# Zero

**Type:** Concept  
**Domain:** Number Theory · Indian Mathematics  
**Codex URL:** /mathematics/number-theory/zero/  
**Entry status:** Live — v1.0 (2026-05-27)

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## Curious Level

Zero plays two distinct roles: as a **placeholder** (in 304, distinguishing it from 34) and as a **number** (that can be calculated with). These were recognised at different times.

**Babylonian placeholder (c. 300 BCE):** Two-wedge symbol for empty sexagesimal position. Not used at end of numbers; not a number in calculations.

**Mayan zero (c. 350 CE):** Shell glyph in base-20 system. Used at end of numbers (improving on Babylon). No arithmetic treatment.

**Indian zero — śūnya:** Brahmagupta (628 CE), *Brāhmasphuṭasiddhānta* Ch. 18 — first systematic arithmetic rules: a + 0 = a, a × 0 = 0, 0 − a = −a. His rule 0 ÷ 0 = 0 was incorrect.

**Oldest written zero symbol:** Bakhshali manuscript, oldest folia carbon-dated 224–383 CE (Bodleian Library, 2017).

**Word etymology:** Sanskrit *śūnya* (void) → Arabic *ṣifr* → Medieval Latin *zephirum* → Italian *zero*.

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## Exploring Level

### Brahmagupta's rules (628 CE)

- a + 0 = a ✓
- a − 0 = a ✓
- a × 0 = 0 ✓
- 0 − a = −a ✓ (also introduces negative numbers)
- 0 ÷ 0 = 0 ✗ (incorrect — division by zero is undefined)

### Historical progression in India

- Āryabhaṭa (499 CE): decimal place-value system requires zero implicitly
- Brahmagupta (628 CE): zero as number with arithmetic rules
- Mahāvīra (850 CE): a/0 leaves a unchanged (incorrect)
- Bhāskara II (1150 CE): a/0 = ∞ (*khahara*) — precursor to limit concept

### Bakhshali manuscript

Discovered 1881. Contains hollow dot for zero. Bodleian carbon dating (2017) dates oldest folia to 224–383 CE. A compilation from multiple periods; scholarly debate on whether all folia share the same mathematical tradition.

### Transmission to Europe

Al-Khwārizmī (c. 820 CE) → Arabic numerals including zero → Fibonacci, *Liber Abaci* (1202 CE) → Europe.

### Modern algebraic role

Zero is the **additive identity** in any ring, group, or field: a + 0 = 0 + a = a. Also: 0 · a = 0 for all a (provable from distributive law). Zero is neither positive nor negative.

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## Deep Dive

### Why division by zero is undefined

In any non-trivial ring, 0 · a = 0 for all a (proof: 0 · a = (0+0) · a = 0 · a + 0 · a; cancel 0 · a). So no element x satisfies 0 · x = 1 (since 0 ≠ 1). Division by zero is undefined in all fields and integral domains.

### Brahmagupta's text (Ch. 18, verses 18.29–18.35)

Sanskrit verses giving arithmetic of positive, negative, and zero quantities. The term *kha* (sky) used for zero. Available in Colebrooke 1817 translation (primary source) and Plofker 2009 analysis.

### Bakhshali dating controversy

Radiocarbon dating (Bodleian, 2017) assigns dates 224–383 CE, 680–779 CE, 885–993 CE to different folia. If oldest folia date is correct, this predates other written zero symbols. Historians note the manuscript is a compilation; the mathematical tradition may not all be as old as the oldest pages.

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## Sources

### Tier 1
- Plofker, K. (2009). *Mathematics in India*. Princeton University Press. pp. 150–156.
- Colebrooke, H.T. (1817). *Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara*. John Murray, London. [Primary source translation]

### Tier 2
- Kaplan, R. (1999). *The Nothing That Is: A Natural History of Zero*. Oxford University Press.
- Neugebauer, O. (1969). *The Exact Sciences in Antiquity*. 2nd ed. Dover.

### Tier 3
- Joseph, G.G. (2011). *The Crest of the Peacock*. 3rd ed. Princeton University Press. pp. 214–223.

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