# Brāhmasphuṭasiddhānta

**Type:** Work  
**Domain:** Indian Mathematics  
**Composed:** 628 CE  
**Author:** Brahmagupta (598 – c. 668 CE)  
**Language:** Sanskrit verse, 25 chapters  
**Codex URL:** /mathematics/indian-mathematics/brahmasphutasiddhanta/  
**Entry status:** Live — v1.0 (2026-05-27)

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## Summary

The first text to give systematic rules for arithmetic with zero and negative numbers (Chapter 18). Contains Brahmagupta's formula for cyclic quadrilateral area (Chapter 12) and methods for indeterminate equations. Translated into Arabic c. 771 CE; influenced Al-Khwārizmī (c. 820 CE).

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## Key Content

**Chapter 18 — Zero and negative arithmetic:**
- a + 0 = a; a × 0 = 0 ✓
- Debt × debt = fortune (negative × negative = positive) ✓
- Zero ÷ zero = zero ✗ (known error)

**Chapter 12 — Brahmagupta's formula:**  
Area = √[(s−a)(s−b)(s−c)(s−d)] for cyclic quadrilateral with sides a,b,c,d and semi-perimeter s.

**Brahmagupta–Fibonacci identity:**  
(a²+b²)(c²+d²) = (ac−bd)² + (ad+bc)²

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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*. John Murray. [Chs. 12, 18]

### Tier 2
- Dvivedin, S. (1902). *Brāhmasphuṭasiddhānta*. Benares. [Sanskrit critical edition]

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

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