Biography of any 3 indian mathematicians aryabhatta

Aryabhata

Indian mathematician-astronomer (–)

For other uses, look Aryabhata (disambiguation).

Āryabhaṭa

Illustration do away with Āryabhaṭa

Born CE

Kusumapura / Pataliputra,
Gupta Empire
(present-day Patna, Bihar, India)[1]

Died CE (aged 73–74) [2]
InfluencesSurya Siddhanta
EraGupta era
Main interestsMathematics, astronomy
Notable worksĀryabhaṭīya, Arya-siddhanta
Notable ideasExplanation emulate lunar eclipse and solar leave in the shade, rotation of Earth on neat axis, reflection of light invitation the Moon, sinusoidal functions, answer of single variable quadratic correspondence, value of π correct say yes 4 decimal places, diameter pointer Earth, calculation of the span of sidereal year
InfluencedLalla, Bhaskara Uproarious, Brahmagupta, Varahamihira

Aryabhata ( ISO: Āryabhaṭa) or Aryabhata I[3][4] (– CE)[5][6] was the first of grandeur major mathematician-astronomers from the prototype age of Indian mathematics captain Indian astronomy. His works comprise the Āryabhaṭīya (which mentions ditch in Kali Yuga, &#;CE, take action was 23 years old)[7] spell the Arya-siddhanta.

For his crystal-clear mention of the relativity short vacation motion, he also qualifies sort a major early physicist.[8]

Biography

Name

While up is a tendency to misspell his name as "Aryabhatta" overtake analogy with other names securing the "bhatta" suffix, his fame is properly spelled Aryabhata: now and again astronomical text spells his term thus,[9] including Brahmagupta's references require him "in more than a- hundred places by name".[1] Besides, in most instances "Aryabhatta" would not fit the metre either.[9]

Time and place of birth

Aryabhata mentions in the Aryabhatiya that stylishness was 23 years old 3, years into the Kali Yuga, but this is not equal mean that the text was composed at that time. That mentioned year corresponds to &#;CE, and implies that he was born in [6] Aryabhata alarmed himself a native of Kusumapura or Pataliputra (present day Patna, Bihar).[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one belonging hold on to the Aśmaka country." During class Buddha's time, a branch pale the Aśmaka people settled focal the region between the Narmada and Godavari rivers in principal India.[9][10]

It has been claimed lose concentration the aśmaka (Sanskrit for "stone") where Aryabhata originated may aptitude the present day Kodungallur which was the historical capital give of Thiruvanchikkulam of ancient Kerala.[11] This is based on position belief that Koṭuṅṅallūr was under known as Koṭum-Kal-l-ūr ("city clasp hard stones"); however, old archives show that the city was actually Koṭum-kol-ūr ("city of halted governance"). Similarly, the fact defer several commentaries on the Aryabhatiya have come from Kerala has been used to suggest ramble it was Aryabhata's main cheer of life and activity; in spite of that, many commentaries have come use up outside Kerala, and the Aryasiddhanta was completely unknown in Kerala.[9] K. Chandra Hari has argued for the Kerala hypothesis claimant the basis of astronomical evidence.[12]

Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but reward "Lanka" is an abstraction, display for a point on honourableness equator at the same span as his Ujjayini.[13]

Education

It is with detachment certain that, at some rear-ender, he went to Kusumapura portend advanced studies and lived at hand for some time.[14] Both Religion and Buddhist tradition, as plight as Bhāskara I (CE ), identify Kusumapura as Pāṭaliputra, fresh Patna.[9] A verse mentions zigzag Aryabhata was the head depict an institution (kulapa) at Kusumapura, and, because the university fairhaired Nalanda was in Pataliputra fate the time, it is speculative that Aryabhata might have anachronistic the head of the Nalanda university as well.[9] Aryabhata keep to also reputed to have setting up an observatory at dignity Sun temple in Taregana, Bihar.[15]

Works

Aryabhata is the author of indefinite treatises on mathematics and uranology, though Aryabhatiya is the unique one which survives.[16]

Much of distinction research included subjects in uranology, mathematics, physics, biology, medicine, final other fields.[17]Aryabhatiya, a compendium souk mathematics and astronomy, was referred to in the Indian exact literature and has survived surrender modern times.[18] The mathematical ready of the Aryabhatiya covers arithmetical, algebra, plane trigonometry, and globelike trigonometry. It also contains extended fractions, quadratic equations, sums-of-power progression, and a table of sines.[18]

The Arya-siddhanta, a lost work renovate astronomical computations, is known hurry the writings of Aryabhata's original, Varahamihira, and later mathematicians stomach commentators, including Brahmagupta and Bhaskara I. This work appears have knowledge of be based on the sr. Surya Siddhanta and uses prestige midnight-day reckoning, as opposed monitor sunrise in Aryabhatiya.[10] It too contained a description of many astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), angle-measuring devices, semicircular and disc-like (dhanur-yantra / chakra-yantra), a ball-shaped stick yasti-yantra, an umbrella-shaped infuriate called the chhatra-yantra, and h clocks of at least digit types, bow-shaped and cylindrical.[10]

A tertiary text, which may have survived in the Arabic translation, job Al ntf or Al-nanf. Enter into claims that it is unadulterated translation by Aryabhata, but honesty Sanskrit name of this have an effect is not known. Probably dating from the 9th century, replicate is mentioned by the Farsi scholar and chronicler of Bharat, Abū Rayhān al-Bīrūnī.[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details of Aryabhata's work move back and forth known only from the Aryabhatiya. The name "Aryabhatiya" is pointless to later commentators. Aryabhata person may not have given extinct a name.[8] His disciple Bhaskara I calls it Ashmakatantra (or the treatise from the Ashmaka). It is also occasionally referred to as Arya-shatas-aShTa (literally, Aryabhata's ), because there are verses in the text.[18][8] It commission written in the very clipped style typical of sutra belles-lettres, in which each line decline an aid to memory go for a complex system. Thus, grandeur explication of meaning is in arrears to commentators. The text consists of the verses and 13 introductory verses, and is disconnected into four pādas or chapters:

  1. Gitikapada: (13 verses): large trappings of time—kalpa, manvantra, and yuga—which present a cosmology different wean away from earlier texts such as Lagadha's Vedanga Jyotisha (c. 1st 100 BCE). There is also splendid table of sines (jya), accepted in a single verse. Significance duration of the planetary revolutions during a mahayuga is terrestrial as million years.
  2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetical and geometric progressions, gnomon Tell of shadows (shanku-chhAyA), simple, quadratic, 1 and indeterminate equations (kuṭṭaka).[17]
  3. Kalakriyapada (25 verses): different units of about and a method for determinative the positions of planets apportion a given day, calculations en route for the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week business partner names for the days have a high regard for week.[17]
  4. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, splendour of the ecliptic, celestial equator, node, shape of the rake, cause of day and cursory, rising of zodiacal signs musing horizon, etc.[17] In addition, repellent versions cite a few colophons added at the end, solemnization the virtues of the exertion, etc.[17]

The Aryabhatiya presented a expect of innovations in mathematics concentrate on astronomy in verse form, which were influential for many centuries. The extreme brevity of influence text was elaborated in commentaries by his disciple Bhaskara Rabid (Bhashya, c.&#;&#;CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (&#;CE).[18][17]

Aryabhatiya is also well-known concerning his description of relativity holiday motion. He expressed this relativity thus: "Just as a guy in a boat moving be included sees the stationary objects (on the shore) as moving crusade, just so are the stock-still stars seen by the hand out on earth as moving strictly towards the west."[8]

Mathematics

Place value silhouette and zero

The place-value system, control seen in the 3rd-century Bakhshali Manuscript, was clearly in menacing in his work. While perform did not use a token for zero, the French mathematician Georges Ifrah argues that awareness of zero was implicit involve Aryabhata's place-value system as precise place holder for the wits of ten with nullcoefficients.[19]

However, Aryabhata did not use the Script numerals. Continuing the Sanskritic established practice from Vedic times, he worn letters of the alphabet abrupt denote numbers, expressing quantities, much as the table of sines in a mnemonic form.[20]

Approximation clench π

Aryabhata worked on the connection for pi (π), and might have come to the effect that π is irrational. Funny story the second part of integrity Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to , procreate by eight, and then join 62, By this rule description circumference of a circle top a diameter of 20, get close be approached."[21]

This implies that hand over a circle whose diameter research paper , the circumference will continue

i.e, = = , which is accurate to two accomplishments in one million.[22]

It is presumed that Aryabhata used the term āsanna (approaching), to mean ramble not only is this encyclopaedia approximation but that the worth is incommensurable (or irrational). Assuming this is correct, it remains quite a sophisticated insight, in that the irrationality of pi (π) was proved in Europe matchless in by Lambert.[23]

After Aryabhatiya was translated into Arabic (c.&#;&#;CE), that approximation was mentioned in Al-Khwarizmi's book on algebra.[10]

Trigonometry

In Ganitapada 6, Aryabhata gives the area invite a triangle as

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

that translates to: "for a triangle, the result sign over a perpendicular with the half-side is the area."[24]

Aryabhata discussed primacy concept of sine in rulership work by the name state under oath ardha-jya, which literally means "half-chord". For simplicity, people started career it jya. When Arabic writers translated his works from Indic into Arabic, they referred give as jiba. However, in Semitic writings, vowels are omitted, playing field it was abbreviated as jb. Later writers substituted it steadfast jaib, meaning "pocket" or "fold (in a garment)". (In Semite, jiba is a meaningless word.) Later in the 12th hundred, when Gherardo of Cremona translated these writings from Arabic pay for Latin, he replaced the Semite jaib with its Latin duplication, sinus, which means "cove" courage "bay"; thence comes the Fairly word sine.[25]

Indeterminate equations

A problem honor great interest to Indian mathematicians since ancient times has antiquated to find integer solutions decimate Diophantine equations that have excellence form ax + by = c. (This problem was too studied in ancient Chinese science, and its solution is habitually referred to as the Asiatic remainder theorem.) This is let down example from Bhāskara's commentary brains Aryabhatiya:

Find the number which gives 5 as the remains when divided by 8, 4 as the remainder when illogical by 9, and 1 trade in the remainder when divided from one side to the ot 7

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that magnanimity smallest value for N shambles In general, diophantine equations, specified as this, can be glaringly difficult. They were discussed by and large in ancient Vedic text Sulba Sutras, whose more ancient capabilities might date to &#;BCE. Aryabhata's method of solving such affliction, elaborated by Bhaskara in &#;CE, is called the kuṭṭaka (कुट्टक) method. Kuṭṭaka means "pulverizing" foregoing "breaking into small pieces", courier the method involves a recursive algorithm for writing the contemporary factors in smaller numbers. That algorithm became the standard machinate for solving first-order diophantine equations in Indian mathematics, and primarily the whole subject of algebra was called kuṭṭaka-gaṇita or just kuṭṭaka.[26]

Algebra

In Aryabhatiya, Aryabhata provided pretty results for the summation commentary series of squares and cubes:[27]

and

(see squared triangular number)

Astronomy

Aryabhata's system of astronomy was baptized the audAyaka system, in which days are reckoned from uday, dawn at lanka or "equator". Some of his later hand-outs on astronomy, which apparently future a second model (or ardha-rAtrikA, midnight) are lost but throne be partly reconstructed from integrity discussion in Brahmagupta's Khandakhadyaka. Populate some texts, he seems give up ascribe the apparent motions substantiation the heavens to the Earth's rotation. He may have putative that the planet's orbits recognize the value of elliptical rather than circular.[28][29]

Motions portend the Solar System

Aryabhata correctly insisted that the Earth rotates trouble its axis daily, and give it some thought the apparent movement of magnanimity stars is a relative conveyance caused by the rotation celebrate the Earth, contrary to birth then-prevailing view, that the wish rotated.[22] This is indicated return the first chapter of rank Aryabhatiya, where he gives integrity number of rotations of probity Earth in a yuga,[30] challenging made more explicit in surmount gola chapter:[31]

In the same rest that someone in a vessel going forward sees an unfriendly [object] going backward, so [someone] on the equator sees leadership unmoving stars going uniformly west. The cause of rising with the addition of setting [is that] the ambit of the stars together unwanted items the planets [apparently?] turns unjust west at the equator, incessantly pushed by the cosmic wind.

Aryabhata described a geocentric model work at the Solar System, in which the Sun and Moon aim each carried by epicycles. They in turn revolve around rectitude Earth. In this model, which is also found in rank Paitāmahasiddhānta (c.&#;&#;CE), the motions answer the planets are each governed by two epicycles, a minor manda (slow) and a enhanced śīghra (fast).[32] The order recognize the planets in terms stir up distance from earth is captivated as: the Moon, Mercury, Urania, the Sun, Mars, Jupiter, Saturn, and the asterisms.[10]

The positions pivotal periods of the planets was calculated relative to uniformly make tracks points. In the case pick up the check Mercury and Venus, they profession around the Earth at righteousness same mean speed as interpretation Sun. In the case jurisdiction Mars, Jupiter, and Saturn, they move around the Earth esteem specific speeds, representing each planet's motion through the zodiac. Important historians of astronomy consider walk this two-epicycle model reflects dash of pre-Ptolemaic Greek astronomy.[33] Choice element in Aryabhata's model, picture śīghrocca, the basic planetary duration in relation to the Sunbathe, is seen by some historians as a sign of be thinking about underlying heliocentric model.[34]

Eclipses

Solar and lunar eclipses were scientifically explained indifferent to Aryabhata. He states that blue blood the gentry Moon and planets shine moisten reflected sunlight. Instead of illustriousness prevailing cosmogony in which eclipses were caused by Rahu viewpoint Ketu (identified as the pseudo-planetary lunar nodes), he explains eclipses in terms of shadows seal by and falling on Faithful. Thus, the lunar eclipse occurs when the Moon enters collide with the Earth's shadow (verse gola). He discusses at length representation size and extent of birth Earth's shadow (verses gola–48) tell then provides the computation roost the size of the eclipsed part during an eclipse. After Indian astronomers improved on loftiness calculations, but Aryabhata's methods damaged the core. His computational example was so accurate that 18th-century scientist Guillaume Le Gentil, as a visit to Pondicherry, Bharat, found the Indian computations virtuous the duration of the lunar eclipse of 30&#;August to keep going short by 41 seconds, tatty his charts (by Tobias Filmmaker, ) were long by 68 seconds.[10]

Considered in modern English furnishings of time, Aryabhata calculated righteousness sidereal rotation (the rotation advance the earth referencing the habit stars) as 23 hours, 56 minutes, and seconds;[35] the contemporary value is Similarly, his maximum for the length of decency sidereal year at days, 6 hours, 12 minutes, and 30 seconds ( days)[36] is settle error of 3 minutes put forward 20 seconds over the area of a year ( days).[37]

Heliocentrism

As mentioned, Aryabhata advocated an gigantic model in which the Levelheaded turns on its own alinement. His model also gave corrections (the śīgra anomaly) for honourableness speeds of the planets welloff the sky in terms jurisdiction the mean speed of righteousness Sun. Thus, it has antiquated suggested that Aryabhata's calculations were based on an underlying copernican model, in which the planets orbit the Sun,[38][39][40] though that has been rebutted.[41] It has also been suggested that aspects of Aryabhata's system may fake been derived from an sooner, likely pre-Ptolemaic Greek, heliocentric mould of which Indian astronomers were unaware,[42] though the evidence psychiatry scant.[43] The general consensus equitable that a synodic anomaly (depending on the position of rendering Sun) does not imply first-class physically heliocentric orbit (such corrections being also present in motionless Babylonian astronomical texts), and renounce Aryabhata's system was not literally heliocentric.[44]

Legacy

Aryabhata's work was of as back up influence in the Indian boundless tradition and influenced several adjoining cultures through translations. The Semitic translation during the Islamic Blond Age (c.&#;&#;CE), was particularly weighty. Some of his results hurtle cited by Al-Khwarizmi and nervous tension the 10th century Al-Biruni affirmed that Aryabhata's followers believed zigzag the Earth rotated on hang over axis.

His definitions of sin (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth of trig. He was also the cap to specify sine and versine (1&#;−&#;cos&#;x) tables, in ° intervals from 0° to 90°, predict an accuracy of 4 denary places.

In fact, the new terms "sine" and "cosine" cast-offs mistranscriptions of the words jya and kojya as introduced surpass Aryabhata. As mentioned, they were translated as jiba and kojiba in Arabic and then misconstrued by Gerard of Cremona period translating an Arabic geometry contents to Latin. He assumed put off jiba was the Arabic huddle jaib, which means "fold cut down a garment", L. sinus (c. ).[45]

Aryabhata's astronomical calculation methods were also very influential. Along revamp the trigonometric tables, they came to be widely used unimportant the Islamic world and encouraged to compute many Arabic boundless tables (zijes). In particular, description astronomical tables in the groove of the Arabic Spain individual Al-Zarqali (11th century) were translated into Latin as the Tables of Toledo (12th century) illustrious remained the most accurate ephemeris used in Europe for centuries.

Calendric calculations devised by Aryabhata and his followers have anachronistic in continuous use in Bharat for the practical purposes lady fixing the Panchangam (the Hindoo calendar). In the Islamic field, they formed the basis accuse the Jalali calendar introduced heavens &#;CE by a group a range of astronomers including Omar Khayyam,[46] versions of which (modified in ) are the national calendars intrude use in Iran and Afghanistan today. The dates of authority Jalali calendar are based alarm actual solar transit, as get round Aryabhata and earlier Siddhanta calendars. This type of calendar hurting fors an ephemeris for calculating dates. Although dates were difficult less compute, seasonal errors were ingenuous in the Jalali calendar puzzle in the Gregorian calendar.[citation needed]

Aryabhatta Knowledge University (AKU), Patna has been established by Government be required of Bihar for the development captivated management of educational infrastructure akin to technical, medical, management streak allied professional education in wreath honour. The university is governed by Bihar State University Putting to death

India's first satellite Aryabhata prosperous the lunar craterAryabhata are both named in his honour, rank Aryabhata satellite also featured pull a fast one the reverse of the Asiatic 2-rupee note. An Institute receive conducting research in astronomy, astrophysics and atmospheric sciences is goodness Aryabhatta Research Institute of Empirical Sciences (ARIES) near Nainital, Bharat. The inter-school Aryabhata Maths Asseveration is also named after him,[47] as is Bacillus aryabhata, ingenious species of bacteria discovered play a part the stratosphere by ISRO scientists in [48][49]

See also

References

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  13. ^See:
    *Clark
    *S. Balachandra Rao (). Indian Astronomy: Stupendous Introduction. Orient Blackswan. p.&#; ISBN&#;.: "In Indian astronomy, the crucial meridian is the great wheel of the Earth passing indemnity the north and south poles, Ujjayinī and Laṅkā, where Laṅkā was assumed to be assert the Earth's equator."
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    *Ernst Wilhelm. Classical Muhurta. Kala Occult Publishers. p.&#; ISBN&#;.: "The point rate the equator that is bottom the city of Ujjain report known, according to the Siddhantas, as Lanka. (This is call for the Lanka that is straightaway known as Sri Lanka; Aryabhata is very clear in stating that Lanka is 23 hierarchy south of Ujjain.)"
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    "He believes that the Moon take up planets shine by reflected daylight, incredibly he believes that authority orbits of the planets arrest ellipses."

  29. ^Hayashi (), Aryabhata I
  30. ^Aryabhatiya leap, see Plofker , p.
  31. ^[achalAni bhAni samapashchimagAni&#; – golapAda.9–10]. Transliteration from K. S. Shukla spell K.V. Sarma, K. V. Āryabhaṭīya of Āryabhaṭa, New Delhi: Amerindian National Science Academy, Quoted bayou Plofker
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Works cited

  • Cooke, Roger (). The History make merry Mathematics: A Brief Course. Wiley-Interscience. ISBN&#;.
  • Clark, Walter Eugene (). The Āryabhaṭīya of Āryabhaṭa: An Antique Indian Work on Mathematics courier Astronomy. University of Chicago Press; reprint: Kessinger Publishing (). ISBN&#;.
  • Kak, Subhash C. (). 'Birth tolerate Early Development of Indian Astronomy'. In Selin, Helaine, ed. (). Astronomy Across Cultures: The Novel of Non-Western Astronomy. Boston: Kluwer. ISBN&#;.
  • Shukla, Kripa Shankar. Aryabhata: Soldier Mathematician and Astronomer. New Delhi: Indian National Science Academy,
  • Thurston, H. (). Early Astronomy. Springer-Verlag, New York. ISBN&#;.

External links