During the Buddha's time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers in central India.

[9][10] It has been claimed that the aśmaka (Sanskrit for "stone") where Aryabhata originated may be the present day Kodungallur which was the historical capital city of Thiruvanchikkulam of ancient Kerala.

[14] Both Hindu and Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna.

[18] The Arya-siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata's contemporary, Varahamihira, and 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.

[10] It also contained a description of several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular and circular (dhanur-yantra / chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.

Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, Abū Rayhān al-Bīrūnī.

[18][8] It is written in the very terse style typical of sutra literature, in which each line is an aid to memory for a complex system.

While he did not use a symbol for zero, 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.

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.

In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes: caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"[24] Aryabhata discussed the concept of sine in his work by the name of ardha-jya, which literally means "half-chord".

[25] A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to Diophantine equations that have the form ax + by = c. (This problem was also studied in ancient Chinese mathematics, and its solution is usually referred to as the Chinese remainder theorem.)

Kuṭṭaka means "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original factors in smaller numbers.

This algorithm became the standard method for solving first-order diophantine equations in Indian mathematics, and initially the whole subject of algebra was called kuṭṭaka-gaṇita or simply kuṭṭaka.

Some of his later writings on astronomy, which apparently proposed a second model (or ardha-rAtrikA, midnight) are lost but can be partly reconstructed from the discussion in Brahmagupta's Khandakhadyaka.

turns due west at the equator, constantly pushed by the cosmic wind.Aryabhata described a geocentric model of the Solar System, in which the Sun and Moon are each carried by epicycles.

In this model, which is also found in the Paitāmahasiddhānta (c. 425 CE), the motions of the planets are each governed by two epicycles, a smaller manda (slow) and a larger śīghra (fast).

[32] The order of the planets in terms of distance from earth is taken as: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.

In the case of Mars, Jupiter, and Saturn, they move around the Earth at specific speeds, representing each planet's motion through the zodiac.

His computational paradigm was so accurate that 18th-century scientist Guillaume Le Gentil, during a visit to Pondicherry, India, found the Indian computations of the duration of the lunar eclipse of 30 August 1765 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.

Thus, it has been suggested that Aryabhata's calculations were based on an underlying heliocentric model, in which the planets orbit the Sun,[38][39][40] though this has been rebutted.

[41] It has also been suggested that aspects of Aryabhata's system may have been derived from an earlier, likely pre-Ptolemaic Greek, heliocentric model of which Indian astronomers were unaware,[42] though the evidence is scant.

Some of his results are cited by Al-Khwarizmi and in the 10th century Al-Biruni stated that Aryabhata's followers believed that the Earth rotated on its axis.

In fact, the modern terms "sine" and "cosine" are mistranscriptions of the words jya and kojya as introduced by Aryabhata.

Calendric calculations devised by Aryabhata and his followers have been in continuous use in India for the practical purposes of fixing the Panchangam (the Hindu calendar).