Helioseismology

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Helioseismology

Helioseismology is the study of the structure and dynamics of the Sun through its oscillations. These are principally caused by sound waves that are continuously driven and damped by convection near the Sun's surface. It is similar to geoseismology, or asteroseismology, which are respectively the studies of the Earth or stars through their oscillations. While the Sun's oscillations were first detected in the early 1960s, it was only in the mid-1970s that it was realized that the oscillations propagated throughout the Sun and could allow scientists to study the Sun's deep interior. The term was coined by Douglas Gough in the 90s. The modern field is separated into global helioseismology, which studies the Sun's resonant modes directly, and local helioseismology, which studies the propagation of the component waves near the Sun's surface.

Helioseismology has contributed to a number of scientific breakthroughs. The most notable was to show that the anomaly in the predicted neutrino flux from the Sun could not be caused by flaws in stellar models and must instead be a problem of particle physics. The so-called solar neutrino problem was ultimately resolved by neutrino oscillations. The experimental discovery of neutrino oscillations was recognized by the 2015 Nobel Prize for Physics. Helioseismology also allowed accurate measurements of the quadrupole (and higher-order) moments of the Sun's gravitational potential, which are consistent with General Relativity. The first helioseismic calculations of the Sun's internal rotation profile showed a rough separation into a rigidly-rotating core and differentially-rotating envelope. The boundary layer is now known as the tachocline and is thought to be a key component for the solar dynamo. Although it roughly coincides with the base of the solar convection zone — also inferred through helioseismology — it is conceptually distinct, being a boundary layer in which there is a meridional flow connected with the convection zone and driven by the interplay between baroclinicity and Maxwell stresses.

Helioseismology benefits most from continuous monitoring of the Sun, which began first with uninterrupted observations from near the South Pole over the austral summer. In addition, observations over multiple solar cycles have allowed helioseismologists to study changes in the Sun's structure over decades. These studies are made possible by global telescope networks like the Global Oscillations Network Group (GONG) and the Birmingham Solar Oscillations Network (BiSON), which have been operating for over several decades.

Types of solar oscillation

Illustration of a solar pressure mode (p mode) with radial order n=14, angular degree l=20 and azimuthal order m=16. The surface shows the corresponding spherical harmonic. The interior shows the radial displacement computed using a standard solar model. Note that the increase in the speed of sound as waves approach the center of the Sun causes a corresponding increase in the acoustic wavelength.

Solar oscillation modes are interpreted as resonant vibrations of a roughly spherically symmetric self-gravitating fluid in hydrostatic equilibrium. Each mode can then be represented approximately as the product of a function of radius ${\displaystyle r}$ and a spherical harmonic ${\displaystyle Y_l^m(\theta ,\varphi )}$, and consequently can be characterized by the three quantum numbers which label:

• the number of nodal shells in radius, known as the radial order ${\displaystyle n}$;
• the total number of nodal circles on each spherical shell, known as the angular degree ${\displaystyle \ell }$; and
• the number of those nodal circles that are longitudinal, known as the azimuthal order ${\displaystyle m}$.

It can be shown that the oscillations are separated into two categories: interior oscillations and a special category of surface oscillations. More specifically, there are:

Pressure modes (p modes)

Pressure modes are in essence standing sound waves. The dominant restoring force is the pressure (rather than buoyancy), hence the name. All the solar oscillations that are used for inferences about the interior are p modes, with frequencies between about 1 and 5 millihertz and angular degrees ranging from zero (purely radial motion) to order ${\displaystyle {10}^3}$. Broadly speaking, their energy densities vary with radius inversely proportional to the sound speed, so their resonant frequencies are determined predominantly by the outer regions of the Sun. Consequently it is difficult to infer from them the structure of the solar core.

A propagation diagram for a standard solar model showing where oscillations have a g-mode character (blue) or where dipole modes have a p-mode character (orange). The dashed line shows the acoustic cut-off frequency, computed from more precise modelling, and above which modes are not trapped in the star, and roughly-speaking do not resonate.

Gravity modes (g modes)

Gravity modes are confined to convectively stable regions, either the radiative interior or the atmosphere. The restoring force is predominantly buoyancy, and thus indirectly gravity, from which they take their name. They are evanescent in the convection zone, and therefore interior modes have tiny amplitudes at the surface and are extremely difficult to detect and identify. It has long been recognized that measurement of even just a few g modes could substantially increase our knowledge of the deep interior of the Sun. However, no individual g mode has yet been unambiguously measured, although indirect detections have been both claimed and challenged. Additionally, there can be similar gravity modes confined to the convectively stable atmosphere.

For more, see Phoebus group.

Surface gravity modes (f modes)

Surface gravity waves are analogous to waves in deep water, having the property that the Lagrangian pressure perturbation is essentially zero. They are of high degree ${\displaystyle \ell }$, penetrating a characteristic distance ${\displaystyle R/\ell }$, where ${\displaystyle R}$ is the solar radius. To good approximation, they obey the so-called deep-water-wave dispersion law: ${\displaystyle {\omega }^2=g{k}_{\mathrm{h}}}$, irrespective of the stratification of the Sun, where ${\displaystyle \omega }$ is the angular frequency, ${\displaystyle g}$ is the surface gravity and ${\displaystyle k_{\mathrm{h}}=\ell /R}$ is the horizontal wavenumber, and tend asymptotically to that relation as ${\displaystyle k_{\mathrm{h}}\to \mathrm{\infty }}$.

What seismology can reveal

The oscillations that have been successfully utilized for seismology are essentially adiabatic. Their dynamics is therefore the action of pressure forces ${\displaystyle p}$ (plus putative Maxwell stresses) against matter with inertia density ${\displaystyle \rho }$, which itself depends upon the relation between them under adiabatic change, usually quantified via the (first) adiabatic exponent ${\displaystyle {\gamma }_1}$. The equilibrium values of the variables ${\displaystyle p}$ and ${\displaystyle \rho }$ (together with the dynamically small angular velocity ${\displaystyle \mathrm{\Omega }}$ and magnetic field ${\displaystyle \mathrm{B}}$) are related by the constraint of hydrostatic support, which depends upon the total mass ${\displaystyle M}$ and radius ${\displaystyle R}$ of the Sun. Evidently, the oscillation frequencies ${\displaystyle \omega }$ depend only on the seismic variables ${\displaystyle \rho (p,\mathrm{\Omega },\mathrm{B})}$, ${\displaystyle {\gamma }_1}$, ${\displaystyle \mathrm{\Omega }}$ and ${\displaystyle \mathrm{B}}$, or any independent set of functions of them. Consequently it is only about these variables that information can be derived directly. The square of the adiabatic sound speed, ${\displaystyle c^2={\gamma }_{1}p/\rho }$, is such commonly adopted function, because that is the quantity upon which acoustic propagation principally depends. Properties of other, non-seismic, quantities, such as helium abundance, ${\displaystyle Y}$, or main-sequence age ${\displaystyle t_{\odot }}$, can be inferred only by supplementation with additional assumptions, which renders the outcome more uncertain.

Data analysis

Global helioseismology

Power spectrum of the Sun using data from instruments aboard the Solar and Heliospheric Observatory on double-logarithmic axes. The three passbands of the VIRGO/SPM instrument show nearly the same power spectrum. The line-of-sight velocity observations from GOLF are less sensitive to the red noise produced by granulation. All the datasets clearly show the oscillation modes around 3mHz.

Power spectrum of the Sun around where the modes have maximum power, using data from the GOLF and VIRGO/SPM instruments aboard the Solar and Heliospheric Observatory. The low-degree modes (l<4) show a clear comb-like pattern with a regular spacing.

Power spectrum of medium angular degree (${\displaystyle 0\le \ell <300}$) solar oscillations, computed for 144 days of data from the MDI instrument aboard SOHO. The colour scale is logarithmic and saturated at one hundredth the maximum power in the signal, to make the modes more visible. The low-frequency region is dominated by the signal of granulation. As the angular degree increases, the individual mode frequencies converge onto clear ridges, each corresponding to a sequence of low-order modes.

The chief tool for analysing the raw seismic data is the Fourier transform. To good approximation, each mode is a damped harmonic oscillator, for which the power as a function of frequency is a Lorentz function. Spatially resolved data are usually projected onto desired spherical harmonics to obtain time series which are then Fourier transformed. Helioseismologists typically combine the resulting one-dimensional power spectra into a two-dimensional spectrum.

The lower frequency range of the oscillations is dominated by the variations caused by granulation. This must first be filtered out before (or at the same time that) the modes are analysed. Granular flows at the solar surface are mostly horizontal, from the centres of the rising granules to the narrow downdrafts between them. Relative to the oscillations, granulation produces a stronger signal in intensity than line-of-sight velocity, so the latter is preferred for helioseismic observatories.

Local helioseismology

Local helioseismology—a term coined by Charles Lindsey, Doug Braun and Stuart Jefferies in 1993—employs several different analysis methods to make inferences from the observational data.

• The Fourier–Hankel spectral method was originally used to search for wave absorption by sunspots.
• Ring-diagram analysis, first introduced by Frank Hill, is used to infer the speed and direction of horizontal flows below the solar surface by observing the Doppler shifts of ambient acoustic waves from power spectra of solar oscillations computed over patches of the solar surface (typically 15° × 15°). Thus, ring-diagram analysis is a generalization of global helioseismology applied to local areas on the Sun (as opposed to half of the Sun). For example, the sound speed and adiabatic index can be compared within magnetically active and inactive (quiet Sun) regions.
• Time-distance helioseismology aims to measure and interpret the travel times of solar waves between any two locations on the solar surface. Inhomogeneities near the ray path connecting the two locations perturb the travel time between those two points. An inverse problem must then be solved to infer the local structure and dynamics of the solar interior.
• Helioseismic holography, introduced in detail by Charles Lindsey and Doug Braun for the purpose of far-side (magnetic) imaging, is a special case of phase-sensitive holography. The idea is to use the wavefield on the visible disk to learn about active regions on the far side of the Sun. The basic idea in helioseismic holography is that the wavefield, e.g., the line-of-sight Doppler velocity observed at the solar surface, can be used to make an estimate of the wavefield at any location in the solar interior at any instant in time. In this sense, holography is much like seismic migration, a technique in geophysics that has been in use since the 1940s. As another example, this technique has been used to give a seismic image of a solar flare.
• In direct modelling, the idea is to estimate subsurface flows from direct inversion of the frequency-wavenumber correlations seen in the wavefield in the Fourier domain. Woodard demonstrated the ability of the technique to recover near-surface flows the f modes.

Inversion

Introduction

The Sun's oscillation modes represent a discrete set of observations that are sensitive to its continuous structure. This allows scientists to formulate inverse problems for the Sun's interior structure and dynamics. Given a reference model of the Sun, the differences between its mode frequencies and those of the Sun, if small, are weighted averages of the differences between the Sun's structure and that of the reference model. The frequency differences can then be used to infer those structural differences. The weighting functions of these averages are known as kernels.

Structure

The first inversions of the Sun's structure were made using Duvall's law and later using Duvall's law linearized about a reference solar model. These results were subsequently supplemented by analyses that linearize the full set of equations describing the stellar oscillations about a theoretical reference model and are now a standard way to invert frequency data. The inversions demonstrated differences in solar models that were greatly reduced by implementing gravitational settling: the gradual separation of heavier elements towards the solar centre (and lighter elements to the surface to replace them).

Rotation

The internal rotation profile of the Sun inferred using data from the Helioseismic and Magnetic Imager aboard the Solar Dynamics Observatory. The inner radius has been truncated where the measurements are less certain than 1%, which happens around 3/4 of the way to the core. The dashed line indicates the base of the solar convection zone, which happens to coincide with the boundary at which the rotation profile changes, known as the tachocline.

If the Sun were perfectly spherical, the modes with different azimuthal orders m would have the same frequencies. Rotation, however, breaks this degeneracy, and the modes frequencies differ by rotational splittings that are weighted-averages of the angular velocity through the Sun. Different modes are sensitive to different parts of the Sun and, given enough data, these differences can be used to infer the rotation rate throughout the Sun. For example, if the Sun were rotating uniformly throughout, all the p modes would be split by approximately the same amount. Actually, the angular velocity is not uniform, as can be seen at the surface, where the equator rotates faster than the poles. The Sun rotates slowly enough that a spherical, non-rotating model is close enough to reality for deriving the rotational kernels.

Helioseismology has shown that the Sun has a rotation profile with several features:

• a rigidly-rotating radiative (i.e. non-convective) zone, though the rotation rate of the inner core is not well known;
• a thin shear layer, known as the tachocline, which separates the rigidly-rotating interior and the differentially-rotating convective envelope;
• a convective envelope in which the rotation rate varies both with depth and latitude; and
• a final shear layer just beneath the surface, in which the rotation rate slows down towards the surface.

Relationship to other fields

Geoseismology

Main article: Seismology

Helioseismology was born from analogy with geoseismology but several important differences remain. First, the Sun lacks a solid surface and therefore cannot support shear waves. From the data analysis perspective, global helioseismology differs from geoseismology by studying only normal modes. Local helioseismology is thus somewhat closer in spirit to geoseismology in the sense that it studies the complete wavefield.

Asteroseismology

Main article: Asteroseismology

Because the Sun is a star, helioseismology is closely related to the study of oscillations in other stars, known as asteroseismology. Helioseismology is most closely related to the study of stars whose oscillations are also driven and damped by their outer convection zones, known as solar-like oscillators, but the underlying theory is broadly the same for other classes of variable star.

The principal difference is that oscillations in distant stars cannot be resolved. Because the brighter and darker sectors of the spherical harmonic cancel out, this restricts asteroseismology almost entirely to the study of low degree modes (angular degree ${\displaystyle \ell \le 3}$). This makes inversion much more difficult but upper limits can still be achieved by making more restrictive assumptions.

History

Solar oscillations were first observed in the early 1960s as a quasi-periodic intensity and line-of-sight velocity variation with a period of about 5 minutes. Scientists gradually realized that the oscillations might be global modes of the Sun and predicted that the modes would form clear ridges in two-dimensional power spectra. The ridges were subsequently confirmed in observations of high-degree modes in the mid 1970s, and mode multiplets of different radial orders were distinguished in whole-disc observations. At a similar time, Jørgen Christensen-Dalsgaard and Douglas Gough suggested the potential of using individual mode frequencies to infer the interior structure of the Sun. They calibrated solar models against the low-degree data finding two similarly good fits, one with low ${\displaystyle Y}$ and a corresponding low neutrino production rate ${\displaystyle L_{\nu }}$, the other with higher ${\displaystyle Y}$ and ${\displaystyle L_{\nu }}$; earlier envelope calibrations against high-degree frequencies preferred the latter, but the results were not wholly convincing. It was not until Tom Duvall and Jack Harvey connected the two extreme data sets by measuring modes of intermediate degree to establish the quantum numbers associated with the earlier observations that the higher-${\displaystyle Y}$ model was established, thereby suggesting at that early stage that the resolution of the neutrino problem must lie in nuclear or particle physics.

New methods of inversion developed in the 1980s, allowing researchers to infer the profiles sound speed and, less accurately, density throughout most of the Sun, corroborating the conclusion that residual errors in the inference of the solar structure is not the cause of the neutrino problem. Towards the end of the decade, observations also began to show that the oscillation mode frequencies vary with the Sun's magnetic activity cycle.

To overcome the problem of not being able to observe the Sun at night, several groups had begun to assemble networks of telescopes (e.g. the Birmingham Solar Oscillations Network, or BiSON, and the Global Oscillation Network Group) from which the Sun would always be visible to at least one node. Long, uninterrupted observations brought the field to maturity, and the state of the field was summarized in a 1996 special issue of Science magazine. This coincided with the start of normal operations of the Solar and Heliospheric Observatory (SoHO), which began producing high-quality data for helioseismology.

The subsequent years saw the resolution of the solar neutrino problem, and the long seismic observations began to allow analysis of multiple solar activity cycles. The agreement between standard solar models and helioseismic inversions was disrupted by new measurements of the heavy element content of the solar photosphere based on detailed three-dimensional models. Though the results later shifted back towards the traditional values used in the 1990s, the new abundances significantly worsened the agreement between the models and helioseismic inversions. The cause of the discrepancy remains unsolved and is known as the solar abundance problem.

Space-based observations by SoHO have continued and SoHO was joined in 2010 by the Solar Dynamics Observatory (SDO), which has also been monitoring the Sun continuously since its operations began. In addition, ground-based networks (notably BiSON and GONG) continue to operate, providing nearly continuous data from the ground too.

Simple continued fraction

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Simple_continued_fraction

A simple or regular continued fraction is a continued fraction with numerators all equal to one, and denominators built from a sequence ${\displaystyle \{a_i\}}$ of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fraction like

${\displaystyle a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{\ddots +\frac{1}{a_n}}}}}$

or an infinite continued fraction like

${\displaystyle a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{\ddots }}}.}$

Typically, such a continued fraction is obtained through a recursive process which starts by representing a number as the sum of its integer part and its fractional part. The integer is recorded and the reciprocal of the fractional part is then recursively represented by another continued fraction. In the finite case, the recursion is stopped 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 ${\displaystyle a_i}$ are called the coefficients or terms of the continued fraction.

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

Motivation and notation

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

That expression 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 place a semicolon after the first number to indicate that it 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,...] (sequence A010124 in the OEIS). The pattern repeats indefinitely with a period of 6.
• e = [2;1,2,1,1,4,1,1,6,1,1,8,...] (sequence A003417 in the OEIS). The pattern repeats indefinitely with a period of 3 except that 2 is added to one of the terms in each cycle.
• π = [3;7,15,1,292,1,1,1,2,1,3,1,...] (sequence A001203 in the OEIS). No pattern has ever been found in this representation.
• φ = [1;1,1,1,1,1,1,1,1,1,1,1,...] (sequence A000012 in the OEIS). The golden ratio, the irrational number that is the "most difficult" to approximate rationally (see § A property of the golden ratio φ below).
• γ = [0;1,1,2,1,2,1,4,3,13,5,1,...] (sequence A002852 in the OEIS). The Euler–Mascheroni constant, which is expected but not known to be irrational, and whose continued fraction has no apparent pattern.

Continued fractions are, in some ways, more "mathematically natural" representations of a real number than other representations such as decimal representations, and they have several desirable properties:

• The continued fraction representation for a real number is finite if and only if it is a rational number. In contrast, the decimal representation of a rational number may be finite, for example ⁠137/1600⁠ = 0.085625, or infinite with a repeating cycle, for example ⁠4/27⁠ = 0.148148148148...
• Every rational number has an essentially unique simple continued fraction representation. Each rational can be represented in exactly two ways, since [a₀;a₁,... a_n−1,a_n] = [a₀;a₁,... 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".

Formulation

A continued fraction in canonical form is an expression of the form

${\displaystyle a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\phantom{\frac{\mathrm{}}{\mathrm{}}}{}_{\ddots }}}}}$

where a_i are integer numbers, called the coefficients or terms of the 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 continued fraction is called periodic.

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

Examples of finite simple continued fractions

Formula	Numeric	Remarks
${\displaystyle a_0}$	${\displaystyle 2}$	All integers are a degenerate case
${\displaystyle a_0+\frac{1}{a_1}}$	${\displaystyle 2+\frac{1}{3}}$	Simplest possible fractional form
${\displaystyle a_0+\frac{1}{a_1+\frac{1}{a_2}}}$	${\displaystyle -3+\frac{1}{2+\frac{1}{18}}}$	First integer may be negative
${\displaystyle a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3}}}}$	${\displaystyle \frac{1}{15+\frac{1}{1+\frac{1}{102}}}}$	First integer may be zero

For simple continued fractions of the form

${\displaystyle r=a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\phantom{\frac{\mathrm{}}{\mathrm{}}}{}_{\ddots }}}}}$

the ${\displaystyle a_n}$ term can be calculated from the following recursive sequence:

${\displaystyle f_{n+1}=\frac{1}{f_n-\lfloor f_n\rfloor }}$

where ${\displaystyle f_0=r}$ and ${\displaystyle a_n=\lfloor f_n\rfloor }$.

from which it can be understood that the ${\displaystyle a_n}$ sequence stops if ${\displaystyle f_n=\lfloor f_n\rfloor }$ is an integer.

Notations

Consider a continued fraction expressed as

${\displaystyle x=a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{a_4}}}}}$

Because such a continued fraction expression may take a significant amount of vertical space, a number of methods have been tried to shrink it.

Gottfried Leibniz sometimes used the notation

${\displaystyle \begin{array}{r}x=a_0+{\displaystyle \frac{1}{a_1}}\genfrac{}{}{0ex}{}{}{+}\\ \end{array}\begin{array}{r}{\displaystyle \frac{1}{a_2}}\genfrac{}{}{0ex}{}{}{+}\\ \end{array}\begin{array}{r}{\displaystyle \frac{1}{a_3}}\genfrac{}{}{0ex}{}{}{+}\end{array}\begin{array}{r}\\ {\displaystyle \frac{1}{a_4}},\end{array}}$

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

${\displaystyle x=a_0+\genfrac{}{}{0ex}{}{}{|}\frac{1}{a_1}\genfrac{}{}{0ex}{}{|}{}+\genfrac{}{}{0ex}{}{}{|}\frac{1}{a_2}\genfrac{}{}{0ex}{}{|}{}+\genfrac{}{}{0ex}{}{}{|}\frac{1}{a_3}\genfrac{}{}{0ex}{}{|}{}+\genfrac{}{}{0ex}{}{}{|}\frac{1}{a_4}\genfrac{}{}{0ex}{}{|}{},}$

or in more common related notations as

${\displaystyle x=a_0+\frac{1}{a_1+}\frac{1}{a_2+}\frac{1}{a_3+}\frac{1}{a_4}}$

or

${\displaystyle x=a_0+\frac{1}{a_1}\genfrac{}{}{0ex}{}{}{+}\frac{1}{a_2}\genfrac{}{}{0ex}{}{}{+}\frac{1}{a_3}\genfrac{}{}{0ex}{}{}{+}\frac{1}{a_4}.}$

Carl Friedrich Gauss used a notation reminiscent of summation notation,

${\displaystyle x=a_0+\underset{i=1}{\stackrel{4}{\mathrm{K}}}\frac{1}{a_i},}$

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

${\displaystyle x=[a_0;a_1,a_2,a_3,a_4].}$

Sometimes list-style notation uses angle brackets instead,

${\displaystyle x=⟨a_0;a_1,a_2,a_3,a_4⟩.}$

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:

${\displaystyle [a_0;a_1,a_2,a_3,\dots ]=\underset{n\to \mathrm{\infty }}{lim}[a_0;a_1,a_2,\dots ,a_n].}$

This limit exists for any choice of ${\displaystyle a_0}$ and positive integers ${\displaystyle a_1,a_2,\dots }$

Calculating continued fraction representations

Consider a real number ⁠${\displaystyle r}$⁠. Let ${\displaystyle i=\lfloor r\rfloor }$ and let ⁠${\displaystyle f=r-i}$⁠. When ⁠${\displaystyle f\ne 0}$⁠, the continued fraction representation of ${\displaystyle r}$ is ⁠${\displaystyle [i;a_1,a_2,\dots ]}$⁠, where ${\displaystyle [a_1;a_2,\dots ]}$ is the continued fraction representation of ⁠${\displaystyle 1/f}$⁠. When ⁠${\displaystyle r\ge 0}$⁠, then ${\displaystyle i}$ is the integer part of ${\displaystyle r}$, and ${\displaystyle f}$ is the fractional part of ⁠${\displaystyle r}$⁠.

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

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

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

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

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

Finding graphically the continued fraction of a number by repeatedly fitting the largest possible square into an oblong of that aspect ratio

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 ${\displaystyle [a_0;a_1,a_2,\dots ,a_n]}$ and ${\displaystyle [0;a_0,a_1,\dots ,a_n]}$ are reciprocals.

For instance if ${\displaystyle a}$ is an integer and ${\displaystyle x<1}$ then

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

If ${\displaystyle x>1}$ then

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

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

For example,

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

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:

[a₀; a₁, a₂, ..., a_{n − 1}, a_n, 1] = [a₀; a₁, a₂, ..., a_{n − 1}, a_n + 1].

[a₀; 1] = [a₀ + 1].

Infinite continued fractions and convergents

Convergents approaching the golden ratio

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

An infinite continued fraction representation for an irrational number is useful because its initial segments provide rational approximations to the number. These rational numbers are called the convergents of the continued fraction. 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, ...], the first four convergents (numbered 0 through 3) are

${\displaystyle \frac{a_0}{1},\frac{a_1{a}_0+1}{a_1},\frac{a_2(a_1{a}_0+1)+a_0}{a_2{a}_1+1},\frac{a_3(a_2(a_1{a}_0+1)+a_0)+(a_1{a}_0+1)}{a_3(a_2{a}_1+1)+a_1}.}$

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 h₁, h₂, ... and denominators k₁, k₂, ... then the relevant recursive relation is that of Gaussian brackets:

${\displaystyle \begin{array}{rl}{h}_n& =a_n{h}_{n-1}+h_{n-2},\\ k_n& =a_n{k}_{n-1}+k_{n-2}.\end{array}}$

The successive convergents are given by the formula

${\displaystyle \frac{h_n}{k_n}=\frac{a_n{h}_{n-1}+h_{n-2}}{a_n{k}_{n-1}+k_{n-2}}.}$

Thus to incorporate a new term into a rational approximation, only the two previous convergents are necessary. The initial "convergents" (required for the first two terms) are ⁰⁄₁ and ¹⁄₀. For example, here are the convergents for [0;1,5,2,2].

n	−2	−1	0	1	2	3	4
an			0	1	5	2	2
hn	0	1	0	1	5	11	27
kn	1	0	1	1	6	13	32

When using the Babylonian method to generate successive approximations to the square root of an integer, if one starts with the lowest integer as first approximant, the rationals generated all appear in the list of convergents for the continued fraction. Specifically, the approximants will appear on the convergents list in positions 0, 1, 3, 7, 15, ... , 2^k−1, ... For example, the continued fraction expansion for ${\displaystyle \sqrt{3}}$ is [1; 1, 2, 1, 2, 1, 2, 1, 2, ...]. Comparing the convergents with the approximants derived from the Babylonian method:

n	−2	−1	0	1	2	3	4	5	6	7
an			1	1	2	1	2	1	2	1
hn	0	1	1	2	5	7	19	26	71	97
kn	1	0	1	1	3	4	11	15	41	56

x₀ = 1 = ⁠1/1⁠

x₁ = ⁠1/2⁠(1 + ⁠3/1⁠) = ⁠2/1⁠ = 2

x₂ = ⁠1/2⁠(2 + ⁠3/2⁠) = ⁠7/4⁠

x₃ = ⁠1/2⁠(⁠7/4⁠ + ⁠3/⁠7/4⁠⁠) = ⁠97/56⁠

Properties

The 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 ${\displaystyle a_0,}$ ${\displaystyle a_1,}$ ${\displaystyle a_2,}$ ${\displaystyle \dots }$ is an infinite sequence of positive integers, define the sequences ${\displaystyle h_n}$ and ${\displaystyle k_n}$ recursively:

${\displaystyle h_n=a_n{h}_{n-1}+h_{n-2},}$			${\displaystyle h_{-1}=1,}$		${\displaystyle h_{-2}=0;}$
${\displaystyle k_n=a_n{k}_{n-1}+k_{n-2},}$			${\displaystyle k_{-1}=0,}$		${\displaystyle k_{-2}=1.}$

Theorem 1. For any positive real number ${\displaystyle x}$

${\displaystyle \left[a_0;a_1,\dots ,a_{n-1},x\right]=\frac{x{h}_{n-1}+h_{n-2}}{x{k}_{n-1}+k_{n-2}},\left[a_0;a_1,\dots ,a_{n-1}+x\right]=\frac{h_{n-1}+x{h}_{n-2}}{k_{n-1}+x{k}_{n-2}}}$

Theorem 2. The convergents of ${\displaystyle [a_0;}$ ${\displaystyle a_1,}$ ${\displaystyle a_2,}$ ${\displaystyle \dots ]}$ are given by

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

or in matrix form,$${\displaystyle \left[\begin{array}{cc}{h}_n& h_{n-1}\\ k_n& k_{n-1}\end{array}\right]=\left[\begin{array}{cc}{a}_0& 1\\ 1& 0\end{array}\right]\cdots \left[\begin{array}{cc}{a}_n& 1\\ 1& 0\end{array}\right]}$$

Theorem 3. If the ${\displaystyle n}$th convergent to a continued fraction is ${\displaystyle \frac{h_n}{k_n},}$ then

${\displaystyle k_n{h}_{n-1}-k_{n-1}{h}_n=(-1{)}^n,}$

or equivalently

${\displaystyle \frac{h_n}{k_n}-\frac{h_{n-1}}{k_{n-1}}=\frac{(-1{)}^{n+1}}{k_{n-1}{k}_n}.}$

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

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

${\displaystyle \frac{h_n}{k_n}-\frac{h_{n-1}}{k_{n-1}}=\frac{h_n{k}_{n-1}-k_n{h}_{n-1}}{k_n{k}_{n-1}}=\frac{(-1{)}^{n+1}}{k_n{k}_{n-1}}.}$

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

${\displaystyle a_0+\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{k_n{k}_{n+1}}.}$

Corollary 4: The matrix

${\displaystyle \left[\begin{array}{cc}{h}_n& h_{n-1}\\ k_n& k_{n-1}\end{array}\right]=\left[\begin{array}{cc}{a}_0& 1\\ 1& 0\end{array}\right]\cdots \left[\begin{array}{cc}{a}_n& 1\\ 1& 0\end{array}\right]}$

has determinant ${\displaystyle (-1{)}^{n+1}}$, and thus belongs to the group of ${\displaystyle 2\times 2}$ unimodular matrices ${\displaystyle \mathrm{G}\mathrm{L}(2,\mathbb{Z}).}$

Corollary 5: The matrix$${\displaystyle \left[\begin{array}{cc}{h}_n& h_{n-2}\\ k_n& k_{n-2}\end{array}\right]=\left[\begin{array}{cc}{h}_{n-1}& h_{n-2}\\ k_{n-1}& k_{n-2}\end{array}\right]\left[\begin{array}{cc}{a}_n& 0\\ 1& 1\end{array}\right]}$$ has determinant ${\displaystyle (-1{)}^n{a}_n}$, or equivalently,$${\displaystyle \frac{h_n}{k_n}-\frac{h_{n-2}}{k_{n-2}}=\frac{(-1{)}^n}{k_{n-2}{k}_n}{a}_n}$$meaning that the odd terms monotonically decrease, while the even terms monotonically increase.

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

Theorem 4. Each (${\displaystyle s}$th) convergent is nearer to a subsequent (${\displaystyle n}$th) convergent than any preceding (${\displaystyle r}$th) convergent is. In symbols, if the ${\displaystyle n}$th convergent is taken to be ${\displaystyle \left[a_0;a_1,\dots ,a_n\right]=x_n,}$ then

${\displaystyle \left|x_r-x_n\right|>\left|x_s-x_n\right|}$

for all ${\displaystyle r<s<n.}$

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

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

Theorem 5.

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

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

Corollary 2: A convergent obtained by terminating the continued fraction just before a large term is a close approximation to the limit of the continued fraction.

Theorem 6: Consider the set of all open intervals with end-points ${\displaystyle [0;a_1,\dots ,a_n],[0;a_1,\dots ,a_n+1]}$. Denote it as ${\displaystyle \mathcal{C}}$. Any open subset of ${\displaystyle [0,1]\setminus \mathbb{Q}}$ is a disjoint union of sets from ${\displaystyle \mathcal{C}}$.

Corollary: The infinite continued fraction provides a homeomorphism from the Baire space to ${\displaystyle [0,1]\setminus \mathbb{Q}}$.

Semiconvergents

If

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

are consecutive convergents, then any fractions of the form

${\displaystyle \frac{h_{n-1}+m{h}_n}{k_{n-1}+m{k}_n},}$

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

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

If a rational approximation ${\displaystyle \frac{p}{q}}$ to a real number ${\displaystyle x}$ is such that the value ${\displaystyle \left|x-\frac{p}{q}\right|}$ is smaller than that of any approximation with a smaller denominator, then ${\displaystyle \frac{p}{q}}$ is a semiconvergent of the continued fraction expansion of ${\displaystyle x}$. The converse is not true, however.

Best rational approximations

See also: Diophantine approximation and Padé approximant

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

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.

Continued fraction	[0;1]	[0;1,3]	[0;1,4]	[0;1,5]	[0;1,5,2]	[0;1,5,2,1]	[0;1,5,2,2]
Rational approximation	1	⁠3/4⁠	⁠4/5⁠	⁠5/6⁠	⁠11/13⁠	⁠16/19⁠	⁠27/32⁠
Decimal equivalent	1	0.75	0.8	~0.83333	~0.84615	~0.84211	0.84375
Error	+18.519%	−11.111%	−5.1852%	−1.2346%	+0.28490%	−0.19493%	0%

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

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

The "half rule" mentioned above requires that when a_k is even, the halved term a_k/2 is admissible if and only if |x − [a₀ ; a₁, ..., a_{k − 1}]| > |x − [a₀ ; a₁, ..., a_{k − 1}, a_k/2]|. This is equivalent to:

[a_k; a_{k − 1}, ..., a₁] > [a_k; a_{k + 1}, ...].

The convergents to x are "best approximations" in a much stronger sense than the one defined above. Namely, n/d is a convergent for x if and only if |dx − n| has the smallest value among the analogous expressions for all rational approximations m/c with c ≤ d; that is, we have |dx − n| < |cx − m| so long as c < d. (Note also that |d_kx − n_k| → 0 as k → ∞.)

Best rational within an interval

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

x = [a₀; a₁, a₂, ..., a_{k − 1}, a_k, a_{k + 1}, ...]

y = [a₀; a₁, a₂, ..., a_{k − 1}, b_k, b_{k + 1}, ...]

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

z(x,y) = [a₀; a₁, a₂, ..., a_{k − 1}, min(a_k, b_k) + 1]

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

If x is rational, it will have two continued fraction representations that are finite, x₁ and x₂, and similarly a rational y will have two representations, y₁ and y₂. The coefficients beyond the last in any of these representations should be interpreted as +∞; and the best rational will be one of z(x₁, y₁), z(x₁, y₂), z(x₂, y₁), or z(x₂, y₂).

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

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

3.14165 = [3; 7, 16, 1, 3, 4, 2, 3, 1] = [3; 7, 16, 1, 3, 4, 2, 4]

and the best rational between these two is

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

Thus, ⁠355/113⁠ is the best rational number corresponding to the rounded decimal number 3.1416, in the sense that no other rational number that would be rounded to 3.1416 will have a smaller numerator or a smaller denominator.

Interval for a convergent

A rational number, which can be expressed as finite continued fraction in two ways,

z = [a₀; a₁, ..., a_{k − 1}, a_k, 1] = [a₀; a₁, ..., a_{k − 1}, a_k + 1] = ⁠p_k/q_k⁠

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

x = [a₀; a₁, ..., a_{k − 1}, a_k, 2] = ⁠2p_k - p_k-1/2q_k - q_k-1⁠ and

y = [a₀; a₁, ..., a_{k − 1}, a_k + 2] = ⁠p_k + p_k-1/q_k + q_k-1⁠

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

For example, the number ⁠355/113⁠ (Zu's fraction) 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

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

Legendre's theorem on continued fractions

See also: Dirichlet's approximation theorem

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 ${\displaystyle \left|\alpha -\frac{p}{q}\right|<\frac{1}{2q^2}}$, then p/q is a convergent of the continued fraction of α.

Proof

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

Comparison

Consider x = [a₀; a₁, ...] and y = [b₀; b₁, ...]. If k is the smallest index for which a_k is unequal to b_k then x < y if (−1)^k(a_k − b_k) < 0 and y < x otherwise.

If there is no such k, but one expansion is shorter than the other, say x = [a₀; a₁, ..., a_n] and y = [b₀; b₁, ..., b_n, b_{n + 1}, ...] with a_i = b_i for 0 ≤ i ≤ n, then x < y if n is even and y < x if n is odd.

Continued fraction expansion of π and its convergents

To calculate the convergents of π we may set a₀ = ⌊π⌋ = 3, define u₁ = ⁠1/π − 3⁠ ≈ 7.0625 and a₁ = ⌊u₁⌋ = 7, u₂ = ⁠1/u₁ − 7⁠ ≈ 15.9966 and a₂ = ⌊u₂⌋ = 15, u₃ = ⁠1/u₂ − 15⁠ ≈ 1.0034. Continuing like this, one can determine the infinite continued fraction of π as

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

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

Let us suppose that the quotients found are, as above, [3;7,15,1]. The following is a rule by which we can write down at once the convergent fractions which result from these quotients without developing the continued fraction.

The first quotient, supposed divided by unity, will give the first fraction, which will be too small, namely, ⁠3/1⁠. Then, multiplying the numerator and denominator of this fraction by the second quotient and adding unity to the numerator, we shall have the second fraction, ⁠22/7⁠, which will be too large. Multiplying in like manner the numerator and denominator of this fraction by the third quotient, and adding to the numerator the numerator of the preceding fraction, and to the denominator the denominator of the preceding fraction, we shall have the third fraction, which will be too small. Thus, the third quotient being 15, we have for our numerator (22 × 15 = 330) + 3 = 333, and for our denominator, (7 × 15 = 105) + 1 = 106. The third convergent, therefore, is ⁠333/106⁠. We proceed in the same manner for the fourth convergent. The fourth quotient being 1, we say 333 times 1 is 333, and this plus 22, the numerator of the fraction preceding, is 355; similarly, 106 times 1 is 106, and this plus 7 is 113. In this manner, by employing the four quotients [3;7,15,1], we obtain the four fractions:

⁠3/1⁠, ⁠22/7⁠, ⁠333/106⁠, ⁠355/113⁠, ....

The following Maple code will generate continued fraction expansions of pi

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

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

The demonstration of the foregoing properties is deduced from the fact that if we seek the difference between one of the convergent fractions and the next adjacent to it we shall obtain a fraction of which the numerator is always unity and the denominator the product of the two denominators. Thus the difference between ⁠22/7⁠ and ⁠3/1⁠ is ⁠1/7⁠, in excess; between ⁠333/106⁠ and ⁠22/7⁠, ⁠1/742⁠, in deficit; between ⁠355/113⁠ and ⁠333/106⁠, ⁠1/11978⁠, in excess; and so on. The result being, that by employing this series of differences we can express in another and very simple manner the fractions with which we are here concerned, by means of a second series of fractions of which the numerators are all unity and the denominators successively be the product of every two adjacent denominators. Instead of the fractions written above, we have thus the series:

⁠3/1⁠ + ⁠1/1 × 7⁠ − ⁠1/7 × 106⁠ + ⁠1/106 × 113⁠ − ...

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

Non-simple continued fraction

Main article: Continued fraction (non-simple)

A non-simple continued fraction is an expression of the form

${\displaystyle x=b_0+\frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3+\frac{a_4}{b_4+\ddots }}}}}$

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

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

${\displaystyle \pi =[3;7,15,1,292,1,1,1,2,1,3,1,\dots ]}$

or

${\displaystyle \pi =3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{3+\frac{1}{1+\ddots }}}}}}}}}}}}$

However, several non-simple continued fractions for π have a perfectly regular structure, such as:

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

${\displaystyle {\displaystyle \pi =2+\frac{2}{1+\frac{1}{1/2+\frac{1}{1/3+\frac{1}{1/4+\ddots }}}}=2+\frac{2}{1+\frac{1\cdot 2}{1+\frac{2\cdot 3}{1+\frac{3\cdot 4}{1+\ddots }}}}}}$

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

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.

${\displaystyle \pi =3+\frac{1}{6+\frac{1^3+2^3}{6\cdot 1^2+1^2\frac{1^3+2^3+3^3+4^3}{6\cdot 2^2+2^2\frac{1^3+2^3+3^3+4^3+5^3+6^3}{6\cdot 3^2+3^2\frac{1^3+2^3+3^3+4^3+5^3+6^3+7^3+8^3}{6\cdot 4^2+\ddots }}}}}}$

The continued fraction of ${\displaystyle \pi }$ above consisting of cubes uses the Nilakantha series and an exploit from Leonhard Euler.

Other continued fraction expansions

Periodic continued fractions

Main article: Periodic continued fraction

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 does not 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 k can be approximated by infinitely many rational ⁠m/n⁠ with

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

While virtually all real numbers k will eventually have infinitely many convergents ⁠m/n⁠ whose distance from k is significantly smaller than this limit, the convergents for φ (i.e., the numbers ⁠5/3⁠, ⁠8/5⁠, ⁠13/8⁠, ⁠21/13⁠, etc.) consistently "toe the boundary", keeping a distance of almost exactly ${\displaystyle \frac{1}{n^2\sqrt{5}}}$ away from φ, thus never producing an approximation nearly as impressive as, for example, ⁠355/113⁠ for π. It can also be shown that every real number of the form ⁠a + bφ/c + dφ⁠, where a, b, c, and d are integers such that a d − b c = ±1, shares this property with the golden ratio φ; and that all other real numbers can be more closely approximated.

Regular patterns in continued fractions

While there is no discernible pattern in the simple continued fraction expansion of π, there is one for e, the base of the natural logarithm:

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

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

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

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

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

with a special case for n = 1:

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

Other continued fractions of this sort are

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

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

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

with a special case for n = 1:

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

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

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

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

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

with similar formulas for negative rationals; in particular we have

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

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

Typical continued fractions

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

The arithmetic average diverges: ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\sum _{k=1}^n{a}_k=+\mathrm{\infty }}$, and so the coefficients grow arbitrarily large: ${\displaystyle \underset{n}{lim sup}{a}_n=+\mathrm{\infty }}$. In particular, this implies that almost all numbers are well-approximable, in the sense that$${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}\left|x-\frac{p_n}{q_n}\right|q_n^2=0}$$Khinchin proved that the geometric mean of a_i tends to a constant (known as Khinchin's constant):$${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{\left(a_1{a}_2...a_n\right)}^{1/n}=K_0=2.6854520010\dots }$$Paul Lévy proved that the nth root of the denominator of the nth convergent converges to Lévy's constant $${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{q}_n^{1/n}=e^{{\pi }^2/(12\ln 2)}=3.2758\dots }$$Lochs' theorem states that the convergents converge exponentially at the rate of$${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\ln \left|x-\frac{p_n}{q_n}\right|=-\frac{{\pi }^2}{6\ln 2}}$$

Applications

Pell's equation

Continued fractions play an essential role in the solution of Pell's equation. For example, for positive integers p and q, and non-square n, it is true that if p² − nq² = ±1, then ⁠p/q⁠ is a convergent of the regular continued fraction for √n. The converse holds if the period of the regular continued fraction for √n is 1, and in general the period describes which convergents give solutions to Pell's equation.

Dynamical systems

Continued fractions also play a role in the study of dynamical systems, where they tie together the Farey fractions which are seen in the Mandelbrot set with Minkowski's question-mark function and the modular group Gamma.

The backwards shift operator for continued fractions is the map h(x) = 1/x − ⌊1/x⌋ called the Gauss map, which lops off digits of a continued fraction expansion: h([0; a₁, a₂, a₃, ...]) = [0; a₂, a₃, ...]. 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.