to indicate that the preceding number is the whole part.) 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 applied to the numerator and denominator of the number. In particular, it must terminate and produce a finite continued fraction representation of the number. The sequence of integers that occur in this representation is the sequence of successive quotients computed by the Euclidean algorithm. 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:

$\sqrt 19 = [4;2,1,3,1,2,8,2,1,3,1,2,8,...]$ (sequence A010124 in the OEIS). The pattern repeats indefinitely with a period of 6.
$e = [2;1,2,1,1,4,1,1,6,1,1,8,...]$ (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.
$π = [3;7,15,1,292,1,1,1,2,1,3,1,...]$ (sequence A001203 in the OEIS). No pattern has ever been found in this representation.
Φ = [1;1,1,1,1,1,1,1,1,1,1,1,...] (sequence A000012 in the OEIS). The golden ratio, the irrational number that is the "most difficult" to approximate rationally .
$γ = [0;1,1,2,1,2,1,4,3,13,5,1,...]$ (sequence A002852 in the OEIS). The Euler–Mascheroni constant, which is expected but not known to be irrational, and whose continued fraction has no apparent pattern.

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 real number is finite if and only if it is a rational number. 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...
Every rational number has an essentially unique simple continued fraction representation. Each rational can be represented in exactly two ways, since $[a 0; a 1,... a n -1, a n] = [a 0; a 1,... a n -1,(a n -1),1]$ . Usually the first, shorter one is chosen as the canonical representation.
The simple continued fraction representation of an irrational number is unique. (However, additional representations are possible when using generalized continued fractions; see below.)
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, and 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 (generalized) continued fraction is an expression of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0 + \cfrac{b_1}{a_1 + \cfrac{b_2}{a_2 + \cfrac{b_3}{a_3 + {_\ddots}}}}}

where a_i and b_i can be any complex numbers.

When b_i = 1 for all i the expression is called a simple continued fraction. When the expression contains finitely many terms, it is called a finite continued fraction. When the expression contains infinitely many terms, it is called an infinite continued fraction.^[6] When the terms eventually repeat from some point onwards, the expression is called a periodic continued fraction.^[5]

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

Examples of finite simple continued fractions
Formula	Numeric	Remarks
$\ a_{0}$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 2}	All integers are a degenerate case
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a_0 + \cfrac{1}{a_1}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 2 + \cfrac{1}{3}}	Simplest possible fractional form
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ -3 + \cfrac{1}{2 + \cfrac{1}{18}}}	First integer may be negative
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3}}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \cfrac{1}{15 + \cfrac{1}{1 + \cfrac{1}{102}}}}	First integer may be zero

For simple continued fractions of the form

$r=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{_{\ddots }}}}}}}}$

the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n} term can be calculated using the following recursive formula:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n= \left \lfloor \frac{N_n}{N_{n+1}} \right \rfloor}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_{n+1}=N_{n-1} \bmod N_n } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} N_{0}=r \\ N_{1}=1 \end{cases} }

From which it can be understood that the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n} sequence stops if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_{n+1} = 0} .

Calculating continued fraction representations

Consider a real number ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ⁠. Let $i=\lfloor r\rfloor$ and let ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f = r - i} ⁠. When ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\neq 0} ⁠, the continued fraction representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [i;a_1,a_2,\ldots]} ⁠, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a_1;a_2,\ldots]} is the continued fraction representation of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/f} ⁠. When ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\geq 0} ⁠, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} is the integer part of $r$ , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is the fractional part of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ⁠.

In order to calculate a continued fraction representation of a number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} , write down the floor of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} . Subtract this value from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} . If the difference is 0, stop; otherwise find the reciprocal of the difference and repeat. The procedure will halt if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 ⁠ $3.245=649/200$ ⁠:

Step	Real Number	Integer part	Fractional part	Simplified	Reciprocal of $f$
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = \frac{649}{200}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f = \frac{649}{200} - 3 }	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{49}{200}}	${\frac {1}{f}}={\frac {200}{49}}$
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = \frac{200}{49} }	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f = \frac{200}{49} - 4 }	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{4}{49} }	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{f} = \frac{49}{4} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = \frac{49}{4} }	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 12}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f = \frac{49}{4} - 12 }	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{1}{4} }	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{f} = \frac{4}{1} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = 4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 4 }	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f = 4 - 4 }	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 0 }	STOP

The continued fraction for ⁠Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3.245} ⁠ is thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [3; 4,12,4],} or, expanded:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{649}{200} = 3 + \cfrac{1}{4 + \cfrac{1}{12 + \cfrac{1}{4}}}. }

Notations

The integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1} etc., are called the coefficients or terms of the continued fraction.^[2] Because a continued fraction such as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4}}}}}

takes a significant amount of vertical space, a number of methods have been tried to shrink it.

Gottfried Leibniz sometimes used the notation^[7]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align}x = a_0 + \dfrac{1}{a_1} {{}\atop+} \\[28mu]\ \end{align} \! \begin{align} \dfrac{1}{a_2} {{}\atop+} \\[2mu] \ \end{align} \! \begin{align}\dfrac{1}{a_3} {{}\atop+}\end{align} \! \begin{align}\\[2mu] \dfrac{1}{a_4},\end{align} }

and later the same idea was taken even further with the nested fraction bars drawn aligned, for example by Alfred Pringsheim as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = a_0 + \frac{\ \ 1 \ \mid}{\mid\,a_1} + \frac{\ \ 1\,\mid}{\mid\,a_2} + \frac{\ \ 1\,\mid}{\mid\,a_3} + \frac{\ \ 1\,\mid}{\mid\,a_4},}

or in more common related notations as^[8]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = a_0 + {1 \over a_1 +}\, {1 \over a_2 +}\, {1 \over a_3 +}\, {1 \over a_4} }

or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = a_0 + {1 \over a_1} {{}\atop+} {1 \over a_2} {{}\atop+} {1 \over a_3} {{}\atop+} {1 \over a_4}. }

Carl Friedrich Gauss used a notation reminiscent of summation notation,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = a_0 + \underset{i=1}{\overset{4}{\mathrm K}} ~ \frac{1}{a_i}, }

or in cases where the numerator is always 1, eliminated the fraction bars altogether, writing a list-style

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = [a_0; a_1, a_2, a_3, a_4]. }

Sometimes list-style notation uses angle brackets instead,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = \left\langle a_0; a_1, a_2, a_3, a_4 \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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0} and positive integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1,a_2,\ldots} .^[9]^[10]

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 the other coefficients are 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, ..., a n - 1, a n, 1] = [a 0; a 1, a 2, ..., a n - 1, a n + 1]

.

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

.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a_0;a_1,a_2,\ldots,a_n]} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0;a_0,a_1,\ldots,a_n]} are reciprocals.

For instance if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is an integer and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x < 1} then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0 + \frac{1}{a + \frac{1}{b}}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{x} = a + \frac{1}{b}} .

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x>1} then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = a + \frac{1}{b}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and its reciprocal.

For example,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2.25 = \frac{9}{4} = [2;4]} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2.25} = \frac{4}{9} = [0;2,4]} .

Infinite continued fractions and convergents

Convergents approaching the golden ratio

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.^[11]^[12] 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 $[a 0; a 1, a 2, ...]$ , the first four convergents (numbered 0 through 3) are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{a_0}{1},\, \frac{a_1a_0 + 1}{a_1},\, \frac{a_2(a_1a_0 + 1) + a_0}{a_2a_1 + 1},\, \frac{a_3\bigl(a_2(a_1a_0 + 1) + a_0\bigr) + (a_1a_0 + 1)}{a_3(a_2a_1 + 1) + a_1}. }

The numerator of the third convergent is formed by multiplying the numerator of the second convergent by the third coefficient, 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 that of Gaussian brackets:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} h_n &= a_nh_{n-1} + h_{n-2}, \\[3mu] k_n &= a_nk_{n-1} + k_{n-2}. \end{align}}

The successive convergents are given by the formula

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{h_n}{k_n} = \frac{a_nh_{n-1} + h_{n-2}}{a_nk_{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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt3} 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

A Baire space is a topological space on infinite sequences of natural numbers. The infinite continued fraction provides a homeomorphism from the 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 Möbius transformations 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.

The limit probability distribution of the coefficients in the continued fraction expansion of a random variable uniformly distributed in (0, 1) is the Gauss–Kuzmin distribution.

Some useful theorems

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a_0\ ,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1\ ,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_2\ ,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \ldots\ } is an infinite sequence of positive integers, define the sequences Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ h_n\ } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ k_n\ } recursively:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{n} = a_n\ h_{n-1} + h_{n-2}\ ,}			Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{-1} = 1\ ,}		Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{-2} = 0\ ;}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{n}= a_n\ k_{n-1} + k_{n-2}\ ,}			Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{-1} = 0\ ,}		Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{-2} = 1 ~.}

Theorem 1. For any positive real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ x\ }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\ a_0;\ a_1,\ \dots, a_{n-1}, x\ \right]=\frac{x\ h_{n-1} + h_{n-2}}{\ x\ k_{n-1} + k_{n-2}\ }, \quad \left[\ a_0;\ a_1,\ \dots, a_{n-1} + x\ \right]=\frac{h_{n-1} + xh_{n-2}}{\ k_{n-1} + x k_{n-2}\ }}

Theorem 2. The convergents of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ [\ a_0\ ;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1\ ,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_2\ ,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ldots\ ]\ } are given by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\ a_0;\ a_1,\ \dots, a_n\ \right] = \frac{h_n}{\ k_n\ } ~.}

or in matrix form,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} h_n & h_{n-1} \\ k_n & k_{n-1} \end{bmatrix} = \begin{bmatrix} a_0 & 1 \\ 1 & 0 \end{bmatrix} \cdots \begin{bmatrix} a_n & 1 \\ 1 & 0 \end{bmatrix}}
Theorem 3. If the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ n} th convergent to a continued fraction is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \frac{ h_n}{ k_n }\ ,} then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_n\ h_{n-1} - k_{n-1}\ h_n = (-1)^n\ ,}

or equivalently

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{ h_n }{\ k_n\ } - \frac{ h_{n-1} }{\ k_{n-1}\ } = \frac{ (-1)^{n+1} }{\ k_{n-1}\ k_n\ } ~.}

Corollary 1: Each convergent is in its lowest terms (for if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ h_n\ } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ k_n\ } had a nontrivial common divisor it would divide Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0 + \sum_{n=0}^\infty \frac{ (-1)^n }{\ k_{n}\ k_{n+1}\ } ~.}

Corollary 4: The matrix

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} h_n & h_{n-1} \\ k_n & k_{n-1} \end{bmatrix} = \begin{bmatrix} a_0 & 1 \\ 1 & 0 \end{bmatrix} \cdots \begin{bmatrix} a_n & 1 \\ 1 & 0 \end{bmatrix}}

has determinant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-1)^{n+1}} , and thus belongs to the group of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 2\times 2\ } unimodular matrices Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathrm{GL}(2,\mathbb{Z}) ~.}

Corollary 5: The matrixFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} h_n & h_{n-2} \\ k_n & k_{n-2} \end{bmatrix} = \begin{bmatrix} h_{n-1} & h_{n-2} \\ k_{n-1} & k_{n-2} \end{bmatrix} \begin{bmatrix} a_{n} & 0 \\ 1 & 1 \end{bmatrix}} has determinant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-1)^na_n} , or equivalently,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{ h_n }{\ k_n\ } - \frac{ h_{n-2} }{\ k_{n-2}\ } = \frac{ (-1)^{n} }{\ k_{n-2 }\ k_n\ }a_n} meaning that the odd terms monotonically decrease, while the even terms monotonically increase.

Corollary 6: The denominator sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_0, k_1, k_2, \dots} satisfies the recurrence relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{-1} = 0, k_0 = 1, k_n = k_{n-1}a_n + k_{n-2}} , and grows at least as fast as the Fibonacci sequence, which itself grows like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O(\phi^n)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi= 1.618\dots} is the golden ratio.

Theorem 4. Each (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ s} th) convergent is nearer to a subsequent (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ n} th) convergent than any preceding (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ r} th) convergent is. In symbols, if the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ n} th convergent is taken to be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \left[\ a_0;\ a_1,\ \ldots,\ a_n\ \right] = x_n\ ,} then
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\ x_r - x_n\ \right| > \left|\ x_s - x_n\ \right| }
for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ r < s < n ~.}

Corollary 1: The even convergents (before the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ n} th) continually increase, but are always less than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ x_n ~.}

Corollary 2: The odd convergents (before the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ n} th) continually decrease, but are always greater than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ x_n ~.}

Theorem 5.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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: 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.

Theorem 6: Consider the set of all open intervals with end-points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0;a_1, \dots, a_n], [0;a_1, \dots, a_n+1]} . Denote it as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal C} . Any open subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0, 1] \setminus \Q} is a disjoint union of sets from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal C} .

Corollary: The infinite continued fraction provides a homeomorphism from the Baire space to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0, 1] \setminus \Q} .

Semiconvergents

If

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{h_{n-1}}{k_{n-1}},\frac{h_n}{k_n} }

are consecutive convergents, then any fractions of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{h_{n-1} + mh_n}{k_{n-1} + mk_n},}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} is an integer such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\leq m\leq a_{n+1}} , are called semiconvergents, secondary convergents, or intermediate fractions. The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (m+1)} -st semiconvergent equals the mediant of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} -th one and the convergent Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{h_n}{k_n}} . Sometimes the term is taken to mean that being a semiconvergent excludes the possibility of being a convergent (i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<m<a_{n+1}} ), rather than that a convergent is a kind of semiconvergent.

It follows that semiconvergents represent a monotonic sequence of fractions between the convergents Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{h_{n-1}}{k_{n-1}}} (corresponding to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=0} ) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{h_{n+1}}{k_{n+1}}} (corresponding to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=a_{n+1}} ). The consecutive semiconvergents Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{a}{b}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{c}{d}} satisfy the property Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ad - bc = \pm 1} .

If a rational approximation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{p}{q}} to a real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is such that the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|x-\tfrac{p}{q}\right|} is smaller than that of any approximation with a smaller denominator, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{p}{q}} is a semiconvergent of the continued fraction expansion of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} . The converse is not true, however.

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 $x$ can be used to generate all of the best rational approximations for $x$ 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.

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%

Best rational approximants for

π

(green circle), e (blue diamond), ϕ (pink oblong), (√3)/2 (grey hexagon), 1/√2 (red octagon) and 1/√3 (orange triangle) calculated from their continued fraction expansions, plotted as slopes y/x with errors from their true values (black dashes)

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 $a$ _$k$ is even, the halved term $a$ _$k$/2 is admissible if and only if $| x - [a 0; a 1, ..., a k - 1]| > | x - [a 0; a 1, ..., a k - 1, a k /2]|$ ^[13] This is equivalent ^[13] to: Shoemake (1995).

[a k; a k - 1, ..., a 1] > [a k; a k + 1, ...]

.

The convergents to $x$ are "best approximations" in a much stronger sense than the one defined above. Namely, $n$ / $d$ is a convergent for $x$ if and only if $| dx - n |$ has the smallest value among the analogous expressions for all rational 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, ..., a k - 1, a k, a k + 1, ...]

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

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, ..., 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.^[14]^[15]

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, ..., a k - 1, a k, 1] = [a 0; a 1, ..., a k - 1, a k + 1] = ⁠ p k / q k ⁠

will be one of the convergents for the continued fraction expansion of a number, if and only if the number is strictly between (see this proof)

x = [a 0; a 1, ..., a k - 1, a k, 2] = ⁠ 2 p k - p k-1 / 2 q k - q k-1 ⁠

and

y = [a 0; a 1, ..., a k - 1, a k + 2] = ⁠ p k + p k-1 / q k + q k-1 ⁠

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

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

Legendre's theorem on continued fractions

In his Essai sur la théorie des nombres (1798), Adrien-Marie Legendre derives a necessary and sufficient condition for a rational number to be a convergent of the continued fraction of a given real number.^[16] A consequence of this criterion, often called Legendre's theorem within the study of continued fractions, is as follows:^[17]

Theorem. If α is a real number and p, q are positive integers such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\alpha - \frac{p}{q}\right| < \frac{1}{2q^2}} , then p/q is a convergent of the continued fraction of α.

Proof

Proof. We follow the proof given in An Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright.^[18]

Suppose α, p, q are such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\alpha - \frac{p}{q}\right| < \frac{1}{2q^2}} , and assume that α > p/q. Then we may write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha - \frac{p}{q} = \frac{\theta}{q^2}} , where 0 < θ < 1/2. We write p/q as a finite continued fraction [a₀; a₁, ..., a_n], where due to the fact that each rational number has two distinct representations as finite continued fractions differing in length by one (namely, one where a_n = 1 and one where a_n ≠ 1), we may choose n to be even. (In the case where α < p/q, we would choose n to be odd.)

Let p₀/q₀, ..., p_n/q_n = p/q be the convergents of this continued fraction expansion. Set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega := \frac{1}{\theta} - \frac{q_{n-1}}{q_n}} , so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta = \frac{q_n}{q_{n-1} + \omega q_n}} and thus,

\alpha ={\frac {p}{q}}+{\frac {\theta }{q^{2}}}={\frac {p_{n}}{q_{n}}}+{\frac {1}{q_{n}(q_{n-1}+\omega q_{n})}}={\frac {(p_{n}q_{n-1}+1)+\omega p_{n}q_{n}}{q_{n}(q_{n-1}+\omega q_{n})}}={\frac {p_{n-1}q_{n}+\omega p_{n}q_{n}}{q_{n}(q_{n-1}+\omega q_{n})}}={\frac {p_{n-1}+\omega p_{n}}{q_{n-1}+\omega q_{n}}},

where we have used the fact that p_n_-1 q_n - p_n q_n_-1 = (-1)ⁿ and that n is even.

Now, this equation implies that α = [a₀; a₁, ..., a_n, ω]. Since the fact that 0 < θ < 1/2 implies that ω > 1, we conclude that the continued fraction expansion of α must be [a₀; a₁, ..., a_n, b₀, b₁, ...], where [b₀; b₁, ...] is the continued fraction expansion of ω, and therefore that p_n/q_n = p/q is a convergent of the continued fraction of α.

This theorem forms the basis for Wiener's attack, a polynomial-time exploit of the RSA cryptographic protocol that can occur for an injudicious choice of public and private keys (specifically, this attack succeeds if the prime factors of the public key n = pq satisfy p < q < 2p and the private key d is less than (1/3)n^1/4).^[19]

Comparison

Consider $x = [a 0; a 1, ...]$ and $y = [b 0; b 1, ...]$ . 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 $x = [a 0; a 1, ..., a n]$ and $y = [b 0; b 1, ..., b n, b n + 1, ...]$ 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 expansion of $π$ and its convergents

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, ....

The following Maple code will generate continued fraction expansions of pi

To sum up, the pattern is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Numerator}_i = \text{Numerator}_{(i-1)} \cdot \text{Quotient}_i + \text{Numerator}_{(i-2)} } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Denominator}_i = \text{Denominator}_{(i-1)} \cdot \text{Quotient}_i + \text{Denominator}_{(i-2)} }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi=[3;7,15,1,292,1,1,1,2,1,3,1,\ldots]}

or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}}}} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}}}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \pi=2+\cfrac{4}{3+\cfrac{1\cdot3}{4+\cfrac{3\cdot5}{4+\cfrac{5\cdot7}{4+\ddots}}}}}

The first two of these are special cases of the arctangent function with $π$ = 4 arctan (1) and the fourth and fifth one can be derived using the Wallis product.^[20]^[21]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi=3+\cfrac{1}{6+\cfrac{1^3+2^3}{6\cdot1^2+1^2\cfrac{1^3+2^3+3^3+4^3}{6\cdot2^2+2^2\cfrac{1^3+2^3 +3^3+4^3+5^3+6^3}{6\cdot3^2+3^2\cfrac{1^3+2^3+3^3+4^3+5^3+6^3+7^3+8^3}{6\cdot4^2+\ddots}}}}} }

The continued fraction of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} above consisting of cubes uses the Nilakantha series and an exploit from Leonhard Euler.^[22]

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 ^[23] states that any irrational number $k$ can be approximated by infinitely many rational ⁠m/n⁠ with

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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 discernible pattern in the simple continued fraction expansion of $π$ , there is one for $e$ , the base of the natural logarithm:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 $n$ :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 $n$ :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 $n = 1$ :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tanh(1/n) = [0; n, 3n, 5n, 7n, 9n, 11n, 13n, 15n, 17n, 19n, \dots] }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 $n = 1$ :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(p/q) = [p+q; p+2q, p+3q, p+4q, \dots],}

with similar formulas for negative rationals; in particular we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, for almost all numbers on the unit interval, they have the same limit behavior.

The arithmetic average diverges: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\to\infty}\frac 1n \sum_{k=1}^n a_k = +\infty} , and so the coefficients grow arbitrarily large: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \limsup_n a_n = +\infty} . In particular, this implies that almost all numbers are well-approximable, in the sense thatFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \liminf_{n\to\infty} \left| x - \frac{p_n}{q_n} \right| q_n^2 = 0} Khinchin proved that the geometric mean of $a i$ tends to a constant (known as Khinchin's constant):Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \rightarrow \infty } \left( a_1 a_2 ... a_n \right) ^{1/n} = K_0 = 2.6854520010\dots} Paul Lévy proved that the $n$ th root of the denominator of the $n$ th convergent converges to Lévy's constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \rightarrow \infty } q_n^{1/n} = e^{\pi^2/(12\ln2)} = 3.2758\ldots} Lochs' theorem states that the convergents converge exponentially at the rate ofFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\to\infty}\frac 1n \ln\left| x - \frac{p_n}{q_n} \right| = -\frac{\pi^2}{6\ln 2} }

Applications

Square roots

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

The identity

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\sqrt {x}}=1+{\frac {x-1}{1+{\sqrt {x}}}}}

(1)

leads via recursion to the generalized continued fraction for any square root:^[24]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 if $p 2 - nq 2 = \pm1$ , then $⁠ p / q ⁠$ is a convergent of the regular continued fraction for √ $n$ . The converse holds if the period of the regular continued fraction for √ $n$ is 1, and in general the period describes which convergents give solutions to Pell's equation.^[25]

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: $h ([0; a 1, a 2, a 3, ...]) = [0; a 2, a 3, ...]$ . 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.^[26]

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.^[27]

Examples of rational and irrational numbers


Number	r	0	1	2	3	4	5	6	7	8	9	10
123	a_r	123
123	ra	123
12.3	a_r	12	3	3
12.3	ra	12	⁠37/3⁠	⁠123/10⁠
1.23	a_r	1	4	2	1	7
1.23	ra	1	⁠5/4⁠	⁠11/9⁠	⁠16/13⁠	⁠123/100⁠
0.123	a_r	0	8	7	1	2	5
0.123	ra	0	⁠1/8⁠	⁠7/57⁠	⁠8/65⁠	⁠23/187⁠	⁠123/1 000⁠
Φ = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1 + \sqrt{5}}{2}}	a_r	1	1	1	1	1	1	1	1	1	1	1
	ra	1	2	⁠3/2⁠	⁠5/3⁠	⁠8/5⁠	⁠13/8⁠	⁠21/13⁠	⁠34/21⁠	⁠55/34⁠	⁠89/55⁠	⁠144/89⁠
-Φ = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1 + \sqrt{5}}{2}}	a_r	-2	2	1	1	1	1	1	1	1	1	1
	ra	-2	-⁠3/2⁠	-⁠5/3⁠	-⁠8/5⁠	-⁠13/8⁠	-⁠21/13⁠	-⁠34/21⁠	-⁠55/34⁠	-⁠89/55⁠	-⁠144/89⁠	-⁠233/144⁠
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{2}}	a_r	1	2	2	2	2	2	2	2	2	2	2
	ra	1	⁠3/2⁠	⁠7/5⁠	⁠17/12⁠	⁠41/29⁠	⁠99/70⁠	⁠239/169⁠	⁠577/408⁠	⁠1 393/985⁠	⁠3 363/2 378⁠	⁠8 119/5 741⁠
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\sqrt{2}}}	a_r	0	1	2	2	2	2	2	2	2	2	2
	ra	0	1	⁠2/3⁠	⁠5/7⁠	⁠12/17⁠	⁠29/41⁠	⁠70/99⁠	⁠169/239⁠	⁠408/577⁠	⁠985/1 393⁠	⁠2 378/3 363⁠
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{3}}	a_r	1	1	2	1	2	1	2	1	2	1	2
	ra	1	2	⁠5/3⁠	⁠7/4⁠	⁠19/11⁠	⁠26/15⁠	⁠71/41⁠	⁠97/56⁠	⁠265/153⁠	⁠362/209⁠	⁠989/571⁠
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\sqrt{3}}}	a_r	0	1	1	2	1	2	1	2	1	2	1
	ra	0	1	⁠1/2⁠	⁠3/5⁠	⁠4/7⁠	⁠11/19⁠	⁠15/26⁠	⁠41/71⁠	⁠56/97⁠	⁠153/265⁠	⁠209/362⁠
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\sqrt{3}}{2}}	a_r	0	1	6	2	6	2	6	2	6	2	6
	ra	0	1	⁠6/7⁠	⁠13/15⁠	⁠84/97⁠	⁠181/209⁠	⁠1 170/1 351⁠	⁠2 521/2 911⁠	⁠16 296/18 817⁠	⁠35 113/40 545⁠	⁠226 974/262 087⁠
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt[3]{2}}	a_r	1	3	1	5	1	1	4	1	1	8	1
	ra	1	⁠4/3⁠	⁠5/4⁠	⁠29/23⁠	⁠34/27⁠	⁠63/50⁠	⁠286/227⁠	⁠349/277⁠	⁠635/504⁠	⁠5 429/4 309⁠	⁠6 064/4 813⁠
e	a_r	2	1	2	1	1	4	1	1	6	1	1
e	ra	2	3	⁠8/3⁠	⁠11/4⁠	⁠19/7⁠	⁠87/32⁠	⁠106/39⁠	⁠193/71⁠	⁠1 264/465⁠	⁠1 457/536⁠	⁠2 721/1 001⁠
π	a_r	3	7	15	1	292	1	1	1	2	1	3
π	ra	3	⁠22/7⁠	⁠333/106⁠	⁠355/113⁠	⁠103 993/33 102⁠	⁠104 348/33 215⁠	⁠208 341/66 317⁠	⁠312 689/99 532⁠	⁠833 719/265 381⁠	⁠1 146 408/364 913⁠	⁠4 272 943/1 360 120⁠
Number	r	0	1	2	3	4	5	6	7	8	9	10

ra: rational approximant obtained by expanding continued fraction up to a_r

History

300 BCE Euclid's Elements contains an algorithm for the greatest common divisor, whose modern version generates a continued fraction as the sequence of quotients of successive Euclidean divisions that occur in it.
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0} & Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{n_1}{d_1 \cdot}} & Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{n_2}{d_2 \cdot}} & Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.^[28]
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.^[29]
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

^ Encyclopaedia Britannica 2013.
^ ^2.0 ^2.1 Pettofrezzo & Byrkit 1970, p. 150.
^ ^3.0 ^3.1 Long 1972, p. 173.
^ ^4.0 ^4.1 Pettofrezzo & Byrkit 1970, p. 152.
^ ^5.0 ^5.1 Weisstein 2022.
^ Collins 2001.
^ Cajori, Florian (1925). "Leibniz, the Master-Builder of Mathematical Notations". Isis. 7 (3): 412–429. doi:10.1086/358328.
^ Swanson, Ellen (1999) [1971]. Mathematics into Type (PDF). Updated by O'Sean, Arlene; Schleyer, Antoinette (Updated ed.). American Mathematical Society. 2.4.1c "Continued fractions", p. 18.
^ Long 1972, p. 183.
^ Pettofrezzo & Byrkit 1970, p. 158.
^ Long 1972, p. 177.
^ Pettofrezzo & Byrkit 1970, pp. 162–163.
^ ^13.0 ^13.1 Thill 2008.
^ Gosper, R. W. (1977). "Appendix 2: Continued Fraction Arithmetic". See "simplest intervening rational", pp. 29–31.
^ Murakami, Hiroshi (February 2015). "Calculation of rational numbers in an interval whose denominator is the smallest by using FP interval arithmetic". ACM Communications in Computer Algebra. 48 (3/4): 134–136. doi:10.1145/2733693.2733711.
^ Legendre, Adrien-Marie (1798). Essai sur la théorie des nombres (in français). Paris: Duprat. pp. 27–29.
^ Barbolosi, Dominique; Jager, Hendrik (1994). "On a theorem of Legendre in the theory of continued fractions". Journal de Théorie des Nombres de Bordeaux. 6 (1): 81–94. doi:10.5802/jtnb.106. JSTOR 26273940 – via JSTOR.
^ Hardy, G. H.; Wright, E. M. (1938). An Introduction to the Theory of Numbers. London: Oxford University Press. pp. 140–141, 153.
^ Wiener, Michael J. (1990). "Cryptanalysis of short RSA secret exponents". IEEE Transactions on Information Theory. 36 (3): 553–558. doi:10.1109/18.54902 – via IEEE.
^ Bunder & Tonien 2017.
^ Scheinerman, Pickett & Coleman 2008.
^ Foster 2015.
^ Hardy & Wright 2008, Theorem 193.
^ Thurston 2012.
^ Niven, Zuckerman & Montgomery 1991.
^ Martin 2004, p. 557.
^ Afifi, Auroux & Karl 2018.
^ Sandifer 2006.
^ Euler 1748.

References

Afifi, Haitham; Auroux, Sébastien; Karl, Holger (April 2018). "MARVELO: Wireless virtual network embedding for overlay graphs with loops". 2018 IEEE Wireless Communications and Networking Conference (WCNC). IEEE. pp. 1–6. arXiv:1712.06676. doi:10.1109/WCNC.2018.8377194. ISBN 978-1-5386-1734-2. S2CID 13447977.

Bunder, Martin W.; Tonien, Joseph (2017). "Closed form expressions for two harmonic continued fractions". The Mathematical Gazette. 101 (552): 439–448. doi:10.1017/mag.2017.125. S2CID 125489697.

Chen, Chen-Fan; Shieh, Leang-San (1969). "Continued fraction inversion by Routh's Algorithm". IEEE Trans. Circuit Theory. 16 (2): 197–202. doi:10.1109/TCT.1969.1082925.

Collins, Darren C. (2001). "Continued Fractions" (PDF). MIT Undergraduate Journal of Mathematics. Archived from the original (PDF) on 2001-11-20.

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.

Encyclopædia Britannica (2013). "Continued fraction – mathematics". Retrieved 26 April 2022.

Euler, Leonhard (1748). "E101 – Introductio in analysin infinitorum, volume 1". The Euler Archive. Retrieved 26 April 2022.

Foster, Tony (22 June 2015). "Theorem of the Day: Theorem no. 203" (PDF). Robin Whitty. Archived (PDF) from the original on 2013-12-11. Retrieved 26 April 2022.

Gragg, William B. (1974). "Matrix interpretations and applications of the continued fraction algorithm". Rocky Mountain J. Math. 4 (2): 213. doi:10.1216/RMJ-1974-4-2-213. S2CID 121378061.

Hardy, Godfrey H.; Wright, Edward M. (December 2008) [1979]. An Introduction to the Theory of Numbers (6 ed.). Oxford University Press. ISBN 9780199219865.

Heilermann, J. B. H. (1846). "Ueber die Verwandlung von Reihen in Kettenbrüche". Journal für die reine und angewandte Mathematik. 33: 174–188.

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. (December 1972). Elementary Introduction to Number Theory (2nd ed.). Lexington: D. C. Heath and Company. ISBN 9780669627039.

Magnus, Arne (1962). "Continued fractions associated with the Padé Table". Math. Z. 78: 361–374. doi:10.1007/BF01195180. S2CID 120535167.

Martin, Richard M. (December 2004). Electronic Structure: Basic Theory and Practical Methods. Cambridge University Press. doi:10.1017/CBO9780511805769. ISBN 9780511805769.

Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L. (1991). An introduction to the theory of numbers (Fifth ed.). New York: Wiley. ISBN 0-471-62546-9.

Perron, Oskar (1950). Die Lehre von den Kettenbrüchen. New York, NY: Chelsea Publishing Company.

Pettofrezzo, Anthony J.; Byrkit, Donald R. (December 1970). Elements of Number Theory. Englewood Cliffs: Prentice Hall. ISBN 9780132683005.

Rieger, Georg Johann (1982). "A new approach to the real numbers (motivated by continued fractions)" (PDF). Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft. 33: 205–217. Archived (PDF) from the original on 2020-12-10.

Rockett, Andrew M.; Szüsz, Peter (1992). Continued Fractions. World Scientific Press. ISBN 981-02-1047-7.

Sandifer, C. Edward (2006). "Chapter 32: Who proved e is irrational?". How Euler Did It (PDF). Mathematical Association of America. pp. 185–190. ISBN 978-0-88385-563-8. LCCN 2007927658. Archived (PDF) from the original on 2014-03-19.

Scheinerman, Ed; Pickett, Thomas J.; Coleman, Ann (2008). "Another Continued Fraction for π". The American Mathematical Monthly. 115 (10): 930–933. doi:10.1080/00029890.2008.11920610. JSTOR 27642639. S2CID 11914017.

Shoemake, Ken (1995). "I.4: Rational Approximation". In Paeth, Alan W. (ed.). Graphics Gems V (IBM Version). San Diego, California: Academic Press. pp. 25–31. ISBN 0-12-543455-3.

Siebeck, H. (1846). "Ueber periodische Kettenbrüche". Journal für die reine und angewandte Mathematik. 33: 68–70.

Thill, M. (2008). "A more precise rounding algorithm for rational numbers". Computing. 82 (2–3): 189–198. doi:10.1007/s00607-008-0006-7. S2CID 45166490.

Thurston, Ben (2012). "Estimating square roots, generalized continued fraction expression for every square root". The Ben Paul Thurston Blog. Retrieved 26 April 2022.

Wall, Hubert Stanley (1948). Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc. ISBN 0-8284-0207-8.

Weisstein, Eric Wolfgang (2022). MathWorld (ed.). "Periodic Continued Fraction". Retrieved 26 April 2022.

External links

"Continued fraction", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Knott, Ron (2018). "Continued fractions (An online Combined Continued Fraction Calculator is available)". Retrieved 26 April 2022.

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 Archived 2009-05-04 at the Wayback Machine
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 sequence A133593 ("Exact" continued fraction for pi)
A view into "fractional interpolation" of a continued fraction ${1; 1, 1, 1, ...$ }
Best rational approximation through continued fractions
CONTINUED FRACTIONS by C. D. Olds

[FOOTNOTEEncyclopaedia_Britannica2013-1] Encyclopaedia Britannica 2013.

[FOOTNOTEPettofrezzoByrkit1970150-2] 2.0 ^2.1 Pettofrezzo & Byrkit 1970, p. 150.

[FOOTNOTELong1972173-3] 3.0 ^3.1 Long 1972, p. 173.

[FOOTNOTEPettofrezzoByrkit1970152-4] 4.0 ^4.1 Pettofrezzo & Byrkit 1970, p. 152.

[FOOTNOTEWeisstein2022-5] 5.0 ^5.1 Weisstein 2022.

[FOOTNOTECollins2001-6] Collins 2001.

[7] Cajori, Florian (1925). "Leibniz, the Master-Builder of Mathematical Notations". Isis. 7 (3): 412–429. doi:10.1086/358328.

[8] Swanson, Ellen (1999) [1971]. Mathematics into Type (PDF). Updated by O'Sean, Arlene; Schleyer, Antoinette (Updated ed.). American Mathematical Society. 2.4.1c "Continued fractions", p. 18.

[FOOTNOTELong1972183-9] Long 1972, p. 183.

[FOOTNOTEPettofrezzoByrkit1970158-10] Pettofrezzo & Byrkit 1970, p. 158.

[FOOTNOTELong1972177-11] Long 1972, p. 177.

[FOOTNOTEPettofrezzoByrkit1970162–163-12] Pettofrezzo & Byrkit 1970, pp. 162–163.

[FOOTNOTEThill2008-13] 13.0 ^13.1 Thill 2008.

[14] Gosper, R. W. (1977). "Appendix 2: Continued Fraction Arithmetic". See "simplest intervening rational", pp. 29–31.

[15] Murakami, Hiroshi (February 2015). "Calculation of rational numbers in an interval whose denominator is the smallest by using FP interval arithmetic". ACM Communications in Computer Algebra. 48 (3/4): 134–136. doi:10.1145/2733693.2733711.

[16] Legendre, Adrien-Marie (1798). Essai sur la théorie des nombres (in français). Paris: Duprat. pp. 27–29.

[17] Barbolosi, Dominique; Jager, Hendrik (1994). "On a theorem of Legendre in the theory of continued fractions". Journal de Théorie des Nombres de Bordeaux. 6 (1): 81–94. doi:10.5802/jtnb.106. JSTOR 26273940 – via JSTOR.

[18] Hardy, G. H.; Wright, E. M. (1938). An Introduction to the Theory of Numbers. London: Oxford University Press. pp. 140–141, 153.

[19] Wiener, Michael J. (1990). "Cryptanalysis of short RSA secret exponents". IEEE Transactions on Information Theory. 36 (3): 553–558. doi:10.1109/18.54902 – via IEEE.

[FOOTNOTEBunderTonien2017-20] Bunder & Tonien 2017.

[FOOTNOTEScheinermanPickettColeman2008-21] Scheinerman, Pickett & Coleman 2008.

[FOOTNOTEFoster2015-22] Foster 2015.

[FOOTNOTEHardyWright2008Theorem_193-23] Hardy & Wright 2008, Theorem 193.

[FOOTNOTEThurston2012-24] Thurston 2012.

[FOOTNOTENivenZuckermanMontgomery1991-25] Niven, Zuckerman & Montgomery 1991.

[FOOTNOTEMartin2004557-26] Martin 2004, p. 557.

[FOOTNOTEAfifiAurouxKarl2018-27] Afifi, Auroux & Karl 2018.

[FOOTNOTESandifer2006-28] Sandifer 2006.

[FOOTNOTEEuler1748-29] Euler 1748.

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Continued fraction

Contents

Motivation and notation

Basic formula

Calculating continued fraction representations

Notations

Finite continued fractions

Reciprocals

Infinite continued fractions and convergents

Properties

Some useful theorems

Semiconvergents

Best rational approximations

Best rational within an interval

Interval for a convergent

Legendre's theorem on continued fractions

Comparison

Continued fraction expansion of $π$ and its convergents

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

Networking applications

Examples of rational and irrational numbers

History

See also

Notes

References

External links

Continued fraction

Motivation and notation

Basic formula

Calculating continued fraction representations

Notations

Finite continued fractions

Reciprocals

Infinite continued fractions and convergents

Properties

Some useful theorems

Semiconvergents

Best rational approximations

Best rational within an interval

Interval for a convergent

Legendre's theorem on continued fractions

Comparison

Continued fraction expansion of π and its convergents

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

Networking applications

Examples of rational and irrational numbers

History

See also

Notes

References

External links

Continued fraction expansion of $π$ and its convergents