{
  "id": "zero",
  "domain": "number-theory",
  "also_in_domain": "indian-mathematics",
  "type": "Concept",
  "title": "Zero",
  "slug": "zero",
  "url": "/mathematics/number-theory/zero/",
  "summary": "The additive identity — the unique number 0 such that a + 0 = a. Functions as both placeholder in positional notation and a number in its own right. First systematic arithmetic rules by Brahmagupta, 628 CE.",
  "formal_definition": "In any ring (R, +, ·), the additive identity element 0 such that a + 0 = 0 + a = a for all a ∈ R. Also: 0 · a = 0 for all a.",
  "civilisational_record": [
    { "tradition": "Babylonian", "role": "placeholder only", "date_ce": -300, "evidence": "Two-wedge symbol in sexagesimal positional notation" },
    { "tradition": "Mayan", "role": "placeholder (including end of number)", "date_ce": 350, "evidence": "Shell glyph in base-20 calendar system" },
    { "tradition": "Indian", "role": "number with arithmetic rules", "date_ce": 628, "evidence": "Brahmagupta, Brāhmasphuṭasiddhānta Ch. 18" },
    { "tradition": "Indian (earliest symbol)", "role": "placeholder dot", "date_ce_range": "224–383", "evidence": "Bakhshali manuscript (oldest folia, carbon dated 2017)" }
  ],
  "added_to_codex": "2026-05-27",
  "last_verified": "2026-05-27",
  "sources": [
    { "tier": 1, "citation": "Plofker, K. (2009). Mathematics in India. Princeton University Press.", "pages": "150–156" },
    { "tier": 1, "citation": "Colebrooke, H.T. (1817). Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara. John Murray, London." },
    { "tier": 2, "citation": "Kaplan, R. (1999). The Nothing That Is: A Natural History of Zero. Oxford University Press." },
    { "tier": 2, "citation": "Neugebauer, O. (1969). The Exact Sciences in Antiquity. 2nd ed. Dover." },
    { "tier": 3, "citation": "Joseph, G.G. (2011). The Crest of the Peacock. 3rd ed. Princeton University Press.", "pages": "214–223" }
  ],
  "relationships": [
    { "type": "algebraic_role", "target": "group", "label": "Additive identity in groups, rings, fields" },
    { "type": "related", "target": "natural-number", "label": "Natural Number" },
    { "type": "part_of_tradition", "target": "indian-mathematics-overview", "label": "Indian Mathematics" },
    { "type": "formalised_by", "target": "brahmagupta", "label": "Brahmagupta (628 CE)" }
  ],
  "entry_status": "live",
  "reading_levels_available": ["curious", "exploring", "deep_dive"],
  "completeness": 1.0,
  "codex_version": "1.1"
}