Brahmagupta writes the Brahmasphutasiddhanta

Brahmagupta, the son of Jishnugupta, states in the treatise that he composed it at the age of thirty, when 550 years of the Shaka era had passed, in the reign of King Vyaghramukha of the Chapa dynasty, whose capital was Bhillamala; the later commentator Prithudaka calls him Bhillamalacharya, ‘the teacher from Bhillamala’. The title means ‘the correctly established doctrine of Brahma’: the work is a revision of the astronomy of the Brahmapaksha school, with much new material. It has 24 chapters with 1,008 verses in the arya metre (25 chapters if the closing versified tables are counted separately). Most of it deals with astronomy — mean and true planetary longitudes, eclipses, risings and settings, conjunctions — and it criticises earlier astronomers, above all Aryabhata. The mathematical chapters cover arithmetic, series and geometry, including the formula for the area of a cyclic quadrilateral that bears Brahmagupta’s name, and algebra: linear and quadratic equations and indeterminate equations of the second degree of the kind later called Pell’s equation.

The treatise gives rules for adding, subtracting, multiplying and dividing with zero and with negative numbers, called ‘debts’ as opposed to positive ‘fortunes’: a debt minus zero is a debt, the product of two debts is a fortune. Most of these rules agree with modern arithmetic, except that Brahmagupta took zero divided by zero to be zero. Later in life he moved to Ujjain, and in 665 he wrote the Khandakhadyaka, a practical manual of astronomy. Under the Abbasid caliph al-Mansur (754–775) an embassy from Sindh brought an astronomer named Kanaka to Baghdad with Indian texts, including Brahmagupta’s; al-Fazari translated them into Arabic as the Sindhind, and through such translations and the works of al-Khwarizmi Indian numerals and methods of calculation spread through the Islamic world and later to Europe.