Brahmagupta

Brahmagupta worked in Bhillamāla (modern Bhinmal, in Rajasthan) and completed his major work, the Brāhmasphuṭasiddhānta, in 628 CE. It is a comprehensive astronomical treatise, but its most consequential chapter, for the history of mathematics generally, is the one devoted to arithmetic rules — because it is there that zero stops being merely a placeholder notation and becomes a number with its own defined behaviour.

Brahmagupta gives explicit rules for adding, subtracting, and multiplying with zero, and — using the language of 'fortunes' and 'debts' — sets out rules for positive and negative numbers that correspond closely to modern sign conventions (a debt subtracted from zero becomes a fortune; two debts added remain a debt, and so on). He also attempts a rule for division by zero, which by his own later account does not fully hold up mathematically — a genuine unresolved edge case rather than a claimed solution, and one that mathematics would not properly resolve until much later analytical developments.

Brahmagupta also solves general linear and certain quadratic indeterminate equations — problems of the type later called Pell's equation in the West, worked with far more limited general methods until centuries afterward — and develops trigonometric techniques for planetary position calculation that extend the astronomical tradition inherited from Āryabhaṭa and Lagadha before him.

Brahmagupta's arithmetic of zero and negative numbers travelled westward through translation into Arabic mathematical scholarship — al-Khwārizmī and other Baghdad-era mathematicians engaged with Indian numerical methods directly — before reaching medieval Europe centuries later via the Islamic world. The zero that underlies the entire modern decimal and algebraic system has a documented lineage running back through this transmission chain to Brahmagupta's 7th-century formal treatment.