Indian mathematician and astronomer (598 – c. 668 CE), born in Bhillamāla (modern Bhinmal, Rajasthan). Author of the Brāhmasphuṭasiddhānta (628 CE) — the first text to treat zero and negative numbers as full participants in arithmetic. His formula for the area of a cyclic quadrilateral remains a landmark of geometry.
Brahmagupta was born in 598 CE in Bhillamāla, a city in what is now Rajasthan, India. He became the head of the astronomical observatory at Ujjain — at that time, one of the most important centres of learning in the world. He wrote his most important work, the Brāhmasphuṭasiddhānta, in 628 CE, when he was 30 years old.
The Brāhmasphuṭasiddhānta was translated into Arabic in Baghdad around 771 CE — one of the first Indian scientific texts to reach the Islamic world. This translation directly influenced Al-Khwārizmī, whose work would later bring Indian numerals (including zero) to Europe. When you write the number 0 today, you are using a concept whose rules were first written down by Brahmagupta in 628 CE.
| Born | 598 CE, Bhillamāla (Bhinmal), Rajasthan, India |
| Died | c. 668 CE |
| Position | Head of astronomical observatory, Ujjain |
| Major work | Brāhmasphuṭasiddhānta (628 CE) — "The Correctly Established Doctrine of Brahma" |
| Second work | Khaṇḍakhādyaka (665 CE) — astronomical handbook |
Brahmagupta stated these rules for "fortunes" (positive), "debts" (negative), and "cipher" (zero):
His one error: he stated that zero divided by zero is zero. The correct answer is that division by zero is undefined.
For a cyclic quadrilateral (a four-sided polygon inscribed in a circle) with consecutive sides a, b, c, d and semi-perimeter s = (a+b+c+d)/2:
This generalises Heron's formula for triangles (which is the case when d = 0). It was one of the most elegant geometric results of its era.
This shows that the product of two numbers, each expressible as a sum of two squares, is itself a sum of two squares. It was rediscovered independently by Fibonacci (1202 CE) and plays a role in Fermat's theorem on sums of two squares.
Brahmagupta studied equations of the form Nx² + 1 = y² (now called Pell's equation, though Pell had little to do with it). He developed the chakravāla (cyclic method) approach, later completed by Bhāskara II (1150 CE), to find integer solutions. This predated European work on the problem by centuries.
The BSS survives in multiple Sanskrit manuscripts. The standard modern edition and translation is by Plofker (2009), building on earlier work by Colebrooke (1817) and Dvivedin (1902). The text has 25 chapters. Chapter 18 (Kuṭṭākadhyāya — "pulveriser") contains the arithmetic of zero and negatives in verses 18.29–18.35, and rules for handling indeterminate equations.
The Abbasid Caliph Al-Manṣūr sent scholars to India to bring back scientific texts. The BSS was translated into Arabic as the Sindhind. This translation reached Al-Khwārizmī, whose Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala (c. 820 CE) — source of the words "algebra" and "algorithm" — incorporates Indian place-value arithmetic including zero. The transmission chain: Brahmagupta (628 CE) → Baghdad translation (771 CE) → Al-Khwārizmī (820 CE) → Fibonacci's Liber Abaci (1202 CE) → European arithmetic.
Brahmagupta explicitly and sharply criticised Āryabhaṭa's claim that Earth rotates on its axis (BSS Ch. 11). This is notable: two of the greatest Indian mathematicians disagreed on heliocentric models. Brahmagupta's geocentric critique held sway in Indian astronomy for several generations, while Āryabhaṭa's rotation claim was vindicated by modern science.
Earlier Indian texts (Āryabhaṭa, 499 CE) used zero implicitly as a placeholder. Brahmagupta's innovation was treating it arithmetically — as something you can add, subtract, and multiply, with explicit rules for every combination of positive, negative, and zero. This is conceptually distinct from a placeholder and represents a genuine mathematical advance. The only predecessor arguably close is the Bakhshali manuscript's hollow dot (possibly 3rd–4th century CE), but no arithmetic rules for zero appear in that text.