Āryabhaṭa (Devanāgarī: आर्यभट) (476 – 550) is the first of the great mathematician-astronomers of the classical age of Indian mathematics. There exists no documentation to ascertain his exact birthplace. Available evidence suggest that he went to Kusumapura for higher studies. He lived in Kusumapura, which his commentator Bhāskara I (629 AD) identifies as Pataliputra (modern Patna).
Aryabhata was the first in the line of brilliant mathematician-astronomers of classical Indian mathematics, whose major work was the Aryabhatiyam and the Aryabhatta-siddhanta. Although his exact place of birth is not documented but he is believed to be from Kashmir. Aryabhatiyam 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, c.600 AD) and by Nilakantha Somayaji in his Aryabhatiya Bhasya, (1465 AD). The number place-value system, first seen in the 3rd century Bakhshali Manuscript was clearly in place in his work [1]. He may have been the first mathematician to use letters of the alphabet to denote unknown quantities. [2]
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. He was perhaps the first to ascribe the motion of the moon to the earth's rotation and the first to develop an elliptical model of the heliocentric planetary system (see below).

[edit] Pi as Irrational

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

chaturadhikam śatamaśṭaguṇam dvāśaśṭistathā sahasrāṇām
Ayutadvayaviśkambhasyāsanno vrîttapariṇahaḥ."Add four to 100, multiply by eight and then add sixty-two thousand. By this rule is the circumference of a circle of diameter 20,000 approximately given"

In other words, , correct to four rounded-off decimal places. The commentator Nilakantha Somayaji, (Kerala School, 15th c.) has argued that the word āsanna (approaching), appearing just before the last word, here means not only that this is 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 (Lambert).

[edit] Mensuration and Trigonometry

In Ganitapada 6, Aryabhata gives the area of triangle astribhujasya falashariram samadalakoti bhujardhasamvargah (for a triangle, the result of a perpendicular with the half-side is the area.) Aryabhata, in his work Aryabhata-Siddhanta, first defined the sine as the modern relationship between half an angle and half a chord. He also defined the cosine, versine, and inverse sine. He used the words jya for sine, kojya for cosine, ukramajya for versine, and otkram jya for inverse sine.
Aryabhata's tables for the sines (from which the rest can be computed), is presented in a single rhyming stanza, with each syllable standing for increments at intervals of 225 minutes of arc or 3 degrees 45'. Using a compact alphabetic code called varga/avarga, he defines the sines for a circle of circumference 21600 (radius 3438). He uses the alphabetic code to define a set of increments :makhi bhakhi fakhi dhakhi Nakhi N~akhi M~akhi hasjha .... Here "makhi" stands for 25 (ma) + 200 (khi), and the corresponding sine value (for 225 minutes of arc) is 225 / 3438. The value corresponding to the eighth term (hasjha, 199 (ha=100 + s=90 + jha=9), is the sum of all the increments before it, totalling 1719. The entire table for 90 degrees is given as follows:225,224,222,219.215,210,205,199,191,183,174,164,154,143,131,119,106,93,79,65,51,37,,22,7 So we see that sin(15) (sum of first four terms) = 890/3438 = 0.258871 (correct value = 0.258819, correct to four significant digits). The value of sin(30) (corresponding to hasjha) is 1719/3438 = 0.5; this is of course, exact. His alphabetic code (there are many such codes in Sanskrit) has come to be known as the Aryabhata cipher.

[edit] Motion of the planets

Aryabhata propunded a heliocentric model of the planets, in which the Earth was taken to be spinning on its axis and the positions and periods of the planets were calculated relative to a stationary Sun (this method was known as "sugrocha"). He states that the Moon and planets shine by reflected sunlight, and that the orbits of the planets are ellipses around the Sun. He also correctly explains the causes of eclipses of the Sun and the Moon.
In the fourth book of his Aryabhatiya, Goladhyaya or Golapada, Aryabhata is dealing with the celestial sphere, shape of the earth, cause of day and night etc. In golapAda.6 he says:bhugolaH sarvato vr.ttaH (The earth is circular everywhere) Another statement, referring to Lanka , 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] In the next verse he says: “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 here is a reference point to mean the equator, which was known to pass through Sri Lanka.
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 improved on the computation by the African mathematician Erastosthenes (c.200 BC), whose exact computation is not known in modern units.

[edit] Length of day

Aryabhata calculated the Sidereal day (the rotation of the earth against 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.

[edit] Heliocentrism

Aryabhata's computations are based on the heliocentric notion of the planets orbiting the sun and the earth spinning on its own axis. While he is not the first to say this, his authority was certainly most influential. The earlier Indian astronomical texts Shatapatha Brahmana (c. 9th-7th century BC), Aitareya Brahmana (c. 9th-7th century BC) and Vishnu Purana (c. 1st century BC) contain early concepts of a heliocentric model. Heraclides of Pontus (4th c. BC) is sometimes credited with a heliocentric theory but it appears that although Heraclides wrote prolifically, he may have been perceived as a vain and pompous man with little influence. Aristarchus of Samos (3rd century BC) is usually credited with a heliocentric theory but his work was not widely influential. The version of Greek astronomy known in ancient India, Paulisha siddhanta (possibly by a Paul of Alexandria) makes no reference to a Heliocentric theory. While none of these texts survived the Middle Ages, the 8th century Arabic edition of the Āryabhatīya was translated into Latin in the 13th century, well before Copernicus. It is possible that some of these ideas may have been influential, though a direct connection with Copernicus cannot be established.

[edit] Diophantine Equations

A problem of great interest to Indian mathematicians since very ancient times concerned diophantine equations. These involve integer solutions to equations such as ax + b = cy. 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 800BC. Aryabhata's method of solving such problems, called the kuttaka 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 AD 621, is the standard method for solving first order Diophantine equations, and it is often referred to as the Aryabhata algorithm. See details of the Kuttaka method in this [1].

[edit] Continued relevance

Aryabhata's methods of astronomical calculations have been in continuous use for practical purposes of fixing the Panchanga Hindu calendar.
Recently Aryabhata was a theme in the RSA Conference 2006. Indocrypt 2005 had an invited talk on Vedic mathematics. The cryptography community seems to be rediscovering more and more interesting results from ancient Indian mathematics, of which Aryabhata is no doubt the leading luminary.
The lunar crater Aryabhata is named in his honour.

[edit] Confusion of identity

There has been some confusion regarding Aryabhatta's identity. Another notable Indian mathematician, Aryabhata II flourished sometime between 950 and 1100 AD and were two famous Indian mathematicians named Aryabhata who lived around 500 AD. The subsequent confusion continued for some time, but in 1926 B Datta showed that al-Biruni's two Aryabhattas were one and the same. However there is a precise mention of the year of birth of Aryabhata in the Aryabhatiya (3-10) which corresponds to 476 AD .