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Encyclopedia > Continued fraction

In mathematics, a continued fraction is an expression such as Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...

$x = a_0 + cfrac{1}{a_1 + cfrac{1}{a_2 + cfrac{1}{a_3 + cfrac{1}{ddots,}}}}$

where a0 is some integer and all the other numbers an are positive integers. Longer expressions are defined analogously. If the partial numerators and partial denominators are allowed to assume arbitrary values, which may in some contexts include functions, the resulting expression is a generalized continued fraction. When it is necessary to distinguish the standard form above from generalized continued fractions, it may be called a simple or regular continued fraction, or is said to be in canonical form. The integers are commonly denoted by the above symbol. ... Partial plot of a function f. ... In mathematics, a generalized continued fraction is a generalization of the concept of continued fraction in which the numerators are allowed to differ from unity. ...

The study of continued fractions is motivated by a desire to have a "mathematically pure" representation for the real numbers. In mathematics, the real numbers may be described informally in several different ways. ...

Most people are familiar with the decimal representation of real numbers: It has been suggested that this article or section be merged with decimal. ...

$r = sum_{i=0}^infty a_i 10^{-i}$

where a0 may be any integer, and each ai 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, ....}. 10 (ten) is an even natural number following 9 and preceding 11. ... The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. ... The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

Continued fraction notation is a representation for the real numbers that avoids 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 redundant parts of the expression 4+1/(2+1/(6+1/7)) gives the abbreviated notation [4; 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 [N; a,...,z] with z>1 there is a less efficient representation ending in 1, [N;a,...,z-1,1]).
• 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 with integer coefficients [1].
• 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 [3; 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, [3; 7, 15, 1] is more than one hundred times more accurate than 3.1416. In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, an irrational number is any real number that is not a rational number, i. ...

## Calculating continued fraction representations

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 [i; …], 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
$3,$ $3.245 - 3,$ $= 0.245,$ $1 / 0.245,$ $= 4.082,$
$4,$ $4.082 - 4,$ $= 0.082,$ $1 / 0.082,$ $= 12.250,$
$12,$ $12.250 - 12,$ $= 0.250,$ $1 / 0.250,$ $= 4.000,$
$4,$ $4.000 - 4,$ $= 0.000,$ STOP
continued fraction form for 3.245 is [3; 4, 12, 4]
$3.245 = 3 + cfrac{1}{4 + cfrac{1}{12 + cfrac{1}{4}}}$

The number 3.245 can also be represented by the continued fraction expansion [3; 4, 12, 3, 1]; refer to Finite continued fractions below. In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

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. In number theory, the Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers). ...

## Notations for continued fractions

One can abbreviate a continued fraction as

$x = [a_0; a_1, a_2, a_3] ;$

or, in the notation of Pringsheim, as Alfred Pringsheim (September 2, 1850 _ June 25, 1941) was a mathematician who was born in Ohlau Lower Silesia (now Olawa Poland) and died in Zurich Switzerland. ...

$x = a_0 + frac{1 mid}{mid a_1} + frac{1 mid}{mid a_2} + frac{1 mid}{mid a_3}$.

Here is another related notation:

$x = a_0 + {1 over a_1 + } {1 over a_2 +} {1 over a_3 +}$ .

Sometimes angle brackets are used, like this:

$x = left langle a_0; a_1, a_2, a_3 right rangle ;$.

When angle brackets are used, the semicolon is optional.

One may also define infinite simple continued fractions as limits: In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...

$[a_{0}; a_{1}, a_{2}, a_{3}, ,ldots ] = lim_{n to infty} [a_{0}; a_{1}, a_{2}, ,ldots, a_{n}].$

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

## Finite continued fractions

Every finite continued fraction is rational, and every rational number can be represented in precisely two different ways as a finite continued fraction, which agree except at the very end. In the longer representation the final term in the continued fraction is 1; the shorter representation drops the 1, but increases the new final term by 1. (The final element in the short representation must therefore be greater than 1, except for representation of integers ≤ 1.) In symbols: In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

$[a_{0}; a_{1}, a_{2}, a_{3}, ,ldots ,a_{n}, 1]=[a_{0}; a_{1}, a_{2}, a_{3}, ,ldots, a_{n} + 1]. ;$

For instance,

$[2; 3, 1] = [2; 4] = 9/4 = 2.25 ;$, or $1 = [1] = [0;1],$.

## Continued fractions of reciprocals

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 $[a_0;a_1,a_2,a_3,ldots,a_n]$ and $[0;a_0,a_1,a_2,ldots,a_n]$ are reciprocals. This is because if $a$ is an integer then if $x<1$ then $x = 0+1/(a+1/b)$ and $1/x = a+1/b$ and if $x>1$ then $x = a+1/b$ and $1/x = 0+1/(a+1/b)$ with the last number that generates the remainder of the continued fraction being the same for both $x$ and its reciprocal. The reciprocal function: y = 1/x. ...

## Infinite continued fractions

Every infinite continued fraction is irrational, and every irrational number can be represented in precisely one way as an infinite continued fraction. In mathematics, an irrational number is any real number that is not a rational number, i. ...

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. A convergent is one of a sequence of rational values obtained by evaluating successive truncations of a continued fraction. ...

For a continued fraction $[a_0;a_1,a_2,ldots]$, 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 $h_1,h_2,ldots$ and denominators $k_1,k_2,ldots$ then the relevant recursive relation is:

$h_n=a_nh_{n-1}+h_{n-2},qquad k_n=a_nk_{n-1}+k_{n-2}.$

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

If a0, a1, a2, ... is an infinite sequence of positive integers, define the sequences hn and kn 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 $xinmathbb{R}$

$left[a_0; a_1, ,dots, a_{n-1}, x right]= frac{x h_{n-1}+h_{n-2}} {x k_{n-1}+k_{n-2}}.$

### Theorem 2

The convergents of [a0; a1, a2, ...] are given by

$left[a_0; a_1, ,dots, a_nright]= frac{h_n} {k_n}.$

### Theorem 3

If the nth convergent to a continued fraction is hn / kn, then

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

Corollary 1: Each convergent is in its lowest terms (for if hn and kn had a nontrivial common divisor it would divide knhn − 1kn − 1hn, 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

$begin{bmatrix} h_n & h_{n-1} k_n & k_{n-1} end{bmatrix}$

has determinant plus or minus one, and thus belongs to the group of 2x2 unimodular matrices $S^*L(2,mathbb{Z})$. In mathematics, a unimodular matrix is a square matrix with determinant +1 or -1. ...

### Theorem 4

Each (sth) convergent is nearer to a subsequent (nth) convergent than any preceding (rth) convergent is. In symbols, if the nth convergent is taken to be $[a_0;a_1,a_2,ldots a_n]=x_n$, then

$left| x_r - x_n right| > left| x_s - x_n right|$

for all r < s < n.

Corollary 1: the odd convergents (before the nth) continually increase, but are always less than xn.

Corollary 2: the even convergents (before the nth) continually decrease, but are always greater than xn.

### 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

If $frac{h_{n-1}}{k_{n-1}}$ and $frac{h_n}{k_n}$ are successive convergents, then any fraction of the form

$frac{h_{n-1} + ah_n}{k_{n-1}+ak_n}$

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 $ad-bc = pm 1$.

## Best rational approximations

A best rational approximation to a real number x is a rational number nd, d > 0, that is closer to x than any approximation with a smaller denominator. The simple continued fraction for x generates all of the best rational approximations for x according to 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 continued fraction [0;1,5,2,2]. Here are all of its best rational approximations.

 [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] 1 3⁄4 4⁄5 5⁄6 11⁄13 16⁄19 27⁄32

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

nk+1 = nk−1 + a nk
dk+1 = dk−1 + a dk

The initial "convergents" (required for the first two terms) are 01 and 10. For example, here are the convergents for [0;1,5,2,2].

 ak 0 1 5 2 2 nk 0 1 0 1 5 11 27 dk 1 0 1 1 6 13 32

One formal description of the half rule is that the halved term, ½ ak, is admissible if and only if

[ak; ak−1, …, a1] > [ak; ak+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 = 2732.

 a0 = ⌊27⁄32⌋ = 0, f0 = 27 − 32a0 = 27 a1 = ⌊32⁄27⌋ = 1, f1 = 32 − 27a1 = 5 a2 = ⌊27⁄5⌋ = 5, f2 = 27 − 5a2 = 2 a3 = ⌊5⁄2⌋ = 2, f3 = 5 − 2a3 = 1 a4 = ⌊2⁄1⌋ = 2, f4 = 2 − 1a4 = 0

Using the f values so generated, the ½ ak admissibility test is dk−2  dk−1 > fk  fk−1. For a3 of the example, d1  d2 = 16 and f3  f2 = 12, so ½ a3 is not admissible; while for a4, d2  d3 = 613 and f4  f3 = 01, so ½ a4 is admissible.

The convergents to x are best approximations in an even stronger sense: nd is a convergent for x if and only if |dxn| is the least relative error among all approximations mc with c ≤ d; that is, we have |dxn| < |cxm| so long as c < d.

## Continued fraction expansions of π

To calculate the convergents of pi we may set $a_0 = lfloor pi rfloor = 3$, define $u_1 = frac {1}{pi - 3}$ $approx frac {113}{16} = 7.0625$ and $a_1 = lfloor u_1 rfloor = 7$, $u_2 = frac {1}{u_1 - 7} approx frac {31993}{2000} = 15.9965$ and $a_2 = lfloor u_2 rfloor = 15$, $u_3 = frac {1}{u_2 - 15}$ $approx frac {1003}{1000} = 1.003$. Continuing like this, one can determine the infinite continued fraction of π as [3; 7, 15, 1, 292, 1, 1, ...]. The third convergent of π is [3; 7, 15, 1] = 355/113 = 3.14159292035..., which is fairly close to the true value of π. When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ...

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:

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

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:

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

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.

For more see π.

This continued fraction is unpredictable and irregular, but there are perfectly regular generalized continued fractions for π, such as: When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ...

$pi=3 + cfrac{1}{6 + cfrac{9}{6 + cfrac{25}{6 + cfrac{49}{6 + cfrac{81}{6 + cfrac{121}{ddots,}}}}}}$

## Other continued fraction expansions

### Periodic continued fractions

The numbers with periodic continued fraction expansion are precisely the solutions of quadratic equations with rational coefficients. 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...]. All square roots of integers have a special form for the period; a symmetrical string, like 1,2,1, followed by the double of the leading integer. In mathematics, a quadratic equation is a polynomial equation of the second degree. ... The golden section is a line segment sectioned into two according to the golden ratio. ...

### Regular patterns in continued fractions

While one cannot discern any pattern in the simple continued fraction expansion of π, this is not true for e, the base of the natural logarithm: e is the unique number such that the value of the derivative (slope of a tangent line) of f (x)=ex (blue curve) at the point x=0 is exactly 1. ...

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

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

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

Another, more complex pattern appears in the continued fraction expansion of exp(2/(2n+1)).

Other continued fractions of this sort are

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

where n is a positive integer; also

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

and, for integral n>1,

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

If In(x) is the modified, or hyperbolic, Bessel function of the first kind, we may define a function on the rationals p/q by In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number Î± (the order). ...

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

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

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

with similar formulas for negative rationals; in particular we have

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

The last two formulas are most easily proven in terms of the Bessel-Clifford function. In mathematical analysis, the Bessel-Clifford function is a an entire function of two complex variables which can be used to provide an alternative development of the theory of Bessel functions. ...

### 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 ai (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. Aleksandr Ya. ... In mathematics, the phrase almost all has a number of specialised uses. ... The geometric mean of a set of positive data is defined as the nth root of the product of all the members of the set, where n is the number of members. ... In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, the infinitely many terms ai of the continued fraction expansion of x have an astonishing property: their geometric mean is a constant, known as Khinchins constant, which is independent of the value of x. ... Paul Pierre LÃ©vy (September 15, 1886 - December 15, 1971) was a French mathematician who was active especially in probability theory, introduced martingales and LÃ©vy flights. ... In mathematics LÃ©vys constant (sometimes known as the Khinchin-LÃ©vy constant) occurs in an expression for the asymptotic behaviour of the denominators of the convergents of continued fractions. ...

## Pell's equation

Continued fractions play an essential role in the solution of Pell's equation. For example, for positive integers p and q, $p^2 - 2q^2 = pm1$ if and only if p / q is a convergent of $sqrt2$. Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ...

## Continued fractions and chaos

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. A plot of the trajectory Lorenz system for values r = 28, Ïƒ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ... In mathematics, a Farey sequence of order n is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. ... Initial image of a Mandelbrot set zoom sequence with continuously colored environment. ... Minkowski question mark function In mathematics, the Minkowski question mark function, sometimes called the slippery devils staircase, is a function, denoted ?(x), possessing various unusual fractal properties. ... In mathematics, the modular group &#915; (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...

The backwards shift operator for continued fractions is the map $h(x)=1/x - lfloor 1/x rfloor,$ called the Gauss map, which lops off digits of a continued fraction expansion: $h([0;a_1,a_2,a_3,dots]) = [0;a_2,a_3,dots]$. The transfer operator of this map is called the Gauss-Kuzmin-Wirsing operator. The distribution of the digits in continued fractions is given by the zero'th eigenvector of this operator, and is called the Gauss-Kuzmin distribution. In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. ... In mathematics, the Gauss-Kuzmin-Wirsing operator occurs in the study of continued fractions; it is also related to the Riemann zeta function. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, the Gauss-Kuzmin distribution gives the probability distribution of the occurrence of a given integer in the continued fraction expansion of an arbitrary real number. ...

## History of continued fractions

• 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
Cataldi represented a continued fraction as $a_0.,$ &$n_1 over d_1.$ &$n_2 over d_2.$ &${n_3 over d_3}$ with the dots indicating where the following fractions went.
• 1695 John Wallis, Opera Mathematica - introduction of the term "continued fraction"
• ca 1780 Joseph Louis Lagrange - provided the general solution to Pell's equation using continued fractions similar to Bombelli's
• 1748 Leonhard Euler, Introductio in analysin infinitorum. Vol. I, Chapter 18 - proved the equivalence of a certain form of continued fraction and a generalized infinite series
• 1813 Karl Friedrich Gauss, Werke, Vol. 3, pp. 134-138 - derived a very general complex-valued continued fraction via a clever identity involving the hypergeometric series

Euclid, is also referred to as Euclid of Alexandria, (Greek: , 330 BC â€“ 275 BC), a Greek mathematician, who lived in the city of Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323â€“283 BC), is often considered to be the father of geometry. His most popular work, Elements... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... Raphael Bombelli (1526-1572) was an Italian mathematician. ... Pietro Antonio but face!!Cataldi (April 15, 1552 - February 11, 1626) was an Italian mathematician. ... Pietro Antonio but face!!Cataldi (April 15, 1552 - February 11, 1626) was an Italian mathematician. ... John Wallis John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. ... Joseph-Louis Lagrange, comte de lEmpire (January 25, 1736 â€“ April 10, 1813; b. ... Euler redirects here. ... Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (Gauß) (April 30, 1777 - February 23, 1855) was a legendary German mathematician, astronomer and physicist with a very wide range of contributions; he is considered to be one of the greatest mathematicians of all time. ...

// Definition The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers such that: Rational numbers have a finite Engel expansion, while irrational numbers have an infinite Engel expansion. ... This is a list of mathematical constants sorted by their representations as continued fractions: (Constants known to be irrational have infinite continued fractions: their last term is . ...

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 math lessons - Continued fraction (2024 words) Continued fractions are motivated by a desire to have a "mathematically pure" representation for the real numbers. 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 the Minkowski question mark function and the modular group Gamma. 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.
 continued fraction - Article and Reference from OnPedia.com (2106 words) Continued fractions are motivated by a desire to have a "mathematically pure" representation for the real numbers. 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 the Minkowski question mark function and the modular group Gamma. 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.
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