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3,300 | The chromosphere, transition region, and corona are much hotter than the surface of the Sun. The reason is not well understood, but evidence suggests that Alfvén waves may have enough energy to heat the corona. | https://en.wikipedia.org/wiki?curid=26751 |
3,301 | Above the temperature minimum layer is a layer about thick, dominated by a spectrum of emission and absorption lines. It is called the "chromosphere" from the Greek root "chroma", meaning color, because the chromosphere is visible as a colored flash at the beginning and end of total solar eclipses. The temperature of t... | https://en.wikipedia.org/wiki?curid=26751 |
3,302 | Above the chromosphere, in a thin (about ) transition region, the temperature rises rapidly from around in the upper chromosphere to coronal temperatures closer to . The temperature increase is facilitated by the full ionization of helium in the transition region, which significantly reduces radiative cooling of the pl... | https://en.wikipedia.org/wiki?curid=26751 |
3,303 | The corona is the next layer of the Sun. The low corona, near the surface of the Sun, has a particle density around 10 m to 10 m. The average temperature of the corona and solar wind is about 1,000,000–2,000,000 K; however, in the hottest regions it is 8,000,000–20,000,000 K. Although no complete theory yet exists to a... | https://en.wikipedia.org/wiki?curid=26751 |
3,304 | The corona is the extended atmosphere of the Sun, which has a volume much larger than the volume enclosed by the Sun's photosphere. A flow of plasma outward from the Sun into interplanetary space is the solar wind. | https://en.wikipedia.org/wiki?curid=26751 |
3,305 | The heliosphere, the tenuous outermost atmosphere of the Sun, is filled with the solar wind plasma. This outermost layer of the Sun is defined to begin at the distance where the flow of the solar wind becomes "superalfvénic"—that is, where the flow becomes faster than the speed of Alfvén waves, at approximately 20 sola... | https://en.wikipedia.org/wiki?curid=26751 |
3,306 | On April 28, 2021, during its eighth flyby of the Sun, NASA's Parker Solar Probe encountered the specific magnetic and particle conditions at 18.8 solar radii that indicated that it penetrated the Alfvén surface, the boundary separating the corona from the solar wind defined as where the coronal plasma's Alfvén speed a... | https://en.wikipedia.org/wiki?curid=26751 |
3,307 | The Sun emits light across the visible spectrum, so its color is white, with a CIE color-space index near (0.3, 0.3), when viewed from space or when the Sun is high in the sky. The Solar radiance per wavelength peaks in the green portion of the spectrum when viewed from space. When the Sun is very low in the sky, atmos... | https://en.wikipedia.org/wiki?curid=26751 |
3,308 | The Sun is a G2V star, with "G2" indicating its surface temperature of approximately , and "V" that it, like most stars, is a main-sequence star. | https://en.wikipedia.org/wiki?curid=26751 |
3,309 | The solar constant is the amount of power that the Sun deposits per unit area that is directly exposed to sunlight. The solar constant is equal to approximately (watts per square meter) at a distance of one astronomical unit (AU) from the Sun (that is, on or near Earth). Sunlight on the surface of Earth is attenuated b... | https://en.wikipedia.org/wiki?curid=26751 |
3,310 | Ultraviolet light from the Sun has antiseptic properties and can be used to sanitize tools and water. It also causes sunburn, and has other biological effects such as the production of vitamin D and sun tanning. It is also the main cause of skin cancer. Ultraviolet light is strongly attenuated by Earth's ozone layer, s... | https://en.wikipedia.org/wiki?curid=26751 |
3,311 | Neutrinos are also released by the fusion reactions in the core, but, unlike photons, they rarely interact with matter, so almost all are able to escape the Sun immediately. For many years measurements of the number of neutrinos produced in the Sun were lower than theories predicted by a factor of 3. This discrepancy w... | https://en.wikipedia.org/wiki?curid=26751 |
3,312 | The Sun has a stellar magnetic field that varies across its surface. Its polar field is , whereas the field is typically in features on the Sun called sunspots and in solar prominences. The magnetic field varies in time and location. The quasi-periodic 11-year solar cycle is the most prominent variation in which the nu... | https://en.wikipedia.org/wiki?curid=26751 |
3,313 | The solar magnetic field extends well beyond the Sun itself. The electrically conducting solar wind plasma carries the Sun's magnetic field into space, forming what is called the interplanetary magnetic field. In an approximation known as ideal magnetohydrodynamics, plasma particles only move along the magnetic field l... | https://en.wikipedia.org/wiki?curid=26751 |
3,314 | At great distances, the rotation of the Sun twists the dipolar magnetic field and corresponding current sheet into an Archimedean spiral structure called the Parker spiral. The interplanetary magnetic field is much stronger than the dipole component of the solar magnetic field. The Sun's dipole magnetic field of 50–400... | https://en.wikipedia.org/wiki?curid=26751 |
3,315 | Sunspots are visible as dark patches on the Sun's photosphere and correspond to concentrations of magnetic field where the convective transport of heat is inhibited from the solar interior to the surface. As a result, sunspots are slightly cooler than the surrounding photosphere, so they appear dark. At a typical solar... | https://en.wikipedia.org/wiki?curid=26751 |
3,316 | An 11-year sunspot cycle is half of a 22-year Babcock–Leighton dynamo cycle, which corresponds to an oscillatory exchange of energy between toroidal and poloidal solar magnetic fields. At solar-cycle maximum, the external poloidal dipolar magnetic field is near its dynamo-cycle minimum strength, but an internal toroida... | https://en.wikipedia.org/wiki?curid=26751 |
3,317 | During the solar cycle's declining phase, energy shifts from the internal toroidal magnetic field to the external poloidal field, and sunspots diminish in number and size. At solar-cycle minimum, the toroidal field is, correspondingly, at minimum strength, sunspots are relatively rare, and the poloidal field is at its ... | https://en.wikipedia.org/wiki?curid=26751 |
3,318 | The Sun's magnetic field leads to many effects that are collectively called solar activity. Solar flares and coronal-mass ejections tend to occur at sunspot groups. Slowly changing high-speed streams of solar wind are emitted from coronal holes at the photospheric surface. Both coronal-mass ejections and high-speed str... | https://en.wikipedia.org/wiki?curid=26751 |
3,319 | Long-term secular change in sunspot number is thought, by some scientists, to be correlated with long-term change in solar irradiance, which, in turn, might influence Earth's long-term climate. The solar cycle influences space weather conditions, including those surrounding Earth. For example, in the 17th century, the ... | https://en.wikipedia.org/wiki?curid=26751 |
3,320 | In December 2019, a new type of solar magnetic explosion was observed, known as forced magnetic reconnection. Previously, in a process called spontaneous magnetic reconnection, it was observed that the solar magnetic field lines diverge explosively and then converge again instantaneously. Forced Magnetic Reconnection w... | https://en.wikipedia.org/wiki?curid=26751 |
3,321 | The Sun today is roughly halfway through the most stable part of its life. It has not changed dramatically for over four billion years and will remain fairly stable for more than five billion more. However, after hydrogen fusion in its core has stopped, the Sun will undergo dramatic changes, both internally and externa... | https://en.wikipedia.org/wiki?curid=26751 |
3,322 | The Sun formed about 4.6 billion years ago from the collapse of part of a giant molecular cloud that consisted mostly of hydrogen and helium and that probably gave birth to many other stars. This age is estimated using computer models of stellar evolution and through nucleocosmochronology. The result is consistent with... | https://en.wikipedia.org/wiki?curid=26751 |
3,323 | The Sun is about halfway through its main-sequence stage, during which nuclear fusion reactions in its core fuse hydrogen into helium. Each second, more than four million tonnes of matter are converted into energy within the Sun's core, producing neutrinos and solar radiation. At this rate, the Sun has so far converted... | https://en.wikipedia.org/wiki?curid=26751 |
3,324 | The Sun is gradually becoming hotter in its core, hotter at the surface, larger in radius, and more luminous during its time on the main sequence: since the beginning of its main sequence life, it has expanded in radius by 15% and the surface has increased in temperature from to , resulting in a 48% increase in luminos... | https://en.wikipedia.org/wiki?curid=26751 |
3,325 | The Sun does not have enough mass to explode as a supernova. Instead, when it runs out of hydrogen in the core in approximately 5 billion years, core hydrogen fusion will stop, and there will be nothing to prevent the core from contracting. The release of gravitational potential energy will cause the luminosity of the ... | https://en.wikipedia.org/wiki?curid=26751 |
3,326 | After the red-giant branch, the Sun has approximately 120 million years of active life left, but much happens. First, the core (full of degenerate helium) ignites violently in the helium flash; it is estimated that 6% of the core—itself 40% of the Sun's mass—will be converted into carbon within a matter of minutes thro... | https://en.wikipedia.org/wiki?curid=26751 |
3,327 | When the helium is exhausted, the Sun will repeat the expansion it followed when the hydrogen in the core was exhausted. This time, however, it all happens faster, and the Sun becomes larger and more luminous, engulfing Venus if it has not already. This is the asymptotic-giant-branch phase, and the Sun is alternately r... | https://en.wikipedia.org/wiki?curid=26751 |
3,328 | According to a 2008 model, Earth's orbit will have initially expanded to at most due to the Sun's loss of mass as a red giant. However, Earth's orbit will later start shrinking due to tidal forces (and, eventually, drag from the lower chromosphere) so that it is engulfed by the Sun during the tip of the red-giant branc... | https://en.wikipedia.org/wiki?curid=26751 |
3,329 | The post-asymptotic-giant-branch evolution is even faster. The luminosity stays approximately constant as the temperature increases, with the ejected half of the Sun's mass becoming ionized into a planetary nebula as the exposed core reaches , as if it is in a sort of blue loop. The final naked core, a white dwarf, wil... | https://en.wikipedia.org/wiki?curid=26751 |
3,330 | The Sun has eight known planets orbiting around it. This includes four terrestrial planets (Mercury, Venus, Earth, and Mars), two gas giants (Jupiter and Saturn), and two ice giants (Uranus and Neptune). The Solar System also has nine bodies generally considered as dwarf planets and some more candidates, an asteroid be... | https://en.wikipedia.org/wiki?curid=26751 |
3,331 | The Sun is moved by the gravitational pull of the planets. The center of the Sun is always within 2.2 solar radii of the barycenter. This motion of the Sun is mainly due to Jupiter, Saturn, Uranus, and Neptune. For some periods of several decades, the motion is rather regular, forming a trefoil pattern, whereas between... | https://en.wikipedia.org/wiki?curid=26751 |
3,332 | The Sun has been an object of veneration in many cultures throughout human history. Humanity's most fundamental understanding of the Sun is as the luminous disk in the sky, whose presence above the horizon causes day and whose absence causes night. In many prehistoric and ancient cultures, the Sun was thought to be a s... | https://en.wikipedia.org/wiki?curid=26751 |
3,333 | In the early first millennium BC, Babylonian astronomers observed that the Sun's motion along the ecliptic is not uniform, though they did not know why; it is today known that this is due to the movement of Earth in an elliptic orbit around the Sun, with Earth moving faster when it is nearer to the Sun at perihelion an... | https://en.wikipedia.org/wiki?curid=26751 |
3,334 | One of the first people to offer a scientific or philosophical explanation for the Sun was the Greek philosopher Anaxagoras. He reasoned that it was not the chariot of Helios, but instead a giant flaming ball of metal even larger than the land of the Peloponnesus and that the Moon reflected the light of the Sun. For te... | https://en.wikipedia.org/wiki?curid=26751 |
3,335 | The theory that the Sun is the center around which the planets orbit was first proposed by the ancient Greek Aristarchus of Samos in the third century BC, and later adopted by Seleucus of Seleucia (see Heliocentrism). This view was developed in a more detailed mathematical model of a heliocentric system in the 16th cen... | https://en.wikipedia.org/wiki?curid=26751 |
3,336 | Observations of sunspots were recorded during the Han Dynasty (206 BC–AD 220) by Chinese astronomers, who maintained records of these observations for centuries. Averroes also provided a description of sunspots in the 12th century. The invention of the telescope in the early 17th century permitted detailed observations... | https://en.wikipedia.org/wiki?curid=26751 |
3,337 | Arabic astronomical contributions include Al-Battani's discovery that the direction of the Sun's apogee (the place in the Sun's orbit against the fixed stars where it seems to be moving slowest) is changing. (In modern heliocentric terms, this is caused by a gradual motion of the aphelion of the "Earth's" orbit). Ibn Y... | https://en.wikipedia.org/wiki?curid=26751 |
3,338 | From an observation of a transit of Venus in 1032, the Persian astronomer and polymath Ibn Sina concluded that Venus is closer to Earth than the Sun. In 1672 Giovanni Cassini and Jean Richer determined the distance to Mars and were thereby able to calculate the distance to the Sun. | https://en.wikipedia.org/wiki?curid=26751 |
3,339 | In 1666, Isaac Newton observed the Sun's light using a prism, and showed that it is made up of light of many colors. In 1800, William Herschel discovered infrared radiation beyond the red part of the solar spectrum. The 19th century saw advancement in spectroscopic studies of the Sun; Joseph von Fraunhofer recorded mor... | https://en.wikipedia.org/wiki?curid=26751 |
3,340 | In the early years of the modern scientific era, the source of the Sun's energy was a significant puzzle. Lord Kelvin suggested that the Sun is a gradually cooling liquid body that is radiating an internal store of heat. Kelvin and Hermann von Helmholtz then proposed a gravitational contraction mechanism to explain the... | https://en.wikipedia.org/wiki?curid=26751 |
3,341 | Not until 1904 was a documented solution offered. Ernest Rutherford suggested that the Sun's output could be maintained by an internal source of heat, and suggested radioactive decay as the source. However, it would be Albert Einstein who would provide the essential clue to the source of the Sun's energy output with hi... | https://en.wikipedia.org/wiki?curid=26751 |
3,342 | The first satellites designed for long term observation of the Sun from interplanetary space were NASA's Pioneers 6, 7, 8 and 9, which were launched between 1959 and 1968. These probes orbited the Sun at a distance similar to that of Earth, and made the first detailed measurements of the solar wind and the solar magnet... | https://en.wikipedia.org/wiki?curid=26751 |
3,343 | In the 1970s, two Helios spacecraft and the Skylab Apollo Telescope Mount provided scientists with significant new data on solar wind and the solar corona. The Helios 1 and 2 probes were U.S.–German collaborations that studied the solar wind from an orbit carrying the spacecraft inside Mercury's orbit at perihelion. Th... | https://en.wikipedia.org/wiki?curid=26751 |
3,344 | In the 1970s, much research focused on the abundances of iron-group elements in the Sun. Although significant research was done, until 1978 it was difficult to determine the abundances of some iron-group elements (e.g. cobalt and manganese) via spectrography because of their hyperfine structures. The first largely comp... | https://en.wikipedia.org/wiki?curid=26751 |
3,345 | Various authors have considered the existence of a gradient in the isotopic compositions of solar and planetary noble gases, e.g. correlations between isotopic compositions of neon and xenon in the Sun and on the planets. Prior to 1983, it was thought that the whole Sun has the same composition as the solar atmosphere.... | https://en.wikipedia.org/wiki?curid=26751 |
3,346 | In 1980, the Solar Maximum Mission probes was launched by NASA. This spacecraft was designed to observe gamma rays, X-rays and UV radiation from solar flares during a time of high solar activity and solar luminosity. Just a few months after launch, however, an electronics failure caused the probe to go into standby mod... | https://en.wikipedia.org/wiki?curid=26751 |
3,347 | Launched in 1991, Japan's Yohkoh ("Sunbeam") satellite observed solar flares at X-ray wavelengths. Mission data allowed scientists to identify several different types of flares and demonstrated that the corona away from regions of peak activity was much more dynamic and active than had previously been supposed. Yohkoh ... | https://en.wikipedia.org/wiki?curid=26751 |
3,348 | One of the most important solar missions to date has been the Solar and Heliospheric Observatory, jointly built by the European Space Agency and NASA and launched on 2 December 1995. Originally intended to serve a two-year mission, a mission extension through 2012 was approved in October 2009. It has proven so useful t... | https://en.wikipedia.org/wiki?curid=26751 |
3,349 | All these satellites have observed the Sun from the plane of the ecliptic, and so have only observed its equatorial regions in detail. The "Ulysses" probe was launched in 1990 to study the Sun's polar regions. It first traveled to Jupiter, to "slingshot" into an orbit that would take it far above the plane of the eclip... | https://en.wikipedia.org/wiki?curid=26751 |
3,350 | Elemental abundances in the photosphere are well known from spectroscopic studies, but the composition of the interior of the Sun is more poorly understood. A solar wind sample return mission, "Genesis", was designed to allow astronomers to directly measure the composition of solar material. | https://en.wikipedia.org/wiki?curid=26751 |
3,351 | The temperature of the photosphere is approximately 6,000 K, whereas the temperature of the corona reaches . The high temperature of the corona shows that it is heated by something other than direct heat conduction from the photosphere. | https://en.wikipedia.org/wiki?curid=26751 |
3,352 | It is thought that the energy necessary to heat the corona is provided by turbulent motion in the convection zone below the photosphere, and two main mechanisms have been proposed to explain coronal heating. The first is wave heating, in which sound, gravitational or magnetohydrodynamic waves are produced by turbulence... | https://en.wikipedia.org/wiki?curid=26751 |
3,353 | Currently, it is unclear whether waves are an efficient heating mechanism. All waves except Alfvén waves have been found to dissipate or refract before reaching the corona. In addition, Alfvén waves do not easily dissipate in the corona. Current research focus has therefore shifted towards flare heating mechanisms. | https://en.wikipedia.org/wiki?curid=26751 |
3,354 | Theoretical models of the Sun's development suggest that 3.8 to 2.5 billion years ago, during the Archean eon, the Sun was only about 75% as bright as it is today. Such a weak star would not have been able to sustain liquid water on Earth's surface, and thus life should not have been able to develop. However, the geolo... | https://en.wikipedia.org/wiki?curid=26751 |
3,355 | However, examination of Archaean sediments appears inconsistent with the hypothesis of high greenhouse concentrations. Instead, the moderate temperature range may be explained by a lower surface albedo brought about by less continental area and the lack of biologically induced cloud condensation nuclei. This would have... | https://en.wikipedia.org/wiki?curid=26751 |
3,356 | The brightness of the Sun can cause pain from looking at it with the naked eye; however, doing so for brief periods is not hazardous for normal non-dilated eyes. Looking directly at the Sun (sungazing) causes phosphene visual artifacts and temporary partial blindness. It also delivers about 4 milliwatts of sunlight to ... | https://en.wikipedia.org/wiki?curid=26751 |
3,357 | Viewing the Sun through light-concentrating optics such as binoculars may result in permanent damage to the retina without an appropriate filter that blocks UV and substantially dims the sunlight. When using an attenuating filter to view the Sun, the viewer is cautioned to use a filter specifically designed for that us... | https://en.wikipedia.org/wiki?curid=26751 |
3,358 | During sunrise and sunset, sunlight is attenuated because of Rayleigh scattering and Mie scattering from a particularly long passage through Earth's atmosphere, and the Sun is sometimes faint enough to be viewed comfortably with the naked eye or safely with optics (provided there is no risk of bright sunlight suddenly ... | https://en.wikipedia.org/wiki?curid=26751 |
3,359 | An optical phenomenon, known as a green flash, can sometimes be seen shortly after sunset or before sunrise. The flash is caused by light from the Sun just below the horizon being bent (usually through a temperature inversion) towards the observer. Light of shorter wavelengths (violet, blue, green) is bent more than th... | https://en.wikipedia.org/wiki?curid=26751 |
3,360 | Solar deities play a major role in many world religions and mythologies. Worship of the Sun was central to civilizations such as the ancient Egyptians, the Inca of South America and the Aztecs of what is now Mexico. In religions such as Hinduism, the Sun is still considered a god, he is known as Surya Dev. Many ancient... | https://en.wikipedia.org/wiki?curid=26751 |
3,361 | The ancient Sumerians believed that the Sun was Utu, the god of justice and twin brother of Inanna, the Queen of Heaven, who was identified as the planet Venus. Later, Utu was identified with the East Semitic god Shamash. Utu was regarded as a helper-deity, who aided those in distress, and, in iconography, he is usuall... | https://en.wikipedia.org/wiki?curid=26751 |
3,362 | From at least the Fourth Dynasty of Ancient Egypt, the Sun was worshipped as the god Ra, portrayed as a falcon-headed divinity surmounted by the solar disk, and surrounded by a serpent. In the New Empire period, the Sun became identified with the dung beetle, whose spherical ball of dung was identified with the Sun. In... | https://en.wikipedia.org/wiki?curid=26751 |
3,363 | The Egyptians portrayed the god Ra as being carried across the sky in a solar barque, accompanied by lesser gods, and to the Greeks, he was Helios, carried by a chariot drawn by fiery horses. From the reign of Elagabalus in the late Roman Empire the Sun's birthday was a holiday celebrated as Sol Invictus (literally "Un... | https://en.wikipedia.org/wiki?curid=26751 |
3,364 | In Proto-Indo-European religion, the Sun was personified as the goddess "*Sehul". Derivatives of this goddess in Indo-European languages include the Old Norse "Sól", Sanskrit "Surya", Gaulish "Sulis", Lithuanian "Saulė", and Slavic "Solntse". In ancient Greek religion, the sun deity was the male god Helios, who in late... | https://en.wikipedia.org/wiki?curid=26751 |
3,365 | In the Bible, mentions the "Sun of Righteousness" (sometimes translated as the "Sun of Justice"), which some Christians have interpreted as a reference to the Messiah (Christ). In ancient Roman culture, Sunday was the day of the sun god. It was adopted as the Sabbath day by Christians who did not have a Jewish backgrou... | https://en.wikipedia.org/wiki?curid=26751 |
3,366 | Tonatiuh, the Aztec god of the sun, was usually depicted holding arrows and a shield and was closely associated with the practice of human sacrifice. The sun goddess Amaterasu is the most important deity in the Shinto religion, and she is believed to be the direct ancestor of all Japanese emperors. | https://en.wikipedia.org/wiki?curid=26751 |
3,367 | In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. | https://en.wikipedia.org/wiki?curid=10918 |
3,368 | The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to West... | https://en.wikipedia.org/wiki?curid=10918 |
3,369 | Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the "Fibonacci Quarterly". Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibona... | https://en.wikipedia.org/wiki?curid=10918 |
3,370 | Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the th Fibonacci number in terms of and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as increases. Fibonacci numbers are also closely related to Lucas numbers, which ... | https://en.wikipedia.org/wiki?curid=10918 |
3,371 | Under some older definitions, the value formula_3 is omitted, so that the sequence starts with formula_4 and the recurrence formula_5 is valid for . | https://en.wikipedia.org/wiki?curid=10918 |
3,372 | The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody. In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L a... | https://en.wikipedia.org/wiki?curid=10918 |
3,373 | Knowledge of the Fibonacci sequence was expressed as early as Pingala ( 450 BC–200 BC). Singh cites Pingala's cryptic formula "misrau cha" ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for beats () is obtained by adding one [S] to the cases and one [L] to the cases... | https://en.wikipedia.org/wiki?curid=10918 |
3,374 | However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135): | https://en.wikipedia.org/wiki?curid=10918 |
3,375 | Variations of two earlier meters [is the variation]... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21]... In this way, the process should be followed in all "mātrā-vṛttas" [prosodic combinations]. | https://en.wikipedia.org/wiki?curid=10918 |
3,376 | Hemachandra (c. 1150) is credited with knowledge of the sequence as well, writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta." | https://en.wikipedia.org/wiki?curid=10918 |
3,377 | Outside India, the Fibonacci sequence first appears in the book "Liber Abaci" ("The Book of Calculation", 1202) by Fibonacci where it is used to calculate the growth of rabbit populations. Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born breeding p... | https://en.wikipedia.org/wiki?curid=10918 |
3,378 | At the end of the th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month ) plus the number of pairs alive last month (month ). The number in the th month is the th Fibonacci number. | https://en.wikipedia.org/wiki?curid=10918 |
3,379 | Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form expression. It has become known as Binet's formula, named after French mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre and Daniel Bernoulli: | https://en.wikipedia.org/wiki?curid=10918 |
3,380 | To see the relation between the sequence and these constants, note that and are both solutions of the equation | https://en.wikipedia.org/wiki?curid=10918 |
3,381 | If and are chosen so that and then the resulting sequence must be the Fibonacci sequence. This is the same as requiring and satisfy the system of equations: | https://en.wikipedia.org/wiki?curid=10918 |
3,382 | for all , the number is the closest integer to formula_22. Therefore, it can be found by rounding, using the nearest integer function: | https://en.wikipedia.org/wiki?curid=10918 |
3,383 | As the floor function is monotonic, the latter formula can be inverted for finding the index of the smallest Fibonacci number that is not less than a positive integer : | https://en.wikipedia.org/wiki?curid=10918 |
3,384 | Since "F" is asymptotic to formula_29, the number of digits in "F" is asymptotic to formula_30. As a consequence, for every integer "d" > 1 there are either 4 or 5 Fibonacci numbers with "d" decimal digits. | https://en.wikipedia.org/wiki?curid=10918 |
3,385 | More generally, in the base "b" representation, the number of digits in "F" is asymptotic to formula_31 | https://en.wikipedia.org/wiki?curid=10918 |
3,386 | Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio formula_32 | https://en.wikipedia.org/wiki?curid=10918 |
3,387 | This convergence holds regardless of the starting values formula_34 and formula_35, unless formula_36. This can be verified using Binet's formula. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, ... The ratio of consecutive terms in this sequence shows t... | https://en.wikipedia.org/wiki?curid=10918 |
3,388 | In general, formula_37, because the ratios between consecutive Fibonacci numbers approaches formula_38. | https://en.wikipedia.org/wiki?curid=10918 |
3,389 | this expression can be used to decompose higher powers formula_40 as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of formula_38 and 1. The resulting recurrence relationships yield Fibonacci numbers as the linear coefficients: | https://en.wikipedia.org/wiki?curid=10918 |
3,390 | These expressions are also true for if the Fibonacci sequence "F" is extended to negative integers using the Fibonacci rule formula_47 | https://en.wikipedia.org/wiki?curid=10918 |
3,391 | Binet's formula provides a proof that a positive integer "x" is a Fibonacci number if and only if at least one of formula_48 or formula_49 is a perfect square. This is because Binet's formula, which can be written as formula_50, can be multiplied by formula_51 and solved as a quadratic equation in formula_40 via the qu... | https://en.wikipedia.org/wiki?curid=10918 |
3,392 | which yields formula_58. The eigenvalues of the matrix are formula_59 and formula_60 corresponding to the respective eigenvectors | https://en.wikipedia.org/wiki?curid=10918 |
3,393 | Equivalently, the same computation may performed by diagonalization of through use of its eigendecomposition: | https://en.wikipedia.org/wiki?curid=10918 |
3,394 | This property can be understood in terms of the continued fraction representation for the golden ratio: | https://en.wikipedia.org/wiki?curid=10918 |
3,395 | The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for , and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1. The matrix representation gives the following closed-form expression for the Fibonacci numbers: | https://en.wikipedia.org/wiki?curid=10918 |
3,396 | For a given , this matrix can be computed in arithmetic operations, using the exponentiation by squaring method. | https://en.wikipedia.org/wiki?curid=10918 |
3,397 | Moreover, since for any square matrix , the following identities can be derived (they are obtained from two different coefficients of the matrix product, and one may easily deduce the second one from the first one by changing into ), | https://en.wikipedia.org/wiki?curid=10918 |
3,398 | These last two identities provide a way to compute Fibonacci numbers recursively in arithmetic operations and in time , where is the time for the multiplication of two numbers of digits. This matches the time for computing the th Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if on... | https://en.wikipedia.org/wiki?curid=10918 |
3,399 | Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that formula_76 can be interpreted as the number of [possibly empty] sequences of 1s and 2s whose sum is formula_77. This can be taken as the definition of formula_78 with the conventions formula_79, meaning no such s... | https://en.wikipedia.org/wiki?curid=10918 |
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