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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. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers $$a_i$$ are called the coefficients or terms of the continued fraction.

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 $$p$⁄$q$$ has two closely related expressions as a finite continued fraction, whose coefficients $a_{i}$ can be determined by applying the Euclidean algorithm to $$(p,q)$$. The numerical value of an infinite continued fraction is irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number $$\alpha$$ is the value of a unique infinite regular continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values $$\alpha$$ and 1. This way of expressing real numbers (rational and irrational) is called their continued fraction representation.

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

Motivation and notation
Consider, for example, the rational number $415⁄93$, which is around 4.4624. As a first approximation, start with 4, which is the integer part; $415⁄93$ = 4 + $43⁄93$. 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].

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:
 * $√19 = [4;2,1,3,1,2,8,2,1,3,1,2,8,...]$ . The pattern repeats indefinitely with a period of 6.
 * $e = [2;1,2,1,1,4,1,1,6,1,1,8,...]$ . The pattern repeats indefinitely with a period of 3 except that 2 is added to one of the terms in each cycle.
 * $π = [3;7,15,1,292,1,1,1,2,1,3,1,...]$ . No pattern has ever been found in this representation.
 * $Φ = [1;1,1,1,1,1,1,1,1,1,1,1,...]$ . The golden ratio, the irrational number that is the "most difficult" to approximate rationally.
 * $γ = [0;1,1,2,1,2,1,4,3,13,5,1,...]$ . 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. 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

 * $$a_0 + \cfrac{b_1}{a_1 + \cfrac{b_2}{a_2 + \cfrac{b_3}{a_3 + {_\ddots}}}}$$

where ai and bi can be any complex numbers.

When bi = 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. When the terms eventually repeat from some point onwards, the expression is called a periodic continued fraction.

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

For simple continued fractions of the form

$$r=a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + {_\ddots}}}}$$

the $$a_n$$ term can be calculated using the following recursive formula:

$$a_n= \left \lfloor \frac{N_n}{N_{n+1}} \right \rfloor$$

where $$N_{n+1}=N_{n-1} \bmod N_n $$ and $$\begin{cases} N_{0}=r \\ N_{1}=1 \end{cases} $$

From which it can be understood that the $$a_n$$ sequence stops if $$N_{n+1} = 0$$.

Calculating continued fraction representations
Consider a real number $r$. Let $$i=\lfloor r \rfloor $$ and let $f = r - i$. When $f\neq 0$, the continued fraction representation of $$r$$ is $[i;a_1,a_2,\ldots]$, where $$[a_1;a_2,\ldots]$$ is the continued fraction representation of $1/f$. When $r\geq 0$, then $$i$$ is the integer part of $$r$$, and $$f$$ is the fractional part of $r$.

In order to calculate a continued fraction representation of a number $$r$$, write down the floor of $$r$$. Subtract this value from $$r$$. If the difference is 0, stop; otherwise find the reciprocal of the difference and repeat. The procedure will halt if and only if $$r$$ is rational. This process can be efficiently implemented using the Euclidean algorithm when the number is rational.

The table below shows an implementation of this procedure for the number $3.245 = 649/200$:

 * {| class="wikitable"

!Step !Real Number !Integer part !Fractional part !Simplified !Reciprocal of $f$ !1 !2 !3 !4
 * $$r = \frac{649}{200}$$
 * $$i = 3$$
 * $$f = \frac{649}{200} - 3 $$
 * $$= \frac{49}{200}$$
 * $$\frac{1}{f} = \frac{200}{49} $$
 * $$r = \frac{200}{49} $$
 * $$i = 4$$
 * $$f = \frac{200}{49} - 4 $$
 * $$ = \frac{4}{49} $$
 * $$ \frac{1}{f} = \frac{49}{4} $$
 * $$r = \frac{49}{4} $$
 * $$i = 12$$
 * $$f = \frac{49}{4} - 12 $$
 * $$= \frac{1}{4} $$
 * $$\frac{1}{f} = \frac{4}{1} $$
 * $$r = 4$$
 * $$i = 4 $$
 * $$f = 4 - 4 $$
 * $$= 0 $$
 * colspan="2"|STOP
 * }

The continued fraction for $3.245$ is thus $$[3; 4,12,4],$$ or, expanded:

$$ \frac{649}{200} = 3 + \cfrac{1}{4 + \cfrac{1}{12 + \cfrac{1}{4}}}. $$

Notations
The integers $$a_0$$, $$a_1$$ etc., are called the coefficients or terms of the continued fraction. Because a continued fraction such as

 * $$x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4}}}}$$

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

Gottfried Leibniz sometimes used the notation

\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

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

x = a_0 + {1 \over a_1 +}\, {1 \over a_2 +}\, {1 \over a_3 +}\, {1 \over a_4} $$

or

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,

 * $$x = a_0 + \underset{i=1}{\overset{4}{\mathrm K}} ~ \frac{1}{a_i}, $$

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

 * $$x = [a_0; a_1, a_2, a_3, a_4]. $$

Sometimes list-style notation uses angle brackets instead,

 * $$x = \left\langle a_0; a_1, a_2, a_3, a_4 \right\rangle.$$

The semicolon in the square and angle bracket notations is sometimes replaced by a comma.

One may also define infinite simple continued fractions as limits:

 * $$[a_0; a_1, a_2, a_3, \,\ldots\, ] = \lim_{n \to \infty}\, [a_0; a_1, a_2, \,\ldots, a_n]. $$

This limit exists for any choice of $$a_0$$ and positive integers $$a_1,a_2,\ldots$$.

Finite continued fractions
Every finite continued fraction represents a rational number, and every rational number can be represented in precisely two different ways as a finite continued fraction, with the conditions that the first coefficient is an integer and 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:

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

For instance if $$a$$ is an integer and $$x < 1$$ then

 * $$x=0 + \frac{1}{a + \frac{1}{b}}$$ and $$\frac{1}{x} = a + \frac{1}{b}$$.

If $$x>1$$ then
 * $$x = a + \frac{1}{b}$$ and $$\frac{1}{x} = 0 + \frac{1}{a + \frac{1}{b}}$$.

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

For example,
 * $$2.25 = \frac{9}{4} = [2;4]$$ and $$\frac{1}{2.25} = \frac{4}{9} = [0;2,4]$$.

Infinite continued fractions and convergents
Every infinite continued fraction is irrational, and every irrational number can be represented in precisely one way as an infinite continued fraction.

An infinite continued fraction representation for an irrational number is useful because its initial segments provide rational approximations to the number. These rational numbers are called the convergents of the continued fraction. 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 $[a0; a1, a2, ..., an − 1, an, 1] = [a0; a1, a2, ..., an − 1, an + 1]$, the first four convergents (numbered 0 through 3) are

\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:

 * $$\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

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

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].

 * {| class="wikitable"

! $n$ ! $[a0; 1] = [a0 + 1]$ ! $[a0; a1, a2, ...]$ ! $an$
 * - align="right"
 * −2|| −1|| 0 || 1 || 2 || 3 || 4
 * - align="right"
 * ||  || 0 || 1 || 5 || 2 || 2
 * - align="right"
 * 0 || 1 || 0 || 1 || 5 || 11 || 27
 * - align="right"
 * 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, ... , $hn$, ... For example, the continued fraction expansion for $\sqrt3$ is $kn$. Comparing the convergents with the approximants derived from the Babylonian method:

 * {| class="wikitable"

! $n$ ! $2^{k}−1$ ! $[1; 1, 2, 1, 2, 1, 2, 1, 2, ...]$ ! $an$
 * - align="right"
 * −2|| −1|| 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7
 * - align="right"
 * ||  || 1 || 1 || 2 || 1 || 2 || 1 || 2 || 1
 * - align="right"
 * 0 || 1 || 1 || 2 || 5 || 7 || 19 || 26 || 71 || 97
 * - align="right"
 * 1 || 0 || 1 || 1 || 3 || 4 || 11 || 15 || 41 || 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 $$\ a_0\ ,$$ $$a_1\ ,$$ $$a_2\ ,$$ $$\ \ldots\ $$ is an infinite sequence of positive integers, define the sequences $$\ h_n\ $$ and $$\ k_n\ $$ recursively:

Theorem 1. For any positive real number $$\ x\ $$

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

Theorem 2. The convergents of $$\ [\ a_0\ ;$$ $$a_1\ ,$$ $$a_2\ ,$$ $$\ldots\ ]\ $$ are given by

 * $$\left[\ a_0;\ a_1,\ \dots, a_n\ \right] = \frac{h_n}{\ k_n\ } ~.$$

or in matrix form,$$\begin{bmatrix} h_n & h_{n-1} \\ k_n & k_{n-1} \end{bmatrix} = \begin{bmatrix} a_0 & 1 \\ 1 & 0 \end{bmatrix} \cdots \begin{bmatrix} a_n & 1 \\ 1 & 0 \end{bmatrix}$$

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

or equivalently
 * $$ \frac{ h_n }{\ k_n\ } - \frac{ h_{n-1} }{\ k_{n-1}\ } = \frac{ (-1)^{n+1} }{\ k_{n-1}\ k_n\ } ~.$$

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

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

 * $$ \frac{h_n}{k_n} - \frac{ h_{n-1} }{ k_{n-1} } = \frac{\ h_n\ k_{n-1} - k_n\ h_{n-1}\ }{\ k_n\ k_{n-1}\ } = \frac{ (-1)^{n+1} }{\ k_n\ k_{n-1}\ } ~.$$

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

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

Corollary 4: The matrix
 * $$\begin{bmatrix}

h_n & h_{n-1} \\ k_n & k_{n-1} \end{bmatrix} = \begin{bmatrix} a_0 & 1 \\ 1 & 0 \end{bmatrix} \cdots \begin{bmatrix} a_n & 1 \\ 1 & 0 \end{bmatrix}$$ has determinant $$(-1)^{n+1}$$, and thus belongs to the group of $$\ 2\times 2\ $$ unimodular matrices $$\ \mathrm{GL}(2,\mathbb{Z}) ~.$$

Corollary 5: The matrix$$\begin{bmatrix} h_n & h_{n-2} \\ k_n & k_{n-2} \end{bmatrix} = \begin{bmatrix} h_{n-1} & h_{n-2} \\ k_{n-1} & k_{n-2} \end{bmatrix} \begin{bmatrix} a_{n} & 0 \\ 1 & 1 \end{bmatrix}$$ has determinant $$(-1)^na_n$$, or equivalently,$$ \frac{ h_n }{\ k_n\ } - \frac{ h_{n-2} }{\ k_{n-2}\ } = \frac{ (-1)^{n} }{\ k_{n-2 }\ k_n\ }a_n$$meaning that the odd terms monotonically decrease, while the even terms monotonically increase.

Corollary 6: The denominator sequence $$k_0, k_1, k_2, \dots$$ satisfies the recurrence relation $$k_{-1} = 0, k_0 = 1, k_n = k_{n-1}a_n + k_{n-2}$$, and grows at least as fast as the Fibonacci sequence, which itself grows like $$O(\phi^n)$$ where $$\phi= 1.618\dots$$ is the golden ratio.

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

for all $$\ r < s < n ~.$$

Corollary 1: The even convergents (before the $$\ n$$th) continually increase, but are always less than $$\ x_n ~.$$

Corollary 2: The odd convergents (before the $$\ n$$th) continually decrease, but are always greater than $$\ x_n ~.$$

Theorem 5.

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

Corollary 1: 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 $[0;a_1, \dots, a_n], [0;a_1, \dots, a_n+1]$. Denote it as $\mathcal C$. Any open subset of $[0, 1] \setminus \Q$ is a disjoint union of sets from $\mathcal C$."Corollary: The infinite continued fraction provides a homeomorphism from the Baire space to $$[0, 1] \setminus \Q$$.

Semiconvergents
If

 * $$ \frac{h_{n-1}}{k_{n-1}},\frac{h_n}{k_n} $$

are consecutive convergents, then any fractions of the form

 * $$ \frac{h_{n-1} + mh_n}{k_{n-1} + mk_n},$$

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

It follows that semiconvergents represent a monotonic sequence of fractions between the convergents $$\tfrac{h_{n-1}}{k_{n-1}}$$ (corresponding to $$m=0$$) and $$\tfrac{h_{n+1}}{k_{n+1}}$$ (corresponding to $$m=a_{n+1}$$). The consecutive semiconvergents $$\tfrac{a}{b}$$ and $$\tfrac{c}{d}$$ satisfy the property $$ad - bc = \pm 1$$.

If a rational approximation $$\tfrac{p}{q}$$ to a real number $$x$$ is such that the value $$\left|x-\tfrac{p}{q}\right|$$ is smaller than that of any approximation with a smaller denominator, then $$\tfrac{p}{q}$$ is a semiconvergent of the continued fraction expansion of $$x$$. 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 $n⁄d$, $hn$, 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:

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

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

 * {| class="wikitable"

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

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

The "half rule" mentioned above requires that when $a$$k$ is even, the halved term $a$$k$/2 is admissible if and only if $kn$ This is equivalent  to:.

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 $x0 = 1 = 1⁄1$ has the smallest value among the analogous expressions for all rational approximations $m$/$c$ with $x1 = 1⁄2(1 + 3⁄1) = 2⁄1 = 2$; that is, we have $x2 = 1⁄2(2 + 3⁄2) = 7⁄4$ so long as $x3 = 1⁄2(7⁄4 + 3⁄7⁄4) = 97⁄56$. (Note also that $d > 0$ as $|x − [a0 ; a1, ..., ak − 1]| > |x − [a0 ; a1, ..., ak − 1, ak/2]|$.)

Best rational within an interval
A rational that falls within the interval $(x, y)$, for $[ak; ak − 1, ..., a1] > [ak; ak + 1, ...]$, can be found with the continued fractions for $x$ and $y$. When both $x$ and $y$ are irrational and

where $x$ and $y$ have identical continued fraction expansions up through $|dx − n|$, a rational that falls within the interval $(x, y)$ is given by the finite continued fraction,

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

If $x$ is rational, it will have two continued fraction representations that are finite, $c ≤ d$ and $|dx − n| < |cx − m|$, and similarly a rational $y$ will have two representations, $c < d$ and $|d_{k}x − n_{k}| → 0$. The coefficients beyond the last in any of these representations should be interpreted as $k → ∞$; and the best rational will be one of $0 < x < y$, $x = [a0; a1, a2, ..., ak − 1, ak, ak + 1, ...]$, $y = [a0; a1, a2, ..., ak − 1, bk, bk + 1, ...]$, or $a_{k−1}$.

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

and the best rational between these two is

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,

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)
 * $z(x,y) = [a0; a1, a2, ..., ak − 1, min(ak, bk) + 1]$ and

The numbers $x$ and $y$ are formed by incrementing the last coefficient in the two representations for $z$. It is the case that $x_{1}$ when $k$ is even, and $x_{2}$ when $k$ is odd.

For example, the number $355⁄113$ has the continued fraction representations
 * $355⁄113$ = [3; 7, 15, 1] = [3; 7, 16]

and thus $355⁄113$ is a convergent of any number strictly between
 * {| cellpadding="2" cellspacing="0"

 * align="right" | $y_{1}$ ||=|| $y_{2}$
 * align="right" | $+∞$ ||=|| $z(x_{1}, y_{1})$
 * }
 * }

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. A consequence of this criterion, often called Legendre's theorem within the study of continued fractions, is as follows:

Theorem. If α is a real number and p, q are positive integers such that $$\left|\alpha - \frac{p}{q}\right| < \frac{1}{2q^2}$$, then p/q is a convergent of the continued fraction of α.

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

Suppose α, p, q are such that $$\left|\alpha - \frac{p}{q}\right| < \frac{1}{2q^2}$$, and assume that α > p/q. Then we may write $$\alpha - \frac{p}{q} = \frac{\theta}{q^2}$$, where 0 < θ < 1/2. We write p/q as a finite continued fraction [a0; a1, ..., an], 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 an = 1 and one where an ≠ 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, ..., pn/qn = p/q be the convergents of this continued fraction expansion. Set $$\omega := \frac{1}{\theta} - \frac{q_{n-1}}{q_n}$$, so that $$\theta = \frac{q_n}{q_{n-1} + \omega q_n}$$ 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 pn-1 qn - pn qn-1 = (-1)n and that n is even.

Now, this equation implies that α = [a0; a1, ..., an, 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, ..., an, b0, b1, ...], where [b0; b1, ...] is the continued fraction expansion of ω, and therefore that pn/qn = 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)n1/4).

Comparison
Consider $z(x_{1}, y_{2})$ and $z(x_{2}, y_{1})$. If $k$ is the smallest index for which $z(x_{2}, y_{2})$ is unequal to $3.14155 = [3; 7, 15, 2, 7, 1, 4, 1, 1] = [3; 7, 15, 2, 7, 1, 4, 2]$ then $3.14165 = [3; 7, 16, 1, 3, 4, 2, 3, 1] = [3; 7, 16, 1, 3, 4, 2, 4]$ if $[3; 7, 16] = 355⁄113 = 3.1415929....$ and $z = [a0; a1, ..., ak − 1, ak, 1] = [a0; a1, ..., ak − 1, ak + 1] = pk⁄qk$ otherwise.

If there is no such $k$, but one expansion is shorter than the other, say $x = [a0; a1, ..., ak − 1, ak, 2] = 2pk - pk-1⁄2qk - qk-1$ and $y = [a0; a1, ..., ak − 1, ak + 2] = pk + pk-1⁄qk + qk-1$ with $x < y$ for $x > y$, then $[3; 7, 15, 2]$ if $n$ is even and $688⁄219 ≈ 3.1415525$ if $n$ is odd.

Continued fraction expansion of $\pi$ and its convergents
To calculate the convergents of π we may set $[3; 7, 17]$, define $377⁄120 ≈ 3.1416667$ and $x = [a0; a1, ...]$, $y = [b0; b1, ...]$ and $ak$, $bk$. Continuing like this, one can determine the infinite continued fraction of π as
 * [3;7,15,1,292,1,1,...].

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:

To sum up, the pattern is $$ \text{Numerator}_i = \text{Numerator}_{(i-1)} \cdot \text{Quotient}_i + \text{Numerator}_{(i-2)} $$ $$ \text{Denominator}_i = \text{Denominator}_{(i-1)} \cdot \text{Quotient}_i + \text{Denominator}_{(i-2)} $$

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 $3⁄1$ is greater than π, but $22⁄7$ − π is less than $333⁄106$ = $355⁄113$ (in fact, $22⁄7$ − π is just more than $22⁄7$ = $1⁄7 × 106$).

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 $1⁄742$ and $22⁄7$ is $1⁄791$, in excess; between $1⁄7 × 113$ and $22⁄7$, $3⁄1$, in deficit; between $1⁄7$ and $333⁄106$, $22⁄7$, 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:

The first term, as we see, is the first fraction; the first and second together give the second fraction, $1⁄742$; the first, the second and the third give the third fraction $355⁄113$, and so on with the rest; the result being that the series entire is equivalent to the original value.

Generalized continued fraction
A generalized continued fraction is an expression of the form

 * $$x = b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots\,}}}}$$

where the an (n > 0) are the partial numerators, the bn are the partial denominators, and the leading term b0 is called the integer part of the continued fraction.

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

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

or

 * $$\pi=3+\cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{292+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{3+\cfrac{1}{1+\ddots}}}}}}}}}}}$$

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

\pi=\cfrac{4}{1+\cfrac{1^2}{2+\cfrac{3^2}{2+\cfrac{5^2}{2+\cfrac{7^2}{2+\cfrac{9^2}{2+\ddots}}}}}} =\cfrac{4}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\cfrac{4^2}{9+\ddots}}}}} =3+\cfrac{1^2}{6+\cfrac{3^2}{6+\cfrac{5^2}{6+\cfrac{7^2}{6+\cfrac{9^2}{6+\ddots}}}}} $$

 * $$\displaystyle \pi=2+\cfrac{2}{1+\cfrac{1}{1/2+\cfrac{1}{1/3+\cfrac{1}{1/4+\ddots}}}}=2+\cfrac{2}{1+\cfrac{1\cdot2}{1+\cfrac{2\cdot3}{1+\cfrac{3\cdot4}{1+\ddots}}}}$$

 * $$ \displaystyle \pi=2+\cfrac{4}{3+\cfrac{1\cdot3}{4+\cfrac{3\cdot5}{4+\cfrac{5\cdot7}{4+\ddots}}}}$$

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.

\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 $$\pi$$ above consisting of cubes uses the Nilakantha series and an exploit from Leonhard Euler.

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 states that any irrational number $333⁄106$ can be approximated by infinitely many rational $1⁄11978$ with

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

While virtually all real numbers $3⁄1$ will eventually have infinitely many convergents $1⁄1 × 7$ whose distance from $1⁄7 × 106$ is significantly smaller than this limit, the convergents for φ (i.e., the numbers $1⁄106 × 113$, $22⁄7$, $333⁄106$, $k$, etc.) consistently "toe the boundary", keeping a distance of almost exactly $${\scriptstyle{1 \over n^2 \sqrt 5}}$$ away from φ, thus never producing an approximation nearly as impressive as, for example, $m⁄n$ for π. It can also be shown that every real number of the form $k$, where $m⁄n$, $k$, $5⁄3$, and $8⁄5$ are integers such that $x < y$, 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 $(−1)k(ak − bk) < 0$, the base of the natural logarithm:

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

which is a special case of this general expression for positive integer $13⁄8$:

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

Another, more complex pattern appears in this continued fraction expansion for positive odd $21⁄13$:

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

with a special case for $y < x$:

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

Other continued fractions of this sort are

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

where $355⁄113$ is a positive integer; also, for integer $a + bφ⁄c + dφ$:

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

with a special case for $x = [a0; a1, ..., an]$:

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

If $y = [b0; b1, ..., bn, bn + 1, ...]$ is the modified, or hyperbolic, Bessel function of the first kind, we may define a function on the rationals $a$ by

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

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

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

with similar formulas for negative rationals; in particular we have

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

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

Typical continued fractions
Most irrational numbers do not have any periodic or regular behavior in their continued fraction expansion. Nevertheless, for almost all numbers on the unit interval, they have the same limit behavior.

The arithmetic average diverges: $$\lim_{n\to\infty}\frac 1n \sum_{k=1}^n a_k = +\infty$$, and so the coefficients grow arbitrarily large: $$\limsup_n a_n = +\infty$$. In particular, this implies that almost all numbers are well-approximable, in the sense that$$\liminf_{n\to\infty} \left| x - \frac{p_n}{q_n} \right| q_n^2 = 0$$Khinchin proved that the geometric mean of $ai = bi$ tends to a constant (known as Khinchin's constant):$$\lim_{n \rightarrow \infty } \left( a_1 a_2 ... a_n \right) ^{1/n} = K_0 = 2.6854520010\dots$$Paul Lévy proved that the $d$th root of the denominator of the $n$th convergent converges to Lévy's constant $$\lim_{n \rightarrow \infty } q_n^{1/n} = e^{\pi^2/(12\ln2)} = 3.2758\ldots$$Lochs' theorem states that the convergents converge exponentially at the rate of$$\lim_{n\to\infty}\frac 1n \ln\left| x - \frac{p_n}{q_n} \right| = -\frac{\pi^2}{6\ln 2} $$

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

The identity

leads via recursion to the generalized continued fraction for any square root:

Pell's equation
Continued fractions play an essential role in the solution of Pell's equation. For example, for positive integers $n$ and $n$, and non-square $n$, it is true that if $0 ≤ i ≤ n$, then $x < y$ is a convergent of the regular continued fraction for √$p⁄q$. The converse holds if the period of the regular continued fraction for √$p$ is 1, and in general the period describes which convergents give solutions to Pell's equation.

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 $y < x$ called the Gauss map, which lops off digits of a continued fraction expansion: $a0 = ⌊\pi⌋ = 3$. 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.

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.

Examples of rational and irrational numbers
ra: rational approximant obtained by expanding continued fraction up to ar

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 $$a_0$$ & $$\frac{n_1}{d_1 \cdot}$$ & $$\frac{n_2}{d_2 \cdot}$$ & $$\frac{n_3}{d_3 \cdot}$$ with the dots indicating where the following fractions went.

 * 1695 John Wallis, Opera Mathematica – introduction of the term "continued fraction"
 * 1737 Leonhard Euler, De fractionibus continuis dissertatio – Provided the first then-comprehensive account of the properties of continued fractions, and included the first proof that the number e is irrational.
 * 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.
 * 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.

title: Continued fraction


text:
Talk:French Revolution

Long-term impact
Hello all

I have summarised some excessive detail in this section. I have cut information already in this article (for example most of the information on the impact of the Revolution on the Church) or the article Influence of the French Revolution. I have removed unsourced content. As most of the content was about the short-term impact of the Revolution on other countries, I have changed the heading to Impact. I think there is a good argument for removing most of the remaining detail and retaining just a few general observations made by historians. One concern is that there is no way of separating the impact of the revolution of 1789-99 from the long-term impact of the Napoleonic era (which was arguably greater). As it is, the section is an invitation for people to write potted histories of every country in the world since 1789.

Happy to discuss. Aemilius Adolphin (talk) 12:08, 31 October 2023 (UTC)

 * One of the long term impacts is that Liberté, égalité, fraternité has the origin in the French Revolution. I suggest that this should be linked here. Zukunft (talk) 09:50, 30 January 2024 (UTC)
 * OK, you're welcome to to link it. WP:BB —GoldRingChip 12:29, 30 January 2024 (UTC)

Semi-protected edit request on 11 March 2024
There are multiple cases in the article where the revolutionary political figure Antoine Barnave's name is mispelled as "Bernave" and action should be taken to rectify these mistakes. By using the text-search tool and entering in "Bernave" you will find these 2 errors. Sarffgadau (talk) 19:49, 11 March 2024 (UTC)
 * ✅ Jamedeus (talk) 19:56, 11 March 2024 (UTC)

Semi-protected edit request on 23 April 2024
Impact: North America The French Revolution had profound effects on the people of San Domingo, modern-day Haiti. The French Revolution served as a guideline and proof a revolution could take place to rework social balance, as well as a massive inspiration to the revolutionaries on the island. White slave owners in San Domingo were terrified of the idea of a slave revolt. This led to a splitting of the island into Patriots and Loyalists. Ultimately tensions would boil to the point of an outright rebellion which would prove effective as the world's only successful slave revolt. JohnOverton1 (talk) 05:20, 23 April 2024 (UTC)

 * This is discussed at length in the section on Slavery and the Colonies. Aemilius Adolphin (talk) 05:48, 23 April 2024 (UTC)

Semi-protected edit request on 8 May 2024
can i edit 24.194.201.151 (talk) 18:25, 8 May 2024 (UTC)
 * Red information icon with gradient background.svg Not done: this is not the right page to request additional user rights. You may reopen this request with the specific changes to be made and someone may add them for you, or if you have an account, you can wait until you are autoconfirmed and edit the page yourself. Jamedeus (talk) 19:39, 8 May 2024 (UTC)

Semi-protected edit request on 16 June 2024
Fix typo. Change the heading "Crisis of theAncien Régime" to "Crisis of the Ancien Régime". 109.255.93.165 (talk) 20:18, 16 June 2024 (UTC)
 * Red information icon with gradient background.svg Not done: There is no such typo. The wikitext is " ". Is there a problem with your browser? Jean-de-Nivelle (talk) 20:47, 16 June 2024 (UTC)
 * This seems to be a markup problem in some browsers that's being resolved at source. See Village_pump_(technical). Jean-de-Nivelle (talk) 07:37, 17 June 2024 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:French Revolution


text:
Talk:Palestinians killed by Israelis

Untitled
This title is mind-bogglingly imprecise. I'd move the article if I knew where to move it to. Koyaanis Qatsi, Saturday, March 30, 2002

There is a major problem with this article, especially when compared to Israelis killed by Palestinians. The problem (in case you can't tell) is that while the other article is very extensive, this article is very lacking and makes it seem as if that is the true proportion of death tolls since the beginning of the conflict.--Node 22:50, 21 Apr 2004 (UTC)

from VfD
On 22 Oct 2004, this article was nominated for deletion. The results of that discussion were ambiguous and the decision defaulted to Keep. However, based on the evidence presented and the edit history of this article, I believe that it would be better as a redirect than as a stand-alone article. (Reminder: Redirect decisions are reversible and therefore to not need a VfD consensus.)  For a record of the full discussion, see Votes for deletion/Palestinians killed by Israelis. Rossami (talk) 00:06, 11 Nov 2004 (UTC)

title: Talk:Palestinians killed by Israelis


text:
Talk:Mohamed Atta

Florida Couple
Surprised not to find this here, relevant to Atta's entry:

A news report about a Saudi family that disappeared from their home in Sarasota, Fla., just two weeks before the 9/11 attacks has brought an angry response from the co-chair of a congressional Sept. 11 committee. [Senator] Graham called Sarasota case "eerily similar" to the FBI's failure to tell the intelligence committee about a former Saudi civil servant, Omar al-Bayoumi, who supported two hijackers while they were living in San Diego..independent journalist Anthony Summers..said on msnbc tv that a hushed-up inquiry found that "three of the (9/11) pilot hijackers had all been in touch with the Saudis in that house." Abdulazzi al-Hiijjii, his wife Anoud and their two children resided in a home owned by Anoud's father, Esam Ghazzawi, in the gated Sarasota subdivision called Prestancia, according to the report...According to the report, the FBI connected the couple to more than a dozen terrorists through telephone records and through their car license tags and driver's licenses as they passed through the subdivision's security gate. Among the terrorists who visited the home or called the couple was 9/11 leader Mohamed Atta, the report said.

Graham said he had no reason to doubt the news report, which said the couple and their two children abruptly abandoned their luxury home, leaving behind a full refrigerator, clothes, furnishings and a new car in the driveway. The news report was based on information from an unnamed counterterrorism official, a neighbor, subdivision administrators, the subdivision security guard and the subdivision lawyer, who said the FBI tried to get him to lure the homeowner back to the United States. According to the report, the Sarasota couple returned to Riyadh, Saudi Arabia, with Anoud's father after abandoning their home. See (above are slightly rearranged and much condensed snippets from) this source and similar reports like

Shared Names
http://www.nytimes.com/2001/09/16/us/after-attacks-missed-cues-saudi-may-have-been-suspected-error-officials-say.html

Mr. Atta is one of three hijackers who has the same name as a person who had received training with the U.S. military

Able Danger
The section on Able Danger to be rewrite to better reflect the Able Danger project, read Able Danger on wiki for a more representative picture. —Preceding unsigned comment added by 66.245.216.205 (talk) 20:45, 26 March 2009 (UTC)

image gallery
At the end of the article there was an image gallery without much sense. Maybe you'd like to use these images in context. --141.30.72.76 (talk) 15:24, 25 February 2008 (UTC)

 * There isn't much "context" to the images as they are just documents, preserved for historical value, more than illustrating a specific fact about his life. Sherurcij (Speaker for the Dead) 15:29, 25 February 2008 (UTC)

 * Images are useful. AdjustShift (talk) 02:51, 1 September 2008 (UTC)

Bot report : Found duplicate references !
In the last revision I edited, I found duplicate named references, i.e. references sharing the same name, but not having the same content. Please check them, as I am not able to fix them automatically :) DumZiBoT (talk) 04:25, 8 August 2008 (UTC)
 * "hooper" :

Fair use images
This article has numerous fair use images. The Non-free_content policy applies here. A key criteria is "Significance. Non-free content is used only if its presence would significantly increase readers' understanding of the topic." Minimal usage of fair use images is one factor to consider.

 * So, we have two images of Mohammed Atta's drivers' license, one in the infobox that is cropped to just show his picture. The second is in further below article text.  I think the infobox photo is justified, since it was widely circulated by the FBI, and demonstrates his "cold" appearance as noted by many.  The picture of his entire drivers license on the other hand, it doesn't add much at all to the readers' understanding of the topic.

 * We also have Atta's college photograph, which again is questionable, since it mainly illustrates or decorates that section of the article.

 * The Huffman Aviation record fits well with the text, illustrating all the hours of training Atta had. It goes beyond what the text says and I think significantly increases readers' understanding.

 * We also have two security camera photos, both which I think are definitely justified to show Atta's whereabouts and activities on 9/10 and 9/11.

 * Atta's visa is public domain, since it's a work of the U.S. government, so no problem with including it.

--Aude (talk) 17:48, 16 September 2008 (UTC)

Also, there is Atta's martydom video. I think a screenshot from the video would significantly add to readers' understanding. --Aude (talk) 17:58, 16 September 2008 (UTC)

 * The simplest solution for the driver's license problem is to include the driver's license, but change our "main" image to another Public Domain image of him if such can be found; thus eliminating the "duplicate" nature. `Sherurcij (speaker for the dead) 18:28, 16 September 2008 (UTC)
 * How does the drivers license add significantly to the readers' understanding of the topic? --Aude (talk) 18:35, 16 September 2008 (UTC)
 * I've erased "Image:AttasFLDriversLic.jpg" from the bio. AdjustShift (talk) 14:59, 24 September 2008 (UTC)

Dubious
Sometime in the last eight months a bunch of dubious templates have been added to this article. These templates are less intrusive than some other editorial templates. But they do direct interested readers to the "Dubious" section of the article. But the concerned wikipedian didn't add one.

The template, when expanded, currently adds two links (1) one to Disputed statement, which explains how the template is supposed to be used; and (2) another to the section of the article entitled Dubious.

The instructions tell concerned wikipedians, when they encountered passages that concerned them, to first see if they could fix the passages on their own. If they couldn't, they were then advised to add a "Disputed" section to the talk page, describing their concern. Only after describing their concern were they supposed to add dubious tags.

AGF -- the first tags were added eight months ago. Maybe the guideline has been changed in the last eight months, and didn't spell out as clearly the concerned wikipedians responsibilities? But the use of this tag clearly doesn't comply with the guideline, and I am going to suggest that any that aren't explained in a reasonable period of time should be removed.

Cheers! Geo Swan (talk) 10:53, 31 March 2009 (UTC)

 * I am mostly on a wikibreak now, but at some point plan to spend more time on the Mohamed Atta article and get it to featured article standards. It's indeed taking a long while for me to get back to working on the article, and not sure when I'll have the time to do so.  The article is fairly close, though there are some unsourced passages and there are sections where the sources (and statements) need checking and replacement with more reliable sources.  Those sources marked as "dubious" are such sources that are questionable. What I could do for now is replace the tags with hidden inline comments that are only visible when in edit mode.

 * As for leaving a note on the talk page, that's just a guideline. Where it's been mostly just me working on an article, as has been the case with the Mohamed Atta article, I sometimes don't bother with notes on the talk page.  But, more than willing to respond if someone does ask on the talk page.  --Aude (talk) 00:45, 1 April 2009 (UTC)

Okina
As I understand it the okina is a purely Hawaiian character. Yet it is in one rendering of Atta's name. ??? Rich Farmbrough, 03:53, 8 August 2009 (UTC).

Redirects for discussion/Log/2010 July 16
Editors may be interested in contributing to this discussion. Bridgeplayer (talk) 01:22, 16 July 2010 (UTC)

Mohamed Atta's identity
A number of reports suggest the possibility that the person purporting to act under the name of Mohamed Atta in the United States was not the same person as the Mohamed el-Amir Atta who was born in Egypt and studied in Hamburg, Germany. Contrary to anonymous intelligence or law-enforcement sources, these reports were published in mainstream media and provide the full names of their sources;

1. Two men who locals recognized to have been Marwan al-Shehhi and Mohamed Atta stayed at a popular resort hotel in Mabalacat, the Philippines, sometime between 1998 and 2000, drank whiskey with Philippine bargirls, dined at a restaurant that specializes in Middle Eastern cuisine and visited at least one of the local flight schools. Philippine and U.S. investigators have been checking out the reported movements of Marwan Al-Shehhi and Mohamed Atta but would not confirm the accused hijackers' presence in the Philippines. Local hotel workers, including Gina Marcelo, a former waitress, Victoria Brocoy, a chambermaid and Ferdinand Abad, a security guard at the Woodland Park Resort Hotel, were however willing to talk to media about these guests. Another person, interviewed by the New York Times, who remembered Mohamed Atta, was Trudis Dago, manager of the Jerusalem Restaurant in Angeles City. [New York Times, October 5, 2001]

2. Spruce Whited, head of security at the [Portland public] library, said he first saw a man he is convinced was Atta in April 2000. He said the man came to the library several times, using the computers. "I only recognized him because he'd been here a few times," he said. Kathy Barry, a reference librarian, also reported seeing Atta, whose photograph has been distributed widely through the media since the Sept. 11 attacks. [Portland Press Herald Report, October 5, 2001, ]

3. An in-depth interview by eminent journalist Brian Ross with Johnelle Bryant broadcast on ABC News on June 6, 2002, where she tells him about her encounter with Mohamed Atta in April or May 2000 in her office at the Department of Agriculture in Florida. and

4. Mohamed Atta, the suspected terrorist who crashed hijacked American Airlines Flight 11 into the World Trade Center, went twice to a small airport in South Florida and asked detailed questions about how to start and fly a crop-duster plane. In Belle Glade, local crop-dusters recounted Sunday what Atta asked about, and speculated on how one of their planes could be used for a terrorist mission. "[Atta] wanted to know how to fly it, how to crank it, how much it would haul, said James Lester, 50, who maintains and loads a 502 Air Tractor crop-duster with as much as 500 gallons of insecticide and fertilizer. Lester said the 33-year-old Atta visited the small airstrip here with several groups of men as recently as last month. "The FBI showed me [Atta's] photo, said Lester, who remembered at least two encounters with Atta -- once in March when he drove up in a green van with two other people, and again in August when he flew into the airport in a single-engine Cessna. "The reason why I recognized him was because he was always walking behind me, being real persistent in asking those questions." [The Miami Herald, October 29, 2001,

These reports provide facts that are incompatible with the official account on Mohamed Atta on three grounds: (a) Mohamed Atta is seen in the Philippines acting as a playboy, contrary to his description as a pious Muslim (b) Mohamed Atta is seen by several independent witnesses in the United States prior to his official entry date into the U.S. (c) Mohamed Atta is described as someone intent to make himself conspicuous as well as suspicious within the U.S.

Whatever the deep significance of these facts, they constitute part and parcel of the public account on Mohamed Atta and must appear on any objective account regarding this person. —Preceding unsigned comment added by Emetvetzedek (talk • contribs) 13:05, 8 August 2010 (UTC)

Challenged without citations
is there a reason why User:JohnCengiz77 feels that these statements need to be in the article without citations? All they need are citations, but he seems willing to reintroduce challenged material without citations and is close to being blocked for it. Please read WP:Verifiability and become accustomed to how things actually work on wikipedia.--Jojhutton (talk) 14:17, 10 August 2010 (UTC)

Martyrdom Video Authenticity
I'm surprised there is no mention here or elsewhere regarding the authenticity of the "martyrdom" video. Specifically, the portion of the video allegedly linking the hijackers to Osama Bin Laden addressing some sort of rally in Afghanistan. Prior to the announcement of this footage, the same rally is featured via an alternate camera angle in the docudrama "The Road to Guantánamo" released earlier during the same year. This footage also contains the same 1/8/2000 date in the datestamp. Coincidentally, the footage in the film supposedly has the detainee inserted via CG so they can try and convince him that he was present at the rally. The datestamp during the rally (in the martydom video) is inconsistent with the edits (including a scene dubbed in which later reverts to the dubbed over scene picking back up where it left off with the new date stamp - camcorders didn't work like this) but I can't find any 'official' sources which state this. Anybody who watches the video can see the inconsistent date stamp for themselves although that would fall under the umbrella of original research I guess. Either way, I think "The Road to Guantánamo" deserves mention since it is the same rally on the same day (according to the date stamp) from the same country that released the martyrdom video and in the same year in a partially fictitious docudrama film (at the 74 minute mark). Ryal-oh (talk) 22:02, 8 April 2011 (UTC)

Was Mohamed Atta
Was Mohamed Atta living with his family in Birmingham, Alabama in 2001? — Preceding unsigned comment added by 72.211.255.221 (talk) 00:25, 5 September 2011 (UTC)

pure terror
Atta was the first face of terror that I came across ever since I was a child. I am seeing his face probably after 10 years today and the mere sight of him has brought back those drastic images of terror back to me. I read his message of 'expanding terror', i wonder how can a person feel gratification of absolutely any kind after having killed another human being. just how !!!!!!!! — Preceding unsigned comment added by 115.119.100.122 (talk) 05:54, 10 September 2011 (UTC)

Mastermind
Atta was a coward, not mastermind. The first sentence should be rephrased. — Preceding unsigned comment added by 174.114.202.71 (talk) 19:26, 14 September 2011 (UTC)
 * That's the term used in reliable sources, not some sort of comment on how good or bad a person he is. In any event one could actually be both a mastermind and a coward--the two are not incompatible. Qwyrxian (talk) 02:13, 15 September 2011 (UTC)

Mohamed Atta's father
WhisperToMe (talk) 17:10, 6 January 2014 (UTC)
 * "Atta's father praises London bombs." (Archive) CNN. Wednesday July 20, 2005.

Section titles
Is there a reason for using "Notes" as the section-title for the citations? I would have expected that to be called "References".

Along the same lines, is there a reason for using "References" as the section-title for the fullsize-books? I would have expected that to be called "Further Reading".

See WP:FNNR and WP:CITEVAR, which explicitly say this: "Editors may use any section title that they choose. ...unless there is consensus to change, defer to the style used by the first major contributor." Current section-titles seem different from what I would expect, but they are not wrong. And I'm not a stickler about "house style" everywhere. If nobody speaks up, then I'll mark this down as no-need-to-change-it.

Worth asking, though: anybody else think changing the section-titles, from notes-to-references, and from references-to-furtherReading, is worthwhile as an exercise in consistency? 74.192.84.101 (talk) 04:16, 1 February 2014 (UTC)

Atta and the CIA
Speculation exists that Atta was handled by the CIA and was used in Kosovo in an attempt to destabilize Milosovich. Claims of his devout Islamic faith have largely been fabricated. — Preceding unsigned comment added by 75.82.92.150 (talk) 22:22, 11 September 2014 (UTC)

External links modified
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Passport Cynicism
' When Atta returned, he claimed that his passport was lost and got a new one, which is a common tactic to erase evidence of travel to places such as Afghanistan. '

This encyclopedic article ought to state facts not cynicism. Beingsshepherd (talk) 16:02, 17 May 2016 (UTC)

External links modified
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EVIL
He is an evil man who deserves nothing but punishment. >:( — Preceding unsigned comment added by 2602:306:34D4:F390:F1E1:FD3A:AE27:EC7C (talk) 23:41, 19 December 2017 (UTC)

External links modified (February 2018)
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Mohamed Attas daughter marries Osama bin Ladens son
See here: http://www.spiegel.de/politik/ausland/osama-bin-ladens-sohn-soll-tochter-von-mohammed-atta-geheiratet-haben-a-1221752.html and also https://www.cbsnews.com/news/osama-bin-laden-son-married-911-hijacker-daughter-reporter-says-today-2018-08-04/ Why is it not mentioned in the article that Mohamed Atta has a daughter? --2003:74:CF49:1D3D:B0BA:73D:BDC4:BDBE (talk) 20:12, 5 August 2018 (UTC)

Umm..
So if Waleed al-Shehri and Wail al-Shehri was on flight 11, then why was Marwan al-Shehhi was on Flight 175? Bacrens (talk) 23:03, 18 January 2020 (UTC)
 * Are you implying all three were brothers? Waleed and Wail were, but Marwan had no relation to either of them (and even his surname was different: al-Shehhi rather than al-Shehri.) --Ismail (talk) 07:02, 3 November 2020 (UTC)

Article was renamed from "Mohamed Atta" to "Muhammad 'Atta"
I notice the article has been renamed, under the theory that the new name "Muhammad 'Atta" is more common than "Mohamed Atta". Certainly my reading on the subject has essentially always found it transliterated as Mohamed Atta. I have literally never seen a news or historical article use the other name. I don't even know what the apostrophe before Atta represents in Arabic. Any comments on this? Tarl N. ( discuss ) 03:35, 6 August 2023 (UTC)
 * The article rename has been undone, and under WP:BRD have reverted the massive changes to the article until discussion takes place here. Tarl N. ( discuss ) 05:33, 6 August 2023 (UTC)

Ringleader vs. Planner and commander
I just reverted an edit changing the description of Atta from "Ringleader" to "Planner and operational commander". I find the change profoundly offensive - he was a mass murderer, and whitewashing descriptions of him is not appropriate. Any disagreements should be discussed here. Tarl N. ( discuss ) 02:40, 27 November 2023 (UTC)
 * Copied from editor's talk page, so discussion can take place here:
 * Isn't a ringleader the same thing as a commander? I can't see how this whitewashes him, especially considering everything else that describes what he did in the article which would make any attempt to "whitewash" him futile without severe vandalization of the article. He is also described as a "commander" in the main Al-Qaeda article (September 11 attacks section, second paragraph, second line). In any case, the main point was to add "planner" to the section. I apologize if you found this edit offensive, and I will not revert your reversion, I will however ask if I can add back the "Planner", as he was heavily involved in the planning of the attack. Enginemen (talk) 13:44, 27 November 2023 (UTC)
 * No. Look the words up. Ringleader and commander are quite different words. One carries the explicit statement of criminal behaviour, the other carries the explicit statement of military command structure. Conflating criminal and military actions (which is characteristic of previous edits by you) is exactly what is so offensive. As for planner, that's implicit in ringleader - leading a criminal gang means planning their actions. The infobox is supposed to be concise, and the term "ringleader" is perfectly adequate, no need to go into great detail there when the rest of the article already has that information.Tarl N. ( discuss ) 00:55, 28 November 2023 (UTC)

title: Talk:Mohamed Atta


text:
Talk:Bioleaching

General
There used to be an article Thiobacillus thio-oxidans. Its entire content was "cu"; it had no history and so no non-vandalized versions, and so I deleted it. If someone here is competent to write about the subject, please do. :-) Koyaanis Qatsi, Saturday, March 30, 2002

I find the description of the process rather confusing; it first deals with copper, then shows a totally different example, dealing with arsenopyrite, and finishes off saying "The gold is now separated from the ore and in solution." - someone should make it a bit clearer! [Moe], Saturday, February 25, 2006

In general,the gold ores mixed with pyrite and arsenopyrite. After pyrite and arsenopyrite oxidized by bacteria, the gold particle is separated from the ore.[macromath],Thursday, Dec 28, 2006

There is a problem with this - This is much cleaner than the traditional heap leaching using cyanide.[1] For bacterial leaching of gold, cyanide is still required to recover the gold, and in fact more is probably required. However, cyanide in an of itself is not really a long term environmental hazard. It has short term toxicity, but is readily broken down by UV light.

I am a specialist in the field and I'll spend some time on this page. The Puppeteer (talk) 03:41, 18 October 2023 (UTC)

Biocatalytic Processes
The physiology and energetic manipulation of metal leaching organisms are extemely significant, especially for gold, copper,uranium leaching and recovery of metals such as arsenic, silver, and mercury.Organisms that thrive in extreme environments such as those described above, are of interest in the production of highly stable enzymes and in the development of certain innovative bioprocesses. One area of environmental/biotechnological research would be the  realization of the biocatalytic potential of these extremophilic microbes, which thrive at very high (boiling) temperatures, high pressures, highly saline or acidic environments. An area of interest would be the development of environmentally relevant (bio)technology based on microbial degradation of the recalcitrant pollutants. This calls for characterization of single microbes and mixed cultures that can survive amidst high concentrations of the pollutants.Essential evaluation of microbial physiology in conjunction with industrially relevant molecular bioprocess design is required for development of these system which can aid as great learning/research tools in arenas of enzyme or cell processing applications alongwith mining, waste-water treatment, bioremediation. Understanding of these "hyperthermophilic anaerobes" that encompass a widely metabolic variety can be utilized for novel applications such as high-temperature 'anaerobic digestors',aiding the conversion of the waste to useful products,molecular engineering of enzymes etc. reshmibanerjee 07:16, 23 June 2008 (UTC)

Nickel Bioleaching
Please would an industrial chemist, mining expert or other suitably knowledgeable person add a section or paragraph on bioleaching nickel, as being carried out by Talvivaara. I think they also produce zinc as a by-product. Dendrotek (talk) 15:15, 11 June 2009 (UTC)

title: Talk:Bioleaching


text:
Long Valley Caldera

Long Valley Caldera is a depression in eastern California that is adjacent to Mammoth Mountain. The valley is one of the Earth's largest calderas, measuring about 20 mi long (east-west), 11 mi wide (north-south), and up to 3000 ft deep.

Long Valley was formed 760,000 years ago when a very large eruption released hot ash that later cooled to form the Bishop tuff that is common to the area. The eruption emptied the magma chamber under the area to the point of collapse. The second phase of the eruption released pyroclastic flows that burned and buried thousands of square miles. Ash from this eruption blanketed much of the western part of what is now the United States.

Geography
The caldera is a giant bowl-shaped depression, approximately 20 mi long, surrounded by mountains except to the southeast. The elevation of the bottom of the bowl ranges from 6500 to 8500 ft, being higher in the west.

Near the center of the bowl, magmatic uplift has formed a resurgent dome. The southeastern slope from the caldera down towards Bishop, California, is filled with the Bishop Tuff, solidified ash that was ejected during the eruption that created the caldera. The Bishop tuff is 1500 m thick in the caldera floor, and is cut by the Owens River Gorge, formed during the Pleistocene when the caldera filled with water and overtopped its rim.

The rim of the caldera is formed from pre-existing rock, rising about 3000 ft above the caldera floor. However, the eastern rim is lower, only about 500 ft.

Mammoth Mountain is a lava dome complex west of the structural rim of the caldera, consisting of about 12 rhyodacite and dacite overlapping domes. These domes formed in a long series of eruptions from 110,000 to 57,000 years ago, building a volcano that reaches 11059 ft in elevation.

The Mono–Inyo Craters are a 25 mi-long volcanic chain situated along a narrow, north–south-trending fissure system extending along the western rim of the caldera from Mammoth Mountain to the north shore of Mono Lake. The Mono-Inyo Craters erupted from 40,000 to 600 years ago, from a magma source separate from the Long Valley Caldera.

The caldera has an extensive hydrothermal system. Casa Diablo Hot Springs at the base of the resurgent dome hosts a geothermal power plant. Hot Creek cuts into part of the resurgent dome and passes through hot springs. The warm water of Hot Creek supports many trout, and is used at the Hot Creek Fish Hatchery. The creek was closed to swimming in 2006 after geothermal activity in the area increased. The area has a number of other hot springs, some of which are open to bathers.

Caldera
The tectonic causes of the Long Valley volcanism are still largely unexplained and are therefore a matter of much ongoing research. Long Valley is not above a hotspot, such as those which fuel Yellowstone Caldera or the volcanoes of Hawaii; nor is it the result of subduction such as that which produces the volcanism of the Cascades.

The known volcanic history of Long Valley Caldera area started a few million years ago when magma began to collect several miles below the surface. Volcanic activity became concentrated in the vicinity of the present site of Long Valley Caldera 3.1 to 2.5 million years ago with eruptions of rhyodacite followed by high-silica rhyolite from 2.1 to 0.8 million years ago. After some time, a cluster of mostly rhyolitic volcanoes formed in the area. All told, about 1500 sqmi were covered by lava.

All but one of these volcanoes, 1–2 million year old Glass Mountain (made of obsidian), were destroyed by the major (volcanic explosivity index (VEI) 7) eruption of the area 760,000 years ago, which released 600. km3 of material from vents just inside the margin of the caldera. (The 1980 Mount St. Helens eruption was a VEI-5 eruption releasing 1.2 km3.) About half of this material was ejected in a series of pyroclastic flows of a very hot (1500 F) mixture of noxious gas, pumice, and volcanic ash that covered the surrounding area hundreds of feet deep. One lobe of this material moved south into Owens Valley, past present-day Big Pine, California. Another lobe moved west over the crest of the Sierra Nevada and into the drainage of the San Joaquin River. The rest of the pyroclastic material, along with 300 km3 of other matter, was blown as far as 25 mi into the air where winds distributed it as far away as eastern Nebraska and Kansas.

The eruption initially produced a caldera 2 - 3 km deep. However, much of the ejecta went straight up, fell down, and filled the initial caldera about two-thirds full.

Eruptions

Subsequent eruptions from the Long Valley magma chamber were confined within the caldera with extrusions of relatively hot (crystal-free) rhyolite 700,000 to 600,000 years ago as the caldera floor was uplifted to form the resurgent dome followed by extrusions of cooler, crystal-rich moat rhyolite at 200,000-year intervals (500,000, 300,000, and 100,000 years ago) in clockwise succession around the dome. The declining volcanic activity and increasingly crystalline lava extruded over the last 650,000 years, as well as other trends, suggest that the magma reservoir under the caldera has now largely crystallized and is unlikely to produce large-scale eruptions in the future.

The Long Valley volcano is unusual in that it has produced eruptions of both basaltic and silicic lava in the same geological place.

Water from the Owens River filled the caldera to a depth of 300 m as of 600,000 years ago. At that time, the lake surface was at an elevation near 7500 ft. The lake drained sometime in the last 100,000 years after it overtopped the southern rim of the caldera, eroded the sill, and created the Owens River Gorge. A human-made dam in the gorge has created Lake Crowley, a partial restoration of the original lake. Since the great eruption, many hot springs developed in the area, and the resurgent dome has uplifted.

During the last ice age, glaciers filled the canyons leading to Long Valley, but the valley floor was clear of ice. Excellent examples of terminal moraines can be seen at Long Valley. Laurel Creek, Convict Creek, and McGee Creek each have prominent moraines.

Recent activity
In May 1980, a strong earthquake swarm that included four Richter magnitude 6 earthquakes struck the southern margin of the Long Valley Caldera. It was associated with a 10 in dome-shaped uplift of the caldera floor. These events marked the onset of the latest period of caldera unrest that is ongoing. This ongoing unrest includes recurring earthquake swarms and continued dome-shaped uplift of the central section of the caldera accompanied by changes in thermal springs and gas emissions. After the quake, a secondary access road was created as a potential escape route for the town of Mammoth Lakes. Its name at first was proposed as the "Mammoth Escape Route" but was changed to the Mammoth Scenic Loop after Mammoth area businesses and land owners complained.

In 1982, the United States Geological Survey under the Volcano Hazards Program began an intensive effort to monitor and study geologic unrest in Long Valley Caldera. The goal is to provide residents and civil authorities with reliable information on the nature of the potential hazards posed by this unrest and timely warning of an impending volcanic eruption, should it develop. Most, perhaps all, volcanic eruptions are preceded and accompanied by geophysical and geochemical changes in the volcanic system. Common precursory indicators of volcanic activity include increased seismicity, ground deformation, and variations in the nature and rate of gas emissions.

Hydrothermal system

The Long Valley Caldera hosts an active hydrothermal system that includes hot springs, fumaroles (steam vents), and mineral deposits. Hot springs exist primarily in the eastern half of the caldera where land-surface elevations are relatively low; fumaroles exist primarily in the western half where elevations are higher. Mineral deposits from thermal activity are found on an uplifted area called the resurgent dome, at Little Hot Creek springs, Hot Creek Gorge, and other locations in the south and east moats of the caldera.

Hot springs discharge primarily in Hot Creek Gorge, along Little Hot Creek, and in the Alkali Lakes area. The largest springs are in Hot Creek Gorge where about 250 L per second of thermal water discharge and account for about 80% of the total thermal water discharge in the caldera. At the other extreme are springs at Hot Creek Fish Hatchery which contain a small component (2–5%) of thermal water that raises water temperatures about 5 C-change higher than background temperatures. Use of the warm spring water in the hatchery has increased fish production because trout growth rates are faster in the warm water than in ambient stream temperatures in Long Valley.

In hydrothermal systems the circulation of groundwater is driven by a combination of topography and heat sources. In Long Valley Caldera, the system is recharged primarily from snow-melt in the highlands around the western and southern rims of the caldera. The water from snow-melt and rainfall infiltrates to depths of a few kilometers (miles) where it is heated to at least 220 C by hot rock near geologically young intrusions. Upflow occurs in the west moat where the heated water with lower density rises along steeply inclined fractures to depths of 1 - 2 km. This hydrothermal fluid flows laterally, down the hydraulic gradient, from the west to the southeast around the resurgent dome and then eastward to discharge points along Hot Creek and around Crowley Lake. Reservoir temperatures in the volcanic fill decline from 220 C near the Inyo Craters to 50 C near Crowley Lake due to a combination of heat loss and mixing with cold water.

Hot Creek has been a popular swimming hole for decades. Over a dozen people have died in Hot Creek since the late 1960s, but most of these deaths happened to people who ignored the numerous warning signs and attempted to use the hydrothermal pools as hot tubs (like the stream portion of the creek, these pools alternate in temperature but the eruptions in the pools are of super-heated water in already very hot water). Recent geothermal instability has led to its temporary closure for swimming. Officials are unsure of when (if ever) Hot Creek will officially reopen for swimming.

Hydrothermal activity has altered many rocks in the caldera, transforming them into travertine and clay. At the Huntley clay mine white chalky clay called kaolinite is mined; the kaolinite is exposed on the resurgent dome and appears as a brilliant white band.

Tourism and hiking
The largest tourist attraction in the caldera is the Mammoth Mountain Ski Area: the area offers skiing and snowboarding in the winter, and mountain biking in the summer. The Hot Creek tourist attraction was closed to swimming in 2006 due to increased geothermal activity.

Hiking and off-road vehicle driving is available throughout the caldera, and in the glacial valleys of the Sherwin Range, immediately to the south of the caldera. Hikers can hike to several lakes in these glacial valleys, including Valentine Lake, Convict Lake, Lake Dorothy, and Laurel Lakes. Crowley Lake, at the south end of the caldera, is noted for its fishing.

The nearest hotel accommodations to the caldera are in Mammoth Lakes, California. There are also campgrounds scattered throughout the caldera, and in the mountains near the edge of the caldera.

Fatalities
In April 2006, three members of the Mammoth Mountain Ski Area ski patrol died while on duty. All three died from suffocation by carbon dioxide when they fell into a fumarole on the slopes of the mountain while attempting to fence it off.

title: Long Valley Caldera