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Āryabhaṭa (Devanāgarī: आर्यभट) (b. 476 AD – 550) is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Aryabhatiya (499) and Arya-Siddhanta.

Contents 1 Biography 2 Works 2.1 Aryabhatiya 3 Mathematics 3.1 Place Value system and zero 3.2 Pi as Irrational 3.3 Mensuration and trigonometry 3.4 Indeterminate Equations 4 Astronomy 4.1 Motions of the Solar System 4.2 Eclipses 4.3 Sidereal periods 4.4 Heliocentrism 5 Legacy 6 See also 7 References 7.1 Other References 8 External links

Aryabhatta was born in Kerala. Asmaka (meaning hard-stone) the place name with which he is referred to by Bhaskara-I is the mythical Jain country surrounding Sravanabelgola to which Camravattam was a part as evidenced by the ancient temple of Bahubali existing even today as Camravattathappan. { {{cite journal

|last=Hari
|first=Chandra.K
|year=2007
|month= August
|title=Eclipse Observed by Aryabhata in Kerala 
|journal=Current Science
|volume=93
|issue=3
|pages=pp. 414-418

Aryabhata is also known as Aryabhata I to distinguish him from the later mathematician of the same name who lived about 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed to believe that there were two different mathematicians called Aryabhata living at the same time. He therefore created a confusion of two different Aryabhatas which was not clarified until 1926 when B Datta showed that al-Biruni's two Aryabhatas were one and the same person.

We know the year of Aryabhata's birth since he tells us that he was twenty-three years of age when he wrote Aryabhatiya which he finished in 499. We have given Kusumapura, thought to be close to Pataliputra (which was refounded as Patna in Bihar in 1541), as the place of Aryabhata's birth but this is far from certain, as is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

   ... no final verdict can be given regarding the locations of Asmakajanapada and Kusumapura. 

We do know that Aryabhata wrote Aryabhatiya in Kusumapura at the time when Pataliputra was the capital of the Gupta empire and a major centre of learning, but there have been numerous other places proposed by historians as his birthplace. Some conjecture that he was born in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that he was born in the north-east of India, perhaps in Bengal. In [8] it is claimed that Aryabhata was born in the Asmaka region of the Vakataka dynasty in South India although the author accepted that he lived most of his life in Kusumapura in the Gupta empire of the north. However, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th century. It is now thought by most historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on the Aryabhatiya.

We should note that Kusumapura became one of the two major mathematical centres of India, the other being Ujjain. Both are in the north but Kusumapura (assuming it to be close to Pataliputra) is on the Ganges and is the more northerly. Pataliputra, being the capital of the Gupta empire at the time of Aryabhata, was the centre of a communications network which allowed learning from other parts of the world to reach it easily, and also allowed the mathematical and astronomical advances made by Aryabhata and his school to reach across India and also eventually into the Islamic world.

As to the texts written by Aryabhata only one has survived. However Jha claims in [21] that:-

   ... Aryabhata was an author of at least three astronomical texts and wrote some free stanzas as well. 

The surviving text is Aryabhata's masterpiece the Aryabhatiya which is a small astronomical treatise written in 118 verses giving a summary of Hindu mathematics up to that time. Its mathematical section contains 33 verses giving 66 mathematical rules without proof. The Aryabhatiya contains an introduction of 10 verses, followed by a section on mathematics with, as we just mentioned, 33 verses, then a section of 25 verses on the reckoning of time and planetary models, with the final section of 50 verses being on the sphere and eclipses.

There is a difficulty with this layout which is discussed in detail by van der Waerden in [35]. Van der Waerden suggests that in fact the 10 verse Introduction was written later than the other three sections. One reason for believing that the two parts were not intended as a whole is that the first section has a different meter to the remaining three sections. However, the problems do not stop there. We said that the first section had ten verses and indeed Aryabhata titles the section Set of ten giti stanzas. But it in fact contains eleven giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have been added and he identifies a small number of verses in the remaining sections which he argues have also been added by a member of Aryabhata's school at Kusumapura.

The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and spherical trigonometry. It also contains continued fractions, quadratic equations, sums of power series and a table of sines. Let us examine some of these in a little more detail.

First we look at the system for representing numbers which Aryabhata invented and used in the Aryabhatiya. See Āryabhaṭa numeration for detail. It consists of giving numerical values to the 33 consonants of the Indian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The higher numbers are denoted by these consonants followed by a vowel to obtain 100, 10000, .... In fact the system allows numbers up to 10¹⁸ to be represented with an alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar with numeral symbols and the place-value system. He writes in [3]:-

   ... it is extremely likely that Aryabhata knew the sign for zero and the numerals of the place value system. This supposition is based on the 
following two facts: first, the invention of his alphabetical counting system would have been impossible without zero or the place-value system; 
secondly, he carries out calculations on square and cubic roots which are impossible if the numbers in question are not written according to the 
place-value system and zero. 

Next we look briefly at some algebra contained in the Aryabhatiya. This work is the first we are aware of which examines integer solutions to equations of the form by = ax + c and by = ax - c, where a, b, c are integers. The problem arose from studying the problem in astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to solve problems of this type. The word kuttaka means "to pulverise" and the method consisted of breaking the problem down into new problems where the coefficients became smaller and smaller with each step. The method here is essentially the use of the Euclidean algorithm to find the highest common factor of a and b but is also related to continued fractions.

Aryabhata gave an accurate approximation for π. He wrote in the Aryabhatiya the following:-

   Add four to one hundred, multiply by eight and then add sixty-two thousand. the result is approximately the circumference of a circle of diameter 
twenty thousand. By this rule the relation of the circumference to diameter is given. 

This gives π = 62832/20000 = 3.1416 which is a surprisingly accurate value. In fact π = 3.14159265 correct to 8 places. If obtaining a value this accurate is surprising, it is perhaps even more surprising that Aryabhata does not use his accurate value for π but prefers to use √10 = 3.1622 in practice. Aryabhata does not explain how he found this accurate value but, for example, Ahmad [5] considers this value as an approximation to half the perimeter of a regular polygon of 256 sides inscribed in the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling of the number of sides. Another interesting paper discussing this accurate value of π by Aryabhata is [22] where Jha writes:-

   Aryabhata I's value of π is a very close approximation to the modern value and the most accurate among those of the ancients. There are reasons to 
believe that Aryabhata devised a particular method for finding this value. It is shown with sufficient grounds that Aryabhata himself used it, and 
several later Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of π is of Greek origin is critically 
examined and is found to be without foundation. Aryabhata discovered this value independently and also realised that π is an irrational number. He 
had the Indian background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit of discovering this exact value of π may be 
ascribed to the celebrated mathematician, Aryabhata I. 

We now look at the trigonometry contained in Aryabhata's treatise. He gave a table of sines calculating the approximate values at intervals of 90degrees/24 = 3degrees 45'. In order to do this he used a formula for sin(n+1)x - sin nx in terms of sin nx and sin (n-1)x. He also introduced the versine (versin = 1 - cosine) into trigonometry.

Other rules given by Aryabhata include that for summing the first n integers, the squares of these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of a circle which are correct, but the formulae for the volumes of a sphere and of a pyramid are claimed to be wrong by most historians. For example Ganitanand in [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V = Ah/2 for the volume of a pyramid with height h and triangular base of area A. He also appears to give an incorrect expression for the volume of a sphere. However, as is often the case, nothing is as straightforward as it appears and Elfering (see for example [13]) argues that this is not an error but rather the result of an incorrect translation.

This relates to verses 6, 7, and 10 of the second section of the Aryabhatiya and in [13] Elfering produces a translation which yields the correct answer for both the volume of a pyramid and for a sphere. However, in his translation Elfering translates two technical terms in a different way to the meaning which they usually have. Without some supporting evidence that these technical terms have been used with these different meanings in other places it would still appear that Aryabhata did indeed give the incorrect formulae for these volumes.

We have looked at the mathematics contained in the Aryabhatiya but this is an astronomy text so we should say a little regarding the astronomy which it contains. Aryabhata gives a systematic treatment of the position of the planets in space. He gave the circumference of the earth as 4 967 yojanas and its diameter as 1 5811/24 yojanas. Since 1 yojana = 5 miles this gives the circumference as 24 835 miles, which is an excellent approximation to the currently accepted value of 24 902 miles. He believed that the apparent rotation of the heavens was due to the axial rotation of the Earth. This is a quite remarkable view of the nature of the solar system which later commentators could not bring themselves to follow and most changed the text to save Aryabhata from what they thought were stupid errors!

Aryabhata gives the radius of the planetary orbits in terms of the radius of the Earth/Sun orbit as essentially their periods of rotation around the Sun. He believes that the Moon and planets shine by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains the causes of eclipses of the Sun and the Moon. The Indian belief up to that time was that eclipses were caused by a demon called Rahu. His value for the length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since the true value is less than 365 days 6 hours.

Bhaskara I who wrote a commentary on the Aryabhatiya about 100 years later wrote of Aryabhata:-

   Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, 
kinematics and spherics, handed over the three sciences to the learned world. 

Region lying between Narmada and Godavari, which was known as Ashmaka,and is now identified with Maharashtra, though early Buddhist texts describe Ashmaka as being further south, dakShiNApath or the Deccan, while other texts describe the Ashmakas as having fought Alexander, which would put them further north.^[1]

Other traditions in India claim that he was from Kerala and that he travelled to the North, or that he was a Maga Brahmin from Gujarat. However, it is fairly certain that at some point, he went to Kusumapura for higher studies, and that he lived here for some time.^[2] Bhāskara I (AD 629) identifies Kusumapura as Pataliputra (modern Patna). He lived there in the dying years of the Gupta empire, the time which is known as the golden age of India, when it was already under Hun attack in the Northeast, during the reign of Buddhagupta and some of the smaller kings before Vishnugupta. Speculations as above may be examined in the light of the new evidences that have got published. Aryabhata was a Jain astronomer of Kerala and Kusumapura was a Jain centre of learning. So it is likely that Aryabhata had been to Kusumapura in later years of his life. His astronomical precepts clearly suggest that he observed the heavens from the latitudes of Kerala including Kanyakumari at 08N00. { {{cite journal

|last=Hari
|first=Chandra.K
|year=2007
|month= October
|title=Critical Evidence to Fix the Place of Aryabhatta-I
|journal=Current Science
|volume=93
|issue=8
|pages=pp. 1177-1186

His first name “Arya” is a term used for respect, such as "Sri", whereas Bhatta is a typical north Indian name -- found today usually among the “Bania” (or trader) community in Bihar and among Brahmin communities in India as their last name.

Modern historians say that Aryabhata was born in 476A.D. Their calculations are based on the following sloka in Aryabhatiya. Shastyabdeenam Shadbhiryada
Vyatheethastrayashcha yugapadah
Adhika vimsathirabdaasthedeha
Mamajanmanotheethah

The following interpretation is baseless and wrong. {{Shadbhih has been taken as shashti by some historians. Then the poem means that Aryabhata was twenty years old in 3600(60 times 60) Kali Era which started in 3102B.C. This calculation will give the year of Aryabhata's birth as 476 A.D. This is wrong for the following reasons.
1.The usage of the word shashti in the above poem is grammatically wrong in Sanskrit.
2.In 478A.D. Kali era was not in vogue. At that time Vikrama and Salivahana eras were widely used. The correct word in the poem is Shadbhih. Aryabhata was 20 years old in 360(6 times 60)Kali Era. This gives Aryabhata's year of birth as 2762B.C., long before the heliocentric theory was even thought of in the West.}}

Aryabhata is the author of several treatises on mathematics and astronomy, some of which are lost. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature, and has survived to modern times.

The Arya-siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata's contemporary Varahamihira, as well as through later mathematicians and commentators including Brahmagupta and Bhaskara I. This work appears to be based on the older Surya Siddhanta, and uses the midnight-day-reckoning, as opposed to sunrise in Aryabhatiya. This also contained a description of several astronomical instruments, the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semi-circle and circle shaped (dhanur-yantra / chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.^[1]

A third text that may have survived in Arabic translation is the Al ntf or Al-nanf, which claims to be a translation of Aryabhata, but the Sanskrit name of this work is not known. Probably dating from the ninth c., it is mentioned by the Persian scholar and chronicler of India, Abū Rayhān al-Bīrūnī.^[1]

Direct details of Aryabhata's work are therefore known only from the Aryabhatiya. The name Aryabhatiya is due to later commentators, Aryabhata himself may not have given it a name; it is referred by his disciple Bhaskara I as Ashmakatantra or the treatise from the Ashmaka. It is also occasionally referred to as Arya-shatas-aShTa, lit., Aryabhata's 108, which is the number of verses in the text. It is written in the very terse style typical of the sutra literature, where each line is an aid to memory for a complex system. Thus, the explication of meaning is due to commentators. The entire text consists of 108 verses, plus an introductory 13, the whole being divided into four pAdas or chapters:

1. gitikApAda: (13 verses) large units of time - kalpa, manvantra, yuga, which present a cosmology that differs from earlier texts such as Lagadha's Vedanga Jyotisha(ca. 1st c. BC). Also includes the table of sines (jya), given in a single verse. For the planetary revolutions during a mahayuga, the number of 4.32mn years is given.
2. gaNitapAda (33 verses), covering mensuration (kShetra vyAvahAra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuTTaka)
3. kAlakriyApAda (25 verses) : different units of time and method of determination of positions of planets for a given day. Calculations concerning the intercalary month (adhikamAsa), kShaya-tithis. Presents a seven-day week, with names for days of week.
4. golapAda (50 verses): Geometric/trigonometric aspects of the celestial sphere, features of the ecliptic, celestial equator, node, shape of the earth, cause of day and night, rising of zodiacal signs on horizon etc.

In addition, some versions cite a few colophons added at the end, extolling the virtues of the work, etc.

The Aryabhatiya presented a number of innovations in mathematics and astronomy in verse form, which were influential for many centuries. The extreme brevity of the text was elaborated in commentaries by his disciple Bhaskara I (Bhashya, ca. 600) and by Nilakantha Somayaji in his Aryabhatiya Bhasya, (1465).

The number place-value system, first seen in the 3rd century Bakhshali Manuscript was clearly in place in his work.^[3] ; he certainly did not use the symbol, but the French mathematician Georges Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients^[4].

However, Aryabhata did not use the brahmi numerals; continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities (such as the table of sines) in a mnemonic form^[5].

Aryabhata worked on the approximation for Pi (π), and may have realized that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:^{[citation needed]}

chaturadhikam śatamaśṭaguṇam dvāśaśṭistathā sahasrāṇām
Ayutadvayaviśkambhasyāsanno vrîttapariṇahaḥ.
"Add four to 100, multiply by eight and then add 62,000. By this rule the circumference of a circle of diameter 20,000 can be approached."

Aryabhata interpreted the word āsanna (approaching), appearing just before the last word, as saying that not only that is this an approximation, but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, for the irrationality of pi was proved in Europe only in 1761 by Lambert)^[6].

After Aryabhatiya was translated into Arabic (ca. 820 AD) this approximation was mentioned in Al-Khwarizmi's book on algebra^[1].

In Ganitapada 6, Aryabhata gives the area of triangle as

tribhujasya phalashariram samadalakoti bhujardhasamvargah

that translates to: for a triangle, the result of a perpendicular with the half-side is the area.^[7] His great contribution to mensuration and trigonometry is used in the current international mathematics.

A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + b = cy, a topic that has come to be known as diophantine equations. Here is an example from Bhaskara's commentary on Aryabhatiya: :

Find the number which gives 5 as the remainder when divided by 8; 4 as the remainder when divided by 9; and 1 as the remainder when divided by 7.

i.e. find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations can be notoriously difficult. Such equations were considered extensively in the ancient Vedic text Sulba Sutras, the more ancient parts of which may date back to 800 BCE. Aryabhata's method of solving such problems, called the kuṭṭaka (कूटटक) method. Kuttaka means pulverizing, that is breaking into small pieces, and the method involved a recursive algorithm for writing the original factors in terms of smaller numbers. Today this algorithm, as elaborated by Bhaskara in AD 621, is the standard method for solving first order Diophantine equations, and it is often referred to as the Aryabhata algorithm^[8].

The diophantine equations are of interest in cryptology, and the RSA Conference, 2006, focused on the kuttaka method and earlier work in the Sulvasutras.

Aryabhata's system of astronomy was called the audAyaka system (days are reckoned from uday, dawn at lanka, equator). Some of his later writings on astronomy, which apparently proposed a second model (ardha-rAtrikA, midnight), are lost, but can be partly reconstructed from the discussion in Brahmagupta's khanDakhAdyaka. In some texts he seems to ascribe the apparent motions of the heavens to the earth's rotation.

Aryabhata appears to have believed that the earth rotates about its axis. This is made clear in the statement, referring to Lanka , which describes the movement of the stars as a relative motion caused by the rotation of the earth:

Like a man in a boat moving forward sees the stationary objects as moving backward, just so are the stationary stars seen by the people in lankA (i.e. on the equator) as moving exactly towards the West. [achalAni bhAni samapashchimagAni - golapAda.9]

But the next verse describes the motion of the stars and planets as real movements: “The cause of their rising and setting is due to the fact the circle of the asterisms together with the planets driven by the provector wind, constantly moves westwards at Lanka”.

Lanka (lit. Sri Lanka) is here a reference point on the equator, which was taken as the equivalent to the reference meridian for astronomical calculations.

Aryabhata described a geocentric model of the solar system, in which the Sun and Moon are each carried by epicycles which in turn revolve around the Earth. In this model, which is also found in the Paitāmahasiddhānta (ca. AD 425), the motions of the planets are each governed by two epicycles, a smaller manda (slow) epicycle and a larger śīghra (fast) epicycle. ^[9] The order of the planets in terms of distance from earth are taken as: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms^[1].

The positions and periods of the planets were calculated relative to uniformly moving points, which in the case of Mercury and Venus, move around the Earth at the same speed as the mean Sun and in the case of Mars, Jupiter, and Saturn move around the Earth at specific speeds representing each planet's motion through the zodiac. Most historians of astronomy consider that this two epicycle model reflects elements of pre-Ptolemaic Greek astronomy.^[10] Another element in Aryabhata's model, the śīghrocca, the basic planetary period in relation to the Sun, is seen by some historians as a sign of an underlying heliocentric model.^[11]

He states that the Moon and planets shine by reflected sunlight. Instead of the prevailing cosmogyny where eclipses were caused by pseudo-planetary nodes Rahu and Ketu, he explains eclipses in terms of shadows cast by and falling on earth. Thus the lunar eclipse occurs when the moon enters into the earth-shadow (verse gola.37), and discusses at length the size and extent of this earth-shadow (verses gola.38-48), and then the computation, and the size of the eclipsed part during eclipses. Subsequent Indian astronomers improved on these calculations, but his methods provided the core. This computational paradigm was so accurate that the 18th century scientist Guillaume le Gentil, during a visit to Pondicherry, found the Indian computations of the duration of the lunar eclipse of 1765-08-30 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.^[1].

Aryabhata's computation of Earth's circumference as 24,835 miles, which was only 0.2% smaller than the actual value of 24,902 miles. This approximation might have improved on the computation by the Greek mathematician Eratosthenes (c.200 BC), whose exact computation is not known in modern units.

Considered in modern English units of time, Aryabhata calculated the sidereal rotation (the rotation of the earth referenced the fixed stars) as 23 hours 56 minutes and 4.1 seconds; the modern value is 23:56:4.091. Similarly, his value for the length of the sidereal year at 365 days 6 hours 12 minutes 30 seconds is an error of 3 minutes 20 seconds over the length of a year. The notion of sidereal time was known in most other astronomical systems of the time, but this computation was likely the most accurate in the period.

Āryabhata claims that the Earth turns on its own axis and some elements of his planetary epicyclic models rotate at the same speed as the motion of the planet around the Sun. This has suggested to some interpreters that Āryabhata's calculations were based on an underlying heliocentric model in which the planets orbit the Sun.^[12][13] A detailed rebuttal to this heliocentric interpretation is in a review which describes B. L. van der Waerden's book as "show[ing] a complete misunderstanding of Indian planetary theory [that] is flatly contradicted by every word of Āryabhata's description,"^[14] although some concede that Āryabhata's system stems from an earlier heliocentric model of which he was unaware.^[15] It has even been claimed that he considered the planet's paths to be elliptical, although no primary evidence for this has been cited.^[16] Though Aristarchus of Samos (3rd century BC) and sometimes Heraclides of Pontus (4th century BC) are usually credited with knowing the heliocentric theory, the version of Greek astronomy known in ancient India, Paulisa Siddhanta (possibly by a Paul of Alexandria) makes no reference to a Heliocentric theory.

Aryabhata's work was of great influence in the Indian astronomical tradition, and influenced several neighbouring cultures through translations. The Arabic translation during the Islamic Golden Age (ca. 820), was particularly influential. Some of his results are cited by Al-Khwarizmi, and he is referred to by the 10th century Arabic scholar Al-Biruni, who states that Āryabhata's followers believed the Earth to rotate on its axis.

His definitions of sine, as well as cosine (kojya), versine (ukramajya), and inverse sine (otkram jya), influenced the birth of trigonometry. He was also the first to specify sine and versine (1 - cosx) tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.

In fact, the modern names "sine" and "cosine", are a mis-transcription of the words jya and kojya as introduced by Aryabhata. They were transcribed as jiba and kojiba in Arabic. They were then misinterpreted by Gerard of Cremona while translating an Arabic geometry text to Latin; he took jiba to be the Arabic word jaib, which means "fold in a garment", L. sinus (c.1150)^[17].

Aryabhata's astronomical calculation methods were also very influential. Along with the trigonometric tables, they came to be widely used in the Islamic world, and were used to compute many Arabic astronomical tables (zijes). In particular, the astronomical tables in the work of the Arabic Spain scientist Al-Zarqali (11th c.), were translated into Latin as the Tables of Toledo (12th c.), and remained the most accurate Ephemeris used in Europe for centuries.

Calendric calculations worked out by Aryabhata and followers have been in continuous use in India for the practical purposes of fixing the Panchanga, or Hindu calendar, These were also transmitted to the Islamic world, and formed the basis for the Jalali calendar introduced 1073 by a group of astronomers including Omar Khayyam^[18], versions of which (modified in 1925) are the national calendars in use in Iran and Afghanistan today. The Jalali calendar determines its dates based on actual solar transit, as in Aryabhata (and earlier Siddhanta calendars). This type of calendar requires an Ephemeris for calculating dates. Although dates were difficult to compute, seasonal errors were lower in the Jalali calendar than in the Gregorian calendar.

India's first satellite Aryabhata, was named after him.

The lunar crater Aryabhata is named in his honour.

The interschool Aryabhatta Maths Competition is named after him.^[19]

• Āryabhaṭa numeration
• Aryabhatiya

• Cooke, Roger (1997). The History of Mathematics: A Brief Course. Wiley-Interscience. ISBN 0471180823. 
• Walter Eugene Clark, The Āryabhaṭīya of Āryabhaṭa, An Ancient Indian Work on Mathematics and Astronomy, University of Chicago Press (1930); reprint: Kessinger Publishing (2006), ISBN 978-1425485993.
• Kak, Subhash C. (2000). 'Birth and Early Development of Indian Astronomy'. In Selin, Helaine (2000), Astronomy Across Cultures: The History of Non-Western Astronomy, Kluwer, Boston, ISBN 0-7923-6363-9

• Shukla, Kripa Shankar. Aryabhata: Indian Mathematician and Astronomer. New Delhi: Indian National Science Academy, 1976.
• Thurston, H. (1994), Early Astronomy, Springer-Verlag, New York, ISBN 0-387-94107-X

Aryabhatta was born in Patna and not in Kerala.

• O'Connor, John J; Edmund F. Robertson "Aryabhata". MacTutor History of Mathematics archive.  
• RSA Conference 2006
• Aryabhata and Diophantus' son, Hindustan Times Storytelling Science column, Nov 2004

v • d • e Indian mathematics
Mathematicians	Achyuta Pisharati · Apastamba · Aryabhata · Aryabhata II · Bhāskara I · Bhāskara II · Baudhayana · Brahmagupta · Jyesthadeva · Katyayana · Madhava · Mahavira · Manava · Melpathur Narayana Bhattathiri · Nilakantha Somayaji · Parameshvara · Pingala · Sripati · Sridhara · Varahamihira · Virasena
Treatises	Aryabhatiya · Bakhshali manuscript · Paulisa Siddhanta · Paitamaha Siddhanta · Romaka Siddhanta · Surya Siddhanta · Śulba Sūtras · Vasishtha Siddhanta · Yavanajataka
Centers	Kerala
Influences	Babylonian mathematics · Greek Mathematics · Chinese mathematics
Influenced	Islamic mathematics · Chinese mathematics