In mathematics, a continued fraction is an expression such as

$x\; =\; a\_0\; +\; \backslash cfrac\{1\}\{a\_1\; +\; \backslash cfrac\{1\}\{a\_2\; +\; \backslash cfrac\{1\}\{a\_3+\backslash ,\backslash cdots\}\}\}$

where a₀ is some integer and all the other numbers a_n are positive integers. Longer expressions are defined analogously. If the numerators are allowed to differ from unity, the resulting expression is a generalized continued fraction. For clarity a non-generalized continued fraction is also called a simple continued fraction.

Motivation

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Continued fractions are motivated by a desire to have a "mathematically pure" representation for the real numbers.

Most people are familar with the decimal representation of real numbers:

$r\; =\; \backslash sum\_\{i=0\}^\backslash infty\; a\_i\; 10^\{-i\}$

where a₀, may be any integer, and each a_i is an element of {0, 1, 2, ..., 9}. In this representation, the number π, for example, is represented by the sequence of integers {3, 1, 4, 1, 5, 9, 2, ...}.

This representation has some problems, however. One problem is the appearance of the arbitrary constant 10 in the formula above. Why 10? This is because of a biological accident, not because of anything related to mathematics. 10 is used because it is the standard base of our number system (10 fingers); we may just as well use base 8 (octal) or base 2 (binary). Another problem is that many rational numbers lack finite representations in this system. For example, the number 1/3 is represented by the infinite sequence {0, 3, 3, 3, 3, ....}.

Continued fraction notation is a representation for the real numbers that evades both these problems. Let us consider how we might describe a number like 415/93, which is around 4.4624. This is approximately 4. Actually it is a little bit more than 4, about 4 + 1/2. But the 2 in the denominator is not correct; the correct denominator is a little bit more than 2, about 2 + 1/6, so 415/93 is approximately 4 + 1/(2 + 1/6). But the 6 in the denominator is not correct; the correct denominator is a little bit more than 6, actually 6+1/7. So 415/93 is actually 4+1/(2+1/(6+1/7)). This is exact.

Dropping the trivial parts of the expression 4+1/(2+1/(6+1/7)) gives the abbreviated notation 2, 6, 7.

The continued fraction representation of real numbers can be defined in this way. It has several desirable properties:

• The continued fraction representation for a number is finite if and only if the number is rational.
• Continued fraction representations for "simple" rational numbers are short.
• The continued fraction representation of any rational number is unique if it has no trailing 1. (For any rational number expressed as a continued fraction a,...,z with z>1 there is a less efficient representation ending in 1, ^*).
• The continued fraction representation of an irrational number is unique.
• The terms of a continued fraction will repeat if and only if it is the continued fraction representation of a quadratic irrational, that is, a real solution to a quadratic equation ^*.
• Truncating the continued fraction representation of a number x early yields a rational approximation for x which is in a certain sense the "best possible" rational approximation (see theorem 5, corollary 1 below for a formal statement).

This last property is extremely important, and is not true of the conventional decimal representation. Truncating the decimal representation of a number yields a rational approximation of that number, but not usually a very good approximation. For example, truncating 1/7 = 0.142857... at various places yields approximations such as 142/1000, 14/100, and 1/10. But clearly the best rational approximation is "1/7" itself. Truncating the decimal representation of π yields approximations such as 31415/10000 and 314/100. The continued fraction representation of π begins 7, 15, 1, 292, .... Truncating this representation yields the excellent rational approximations 3, 22/7, 333/106, 355/113, 103993/33102, ... The denominators of 314/100 and 333/106 are almost the same, but the error in the approximation 314/100 is nineteen times as large as the error in 333/106. As an approximation to π, 7, 15, 1 is more than one hundred times more accurate than 3.1416.

Calculating continued fraction representations

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Consider a real number r. Let i be the integer part and f the fractional part of r. Then the continued fraction representation of r is …, where "…" is the continued fraction representation of 1/f. It is customary to replace the first comma by a semicolon.

To calculate a continued fraction representation of a number r, write down the integer part of r. Subtract this integer part from r. If the difference is 0, stop; otherwise find the reciprocal of the difference and repeat. The procedure will halt if and only if r was rational.

Find the continued fraction for 3.245
$3\backslash ,$	$3.245\; -\; 3\backslash ,$	$=\; 0.245\backslash ,$	$1\; /\; 0.245\backslash ,$	$=\; 4.082\backslash ,$
$4\backslash ,$	$4.082\; -\; 4\backslash ,$	$=\; 0.082\backslash ,$	$1\; /\; 0.082\backslash ,$	$=\; 12.250\backslash ,$
$12\backslash ,$	$12.250\; -\; 12\backslash ,$	$=\; 0.250\backslash ,$	$1\; /\; 0.250\backslash ,$	$=\; 4.000\backslash ,$
$4\backslash ,$	$4.000\; -\; 4\backslash ,$	$=\; 0.000\backslash ,$	STOP
continued fraction form for 3.245 is 4, 12, 4
$3.245\; =\; 3\; +\; \backslash cfrac\{1\}\{4\; +\; \backslash cfrac\{1\}\{12\; +\; \backslash cfrac\{1\}\{4\}\}\}$

The number 3.245 can also be represented by the continued fraction expansion 4, 12, 3, 1; refer to Finite continued fractions below.

This algorithm is suitable for real numbers, but can lead to numerical disaster if implemented with floating point numbers. Instead, any floating point number is an exact rational (the denominator is usually a power of two on modern computers, and a power of ten on electronic calculators), so a variant of Euclid's GCD algorithm can be used to give exact results.

Notations for continued fractions

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One can abbreviate a continued fraction as

$x\; =a\_1,\; a\_2,\; a\_3\backslash ;$

or in the notation of Pringsheim

$x\; =\; a\_0\; +\; \backslash frac\{1\; \backslash mid\}\{\backslash mid\; a\_1\}\; +\; \backslash frac\{1\; \backslash mid\}\{\backslash mid\; a\_2\}\; +\; \backslash frac\{1\; \backslash mid\}\{\backslash mid\; a\_3\}$

or another seldom used notation, similar to the above

$x\; =\; a\_0\; +\; \{1\; \backslash over\; a\_1\; +\; \}\; \{1\; \backslash over\; a\_2\; +\}\; \{1\; \backslash over\; a\_3\; +\}$ .

Sometimes similar notation is used

$x\; =\; \backslash left\; \backslash langle\; a\_0;\; a\_1,\; a\_2,\; a\_3\; \backslash right\; \backslash rangle\; \backslash ;$

with or without semicolon.

One may also define infinite continued fractions as limits:

$a\_\{1\},\; a\_\{2\},\; a\_\{3\},\; \backslash ,\backslash ldots=\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}a\_\{1\},\; a\_\{2\},\; \backslash ,\backslash ldots,\; a\_\{n\}.$

This limit exists for any choice of positive integers a₁, a₂, a₃ ...

Finite continued fractions

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For finite continued fractions, note that

$a\_\{1\},\; a\_\{2\},\; a\_\{3\},\; \backslash ,\backslash ldots\; ,a\_\{n\},\; 1=a\_\{1\},\; a\_\{2\},\; a\_\{3\},\; \backslash ,\backslash ldots,\; a\_\{n\}\; +\; 1.\; \backslash ;$

So, for every finite continued fraction, there is another finite continued fraction that represents the same number, for instance

$3,\; 1=4=\; 9/4\; =\; 2.25.\; \backslash ;$

Every finite continued fraction is rational, and every rational number can be represented in precisely two different ways as a finite continued fraction. In one representation the final term in the continued fraction is 1. The other representation is one element shorter, and the final term must be greater than 1 unless there is only one element.

Continued fractions of reciprocals

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The continued fraction representations of a 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 are reciprocals. This is because if $a\backslash$ is an integer then if then $x\; =\; 0+1/(a+1/b)\backslash$ and $1/x\; =\; a+1/b\backslash$ and if $x>1\backslash$ then $x\; =\; a+1/b\backslash$ and $1/x\; =\; 0+1/(a+1/b)\backslash$ with the last number that generates the remainder of the continued fraction being the same for both $x\backslash$ and its reciprocal.

Infinite continued fractions

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

For a continued fraction $*$, the first four convergents (numbered $0$ through $3$) are

$$

\frac{a_0}{1},\qquad \frac{a_0a_1+1}{a_1},\qquad \frac{ a_2(a_0a_1+1)+a_0}{a_2a_1+1},\qquad \frac{a_3(a_2(a_0a_1+1)+a_0)+(a_0a_1+1)}{a_3(a_2a_1+1)+a_1}.

In words, the numerator of the third convergent is formed by multiplying the numerator of the second convergent by the third quotient, and adding the numerator of the first convergent. The denominators are formed similarly.

If successive convergents are found, with numerators $h\_1,h\_2,\backslash ldots$ and denominators $k\_1,k\_2,\backslash ldots$ then the relevant recursive relation is:

$h\_n=a\_nh\_\{n-1\}+h\_\{n-2\},\backslash qquad\; k\_n=a\_nk\_\{n-1\}+k\_\{n-2\}.$

The successive convergents are given by the formula

$$

\frac{h_n}{k_n}= \frac{a_nh_{n-1}+h_{n-2}}{a_nk_{n-1}+k_{n-2}}.

Some useful theorems

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If a₀, a₁, a₂, ... is an infinite sequence of positive integers, define the sequences $h\_n$ and $k\_n$ recursively:

$h\_\{n\}=a\_nh\_\{n-1\}+h\_\{n-2\}$			$h\_\{-1\}=1$		$h\_\{-2\}=0$
$k\_\{n\}=a\_nk\_\{n-1\}+k\_\{n-2\}$			$k\_\{-1\}=0$		$k\_\{-2\}=1$

Theorem 1

For any positive $x\backslash in\backslash mathbb\{R\}$

$$

\lefta_1, \,\dots, a_{n-1}, x \right= \frac{x h_{n-1}+h_{n-2}} {x k_{n-1}+k_{n-2}}.

Theorem 2

The convergents of a₁, a₂, ... are given by

$$

\lefta_1, \,\dots, a_n\right= \frac{h_n} {k_n}.

Theorem 3

If the nth convergent to a continued fraction is $h\_n/k\_n$, then

$$

k_nh_{n-1}-k_{n-1}h_n=(-1)^n.

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

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

$$

\left|\frac{h_n}{k_n}-\frac{h_{n-1}}{k_{n-1}} \right|= \left|\frac{h_nk_{n-1}-k_nh_{n-1}}{k_nk_{n-1}}\right|= \frac{1}{k_nk_{n-1}}.

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

$$

a_0 + \sum_{n=0}^\infty \frac{(-1)^{n}}{k_{n+1}k_{n}}.

Corollary 4: The matrix

$\backslash begin\{bmatrix\}$

h_n & h_{n-1} \\ k_n & k_{n-1} \end{bmatrix} has determinant plus or minus one, and thus belongs to the group of 2x2 unimodular matrices $S^*L(2,\backslash mathbb\{Z\})$.

Theorem 4

Each convergent is nearer to the nth convergent than any of the preceding convergents. In symbols, if the nth convergent is taken to be $a\_n=x$, then

$$

\left|a_1, a_2, \ldots a_r-x\right|> \left|a_1, a_2, \ldots a_s-x\right| for all

Corollary 1: the odd convergents continually increase, but are always less than $x.$

Corollary 2: the even convergents continually decrease, but are always greater than $x.$

Theorem 5

$$

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

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

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

Semiconvergents

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If $\backslash frac\{h\_\{n-1\}\}\{k\_\{n-1\}\}$ and $\backslash frac\{h\_n\}\{k\_n\}$ are successive convergents, then any fraction of the form

$\backslash frac\{h\_\{n-1\}\; +\; ah\_n\}\{k\_\{n-1\}+ak\_n\}$

where $a$ is a nonnegative integer and the numerators and denominators are between the $n$ and $n+1$ terms inclusive are called semiconvergents, secondary convergents, or intermediate fractions. Often the term is taken to mean that being a semiconvergent excludes the possibility of being a convergent, rather than that a convergent is a kind of semiconvergent.

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

Best rational approximations

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A best rational approximation to a real number x is a rational number $\backslash begin\{matrix\}\; \backslash frac\; \{n\}\{d\}\; \backslash end\{matrix\}$, $d>0$, that is closer to x than any approximation with a smaller denominator. Because a best rational approximation is always a convergent or a semiconvergent, the regular continued fraction for x can be used to generate all of the best rational approximations for x. We may apply these three rules:

1. Truncate the continued fraction, and possibly decrement its last term.
2. The decremented term cannot have less than half its original value.
3. If the final term is even, a special rule decides if half its value is admissible. (See below.)

For example, 0.84375 has regular continued fraction ^*. Here are all of its best rational approximations.

{| border="1" cellspacing="0" ^*" target="_blank" > ^*" target="_blank" > ^* $\backslash begin\{matrix\}\; 1\; \backslash end\{matrix\}$ $\backslash begin\{matrix\}\; \backslash frac\; \{3\}\{4\}\; \backslash end\{matrix\}$ $\backslash begin\{matrix\}\; \backslash frac\; \{4\}\{5\}\; \backslash end\{matrix\}$ $\backslash begin\{matrix\}\; \backslash frac\; \{5\}\{6\}\; \backslash end\{matrix\}$ $\backslash begin\{matrix\}\; \backslash frac\; \{11\}\{13\}\; \backslash end\{matrix\}$ $\backslash begin\{matrix\}\; \backslash frac\; \{16\}\{19\}\; \backslash end\{matrix\}$ $\backslash begin\{matrix\}\; \backslash frac\; \{27\}\{32\}\; \backslash end\{matrix\}$

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.

To incorporate a new term into a rational approximation, only the two previous convergents are necessary. If a is the new term, then the new numerator and denominator are

n_k+1 = n_k−1 + a n_k

d_k+1 = d_k−1 + a d_k

The initial "convergents" (required for the first two terms) are $0\; \backslash atop\; 1$ and $1\backslash atop0$. For example, here are the convergents for ^*.

{| cellspacing="0" cellpadding="8" border="1" a_k 0 1 5 2 2 n_k 0 1 0 1 5 11 27 d_k 1 0 1 1 6 13 32

One formal description of the half rule is that the halved term $\backslash begin\{matrix\}\; \backslash frac\; \{1\}\{2\}\; \backslash end\{matrix\}\; a\_k$ is admissible if and only if

a_k−1, …, a₁ > a_k+1, ….

In practice, something like Euclid's GCD algorithm is often used to generate the terms sequentially, and the auxiliary values it provides allow a more convenient test. For example, here is the term generation for 0.84375 = $\backslash begin\{matrix\}\; \backslash frac\; \{27\}\{32\}\; \backslash end\{matrix\}$ (where $\backslash lfloor\; x\; \backslash rfloor$denotes the floor function).

{| cellspacing="0" cellpadding="0" $a\_0=$ $\backslash lfloor\; \backslash begin\{matrix\}\; \backslash frac\; \{27\}\{32\}\; \backslash end\{matrix\}\; \backslash rfloor$ $=\; 0$, $f\_0$ $=\; 27\; -\; 32a\_0$ $=\; 27$ $a\_1=$ $\backslash lfloor\; \backslash begin\{matrix\}\; \backslash frac\; \{32\}\{27\}\; \backslash end\{matrix\}\; \backslash rfloor$ $=\; 1$, $f\_1$ $=\; 32\; -\; 27a\_1$ $=\; 5$ $a\_2=$ $\backslash lfloor\; \backslash begin\{matrix\}\; \backslash frac\; \{27\}\{5\}\; \backslash end\{matrix\}\; \backslash rfloor$ $=\; 5$, $f\_2$ $=\; 27\; -\; 5a\_2$ $=\; 2$ $a\_3=$ $\backslash lfloor\; \backslash begin\{matrix\}\; \backslash frac\; \{5\}\{2\}\; \backslash end\{matrix\}\; \backslash rfloor$ $=\; 2$, $f\_3$ $=\; 5\; -\; 2a\_3$ $=\; 1$ $a\_4=$ $\backslash lfloor\; \backslash begin\{matrix\}\; \backslash frac\; \{2\}\{1\}\; \backslash end\{matrix\}\; \backslash rfloor$ $=\; 2$, $f\_4$ $=\; 2\; -\; 1a\_4$ $=\; 0$

Using the f values so generated, the $\backslash begin\{matrix\}\; \backslash frac\; \{1\}\{2\}\; \backslash end\{matrix\}\; a\_k$ admissibility test is $\backslash begin\{matrix\}\; \backslash frac\; \{d\_\{k-2\}\}\{d\_\{k-1\}\}\; \backslash end\{matrix\}\; >\; \backslash begin\{matrix\}\; \backslash frac\; \{f\_k\}\{f\_\{k-1\}\}\; \backslash end\{matrix\}$. For $a\_3$ of the example, $\backslash begin\{matrix\}\; \backslash frac\; \{d\_1\}\{d\_2\}\backslash end\{matrix\}\; =\; \backslash begin\{matrix\}\; \backslash frac\; \{1\}\{6\}\; \backslash end\{matrix\}$ and $\backslash begin\{matrix\}\; \backslash frac\; \{f\_3\}\{f\_2\}\backslash end\{matrix\}\; =\; \backslash begin\{matrix\}\; \backslash frac\; \{1\}\{2\}\; \backslash end\{matrix\}$, so $\backslash begin\{matrix\}\; \backslash frac\; \{1\}\{2\}\; \backslash end\{matrix\}\; a\_3$ is not admissible; while for $a\_4$ $\backslash begin\{matrix\}\; \backslash frac\; \{d\_2\}\{d\_3\}\backslash end\{matrix\}\; =\; \backslash begin\{matrix\}\; \backslash frac\; \{6\}\{13\}\; \backslash end\{matrix\}$ and $\backslash begin\{matrix\}\; \backslash frac\; \{f\_4\}\{f\_3\}\backslash end\{matrix\}\; =\; \backslash begin\{matrix\}\; \backslash frac\; \{0\}\{1\}\; \backslash end\{matrix\}$ so $\backslash begin\{matrix\}\; \backslash frac\; \{1\}\{2\}\; \backslash end\{matrix\}\; a\_4$ is admissible.

The convergents to x are best approximations in an even stronger sense: $\backslash begin\{matrix\}\; \backslash frac\; \{n\}\{d\}\; \backslash end\{matrix\}$ is a convergent for x if and only if |dx-n| is the least relative error among all approximations $\backslash begin\{matrix\}\; \backslash frac\; \{m\}\{c\}\; \backslash end\{matrix\}$ with c ≤ d; that is, we have |dx-n| < |cx-m| so long as c

The continued fraction expansion of π

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To calculate the convergents of pi we may set $a\_0\; =\; \backslash lfloor\; \backslash pi\; \backslash rfloor\; =\; 3$, define $u\_1\; =\; \backslash frac\; \{1\}\{\backslash pi\; -\; 3\}$ $\backslash approx\; \backslash frac\; \{113\}\{16\}\; =\; 7.0625$ and $a\_1\; =\; \backslash lfloor\; u\_1\; \backslash rfloor\; =\; 7$, $u\_2\; =\; \backslash frac\; \{1\}\{u\_1\; -\; 7\}\; \backslash approx\; \backslash frac\; \{31993\}\{2000\}\; =\; 15.9965$ and $a\_2\; =\; \backslash lfloor\; u\_2\; \backslash rfloor\; =\; 15$, $u\_3\; =\; \backslash frac\; \{1\}\{u\_2\; -\; 15\}$ $\backslash approx\; \backslash frac\; \{1003\}\{1000\}\; =\; 1.003$. Continuing like this, one can determine the infinite continued fraction of π as 7, 15, 1, 292, 1, 1, .... The third convergent of π is 7, 15, 1 = 355/113 = 3.14159292035..., which is fairly close to the true value of π.

Let us suppose that the quotients found are, as above, 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 7, 15, 1, we obtain the four fractions:

$\backslash frac\{3\}\{1\},\; \backslash frac\{22\}\{7\},\; \backslash frac\{333\}\{106\},\; \backslash frac\{355\}\{113\},\; \backslash ,\backslash ldots$

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), that is 1/742 (in fact, 22/7 − π is just less than 1/790).

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:

$\backslash frac\{3\}\{1\}+\backslash frac\{1\}\{1\; \backslash times\; 7\}-\backslash frac\{1\}\{7\; \backslash times\; 106\}+\backslash frac\{1\}\{106\; \backslash times\; 113\}\; \backslash cdots$

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.

4/π has a generalized continued fraction expansion discovered by Lord Brouncker.

Other continued fraction expansions

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Periodic continued fractions

The numbers with periodic continued fraction expansion are precisely the solutions of quadratic equations with integer coefficients. For example, the golden ratio φ = 1, 1, 1, 1, 1, ... and √ 2 = 2, 2, 2, 2, ....

Regular patterns in continued fractions

While one cannot discern any pattern in the infinite continued fraction expansion of π, this is not true for e, the base of the natural logarithm:

$e\; =\; \backslash exp(1)\; =1,\; 2,\; 1,\; 1,\; 4,\; 1,\; 1,\; 6,\; 1,\; 1,\; 8,\; 1,\; 1,\; 10,\; 1,\; 1,\; 12,\; \backslash dots\backslash ,\backslash !$

We also have, when n is an integer greater than one,

$\backslash exp(1/n)\; =n-1,\; 1,\; 1,\; 3n-1,\; 1,\; 1,\; 5n-1,\; 1,\; 1,\; 7n-1,\; \backslash dots\backslash ,\backslash !$

with a more complex regular pattern for exp(2/(2n+1)).

Other continued fractions of this sort are

$\backslash tanh(1/n)\; \backslash ,\backslash ,\backslash ,\; =n,\; 3n,\; 5n,\; 7n,\; 9n,\; 11n,\; 13n,\; 17n,\; 19n,\; \backslash dots\backslash ,\backslash !$

where n is a postive integer, also

$\backslash tan(1)\; \backslash ,\backslash ,\backslash ,\; =1,\; 1,\; 3,\; 1,\; 5,\; 1,\; 7,\; 1,\; 9,\; 1,\; 11,\; 1,\; 13,\; 1,\; 15\; \backslash dots\backslash ,\backslash !$

and n>1,

$\backslash tan(1/n)\; \backslash ,\backslash ,\backslash ,\; =n-1,\; 1,\; 3n-2,\; 1,\; 5n-2,\; 1,\; 7n-2,\; \backslash dots\backslash ,\backslash !.$

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

$S(p/q)\; =\; \backslash frac\{I\_\{p/q\}(2/q)\}\{I\_\{1+p/q\}(2/q)\},$

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

$S(p/q)\; \backslash ,\backslash ,\backslash ,\; =p+2q,\; p+3q,\; p+4q,\; \backslash dots\backslash ,\backslash !.$

with similar formulas for negative rationals; in particular we have

$S(0)\; =\; S(0/1)\; =\; \backslash ,\backslash ,\backslash ,\; =2,\; 3,\; 4,\; 5,\; 6,\; 7,\; \backslash dots\backslash ,\backslash !.$

The formulas are most easily proven in terms of the Bessel-Clifford function.

Typical continued fractions

Most irrational numbers do not have any periodic or regular behavior in their continued fraction expansion. Nevertheless Khinchin proved that for almost all real numbers x, the a_i (for i = 1, 2, 3, ...) have an astonishing property: their geometric mean is a constant (known as Khinchin's constant, K ≈ 2.6854520010...) independent of the value of x. Paul Lévy showed that the nth root of the denominator of the nth convergent of the continued fraction expansion of almost all real numbers approaches an asymptotic limit, which is known as Lévy's constant.

Pell's equation

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Continued fractions play an essential role in the solution of Pell's equation. For example, for positive integers $p$ and $q$, $p^2\; -\; 2q^2\; =\; \backslash pm1$ if and only if $p/q$ is a convergent of $\backslash sqrt2$.

Continued fractions and chaos

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Continued fractions also play a role in the study of chaos, 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 $h(x)=1/x\; -\; \backslash lfloor\; 1/x\; \backslash rfloor,$ called the Gauss map, which lops off digits of a continued fraction expansion: $h(=\; [0;a\_2,a\_3,\backslash dots$. The transfer operator of this map is called the Gauss-Kuzmin-Wirsing operator. The distribution of the digits in continued fractions is given by the zero'th eigenvector of this operator, and is called the Gauss-Kuzmin distribution.

History of continued fractions

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• 300 BC Euclid, Elements - Algorithm for greatest common divisor which generates a continued fraction as a by-product
• 1579 Rafael Bombelli, L'Algebra Opera - method for the extraction of square roots which is related to continued fractions
• 1613 Pietro Cataldi, Trattato del modo brevissimo di trovar la radice quadra delli numeri - first notation for continued fractions
• 1695 John Wallis, Opera Mathematica - introduction of the term "continued fraction"

Cataldi represented a continued fraction as $a\_0.\backslash ,$ &$n\_1\; \backslash over\; d\_1.$ &$n\_2\; \backslash over\; d\_2.$ &$\{n\_3\; \backslash over\; d\_3\}$ with the dots indicating where the following fractions went.

See also

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• Mathematical constants (sorted by continued fraction representation)

External links

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• Online continued fraction calculator
• Linas Vepstas The Minkowski Question Mark and the Modular Group SL(2,Z) (2004) reviews the isomorphisms of continued fractions.
• Linas Vepstas Continued Fractions and Gaps (2004) reviews chaotic structures in continued fractions.
• Continued Fractions on the Stern-Brocot Tree at cut-the-knot
• Francois Balsalobre cfc - a (cli) continued fraction calculator for POSIX and Cygwin
• Continued Fractions and Fermat's Last Theorem.
• Elementary introduction to continued fractions
• The Antikythera Mechanism I: Gear ratios and continued fractions

References

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• A. Ya. Khinchin, Continued Fractions, 1935, English translation University of Chicago Press, 1961 ISBN 0-486-69630-8
• Oskar Perron, Die Lehre von den Kettenbrüchen, Chelsea Publishing Company, New York, NY 1950.
• Andrew M. Rockett and Peter Szusz, Continued Fractions, World Scientific Press, 1992.

Continued fractions | Mathematical analysis

Kettenbruch | Fracción continua | Fraction continuée | 연분수 | Frazione continua | Kettingbreuk | 連分数 | Ułamek łańcuchowy | Цепная дробь | verižni ulomek | เศษส่วนต่อเนื่อง | 连分数