# Brahmagupta

**Type:** Person  
**Domain:** Indian Mathematics  
**Born:** 598 CE, Bhillamāla (Bhinmal), Rajasthan, India  
**Died:** c. 668 CE  
**Codex URL:** /mathematics/indian-mathematics/brahmagupta/  
**Entry status:** Live — v1.0 (2026-05-27)

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## Summary

Indian mathematician and astronomer. Head of the astronomical observatory at Ujjain. Author of the *Brāhmasphuṭasiddhānta* (628 CE) — the first text to give systematic rules for arithmetic with zero and negative numbers.

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## Key Contributions

- **Zero as a number:** a + 0 = a, a × 0 = 0 (BSS Ch. 18, 628 CE)
- **Negative numbers:** rules for all combinations of positive, negative, zero
- **Brahmagupta's formula:** Area = √[(s−a)(s−b)(s−c)(s−d)] for cyclic quadrilateral
- **Brahmagupta–Fibonacci identity:** (a²+b²)(c²+d²) = (ac−bd)² + (ad+bc)²
- **Pell's equation:** study of Nx² + 1 = y²; chakravāla method

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## Historical significance

The *Brāhmasphuṭasiddhānta* was translated into Arabic in Baghdad c. 771 CE, directly influencing Al-Khwārizmī (c. 820 CE). Transmission chain: Brahmagupta (628 CE) → Baghdad (771 CE) → Al-Khwārizmī (820 CE) → Fibonacci (1202 CE) → European arithmetic. The zero you write today traces its arithmetic rules to Brahmagupta.

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## Sources

### Tier 1
- Plofker, K. (2009). *Mathematics in India*. Princeton University Press. pp. 150–181.
- Colebrooke, H.T. (1817). *Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara*. John Murray.

### Tier 2
- O'Connor, J.J. and Robertson, E.F. (2000). Brahmagupta. MacTutor. https://mathshistory.st-andrews.ac.uk/Biographies/Brahmagupta/

### Tier 3
- Joseph, G.G. (2011). *The Crest of the Peacock*. 3rd ed. Princeton University Press. pp. 223–248.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*