**Brahmagupta** (598–668 CE) was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brahmasphutasiddhanta (Correctly Established Doctrine of Brahma), written in 628 in Bhinmal. Its 25 chapters contain several unprecedented mathematical results.

Brahmagupta is believed to have been born in 598 AD in Bhinmal city in the state of Rajasthan of Northwest India. In ancient times Bhillamala was the seat of power of the Gurjars. His father was Jisnugupta. He likely lived most of his life in Bhillamala (modern Bhinmal in Rajasthan) during the reign (and possibly under the patronage) of King Vyaghramukha. As a result, Brahmagupta is often referred to as Bhillamalacarya, that is, the teacher from Bhillamala. He was the head of the astronomical observatory at Ujjain, and during his tenure there wrote four texts on mathematics and astronomy: the Cadamekela in 624, the Brahmasphutasiddhanta in 628, the Khandakhadyaka in 665, and the Durkeamynarda in 672. The Brahmasphutasiddhanta (Corrected Treatise of Brahma) is arguably his most famous work. The historian al-Biruni (c. 1050) in his book Tariq al-Hind states that the Abbasid caliph al-Ma’mun had an embassy in India and from India a book was brought to Baghdad which was translated into Arabic as Sindhind. It is generally presumed that Sindhind is none other than Brahmagupta’s Brahmasphuta-siddhanta.

Although Brahmagupta was familiar with the works of astronomers following the tradition of Aryabhatiya, it is not known if he was familiar with the work of Bhaskara I, a contemporary. Brahmagupta had a plethora of criticism directed towards the work of rival astronomers, and in his Brahmasphutasiddhanta is found one of the earliest attested schisms among Indian mathematicians. The division was primarily about the application of mathematics to the physical world, rather than about the mathematics itself. In Brahmagupta’s case, the disagreements stemmed largely from the choice of astronomical parameters and theories. Critiques of rival theories appear throughout the first ten astronomical chapters and the eleventh chapter is entirely devoted to criticism of these theories, although no criticisms appear in the twelfth and eighteenth chapters.

#### Mathematics

Brahmagupta was the first to use zero as a number. He gave rules to compute with zero. Brahmagupta used negative numbers and zero for computing. The modern rule that two negative numbers multiplied together equals a positive number first appears in Brahmasputa siddhanta. Brahmagupta’s most famous work is his Brahmasphutasiddhanta. It is composed in elliptic verse, as was common practice in Indian mathematics, and consequently has a poetic ring to it. As no proofs are given, it is not known how Brahmagupta’s mathematics was derived.

He also framed rules to solve a simple equation of the type ax + b = 0 and a quadratic equation of the type ax² + bx + c = 0, as well as methods to sum up a geometric series. Besides, he noted the difference between algebra and arithmetic and so was the first mathematician to treat them as two separate branches of mathematics.

Brahmagupta’s *Karanakhandakhadyaka* is a hand-book on astronomical calculations. In this he effectively used algebra for the first time in calculations. But Brahmagupta always was careful not to anger the priests. His wives were orthodox, in keeping with the beliefs held during those times, and he criticized Aryabhata, who said the earth was not stationary. But he believed that the earth was round.

About gravity he said: “Bodies fall towards the earth as it is in the nature of the earth to attract bodies, just as it is in the nature of water to flow.