{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-complex-number",
  "semantic_family_id": "sf-complex-number",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "複數"
  },
  "summary": {
    "zh-Hant": "複數以實數有序對(a,b)表示a+bi，並以i²=-1定義加減乘法。"
  },
  "statement": {
    "zh-Hant": "複數z=a+bi由實部a與虛部係數b唯一決定，可視為複平面有序對(a,b)。加減逐分量進行，乘法由分配律與i²=-1導出。"
  },
  "formula": {
    "tex": "z=C(a,b)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>z</mi><mo>=</mo><mrow><mi>C</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "symbol",
        "name": "z",
        "glyph": "z"
      },
      "rhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "C",
          "glyph": "C"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "a",
            "glyph": "a"
          },
          {
            "type": "symbol",
            "name": "b",
            "glyph": "b"
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "bdb5fce193a154be64477d71475805a02353236a8544fb406b32addd2c493938"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫限制整數實部與虛部係數，答案以有序對表示。"
    },
    {
      "id": "a2",
      "zh-Hant": "有序對的分量次序是語義資料；不由圖形位置或未標刻度補值，本輪不做複數除法。"
    }
  ],
  "symbols": [
    {
      "token": "z",
      "role": "complex_number",
      "role_zh": "複數",
      "scope": "representative_formula"
    },
    {
      "token": "C",
      "role": "complex_pair_constructor",
      "role_zh": "複數有序對構造",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "實部與虛部係數為實數。"
    },
    {
      "id": "mko-coordinate-plane",
      "title_zh": "座標平面",
      "reason_zh": "複平面以有序對表示。"
    },
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "四則運算",
      "reason_zh": "複數加減乘由實數運算組合。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "a+bi與(c+di)相等當且僅當a=c且b=d。",
      "乘法的實部為ac-bd，虛部係數為ad+bc，負號來自i²=-1。",
      "實數r嵌入為(r,0)；虛部不為零的複數不能和任何實數候選相等。"
    ],
    "common_misconception_zh": "交換實部與虛部，或在乘法時忘記i²=-1。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪只用整數有序對的加減乘與實數嵌入；未涵蓋除法、極式或複分析。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入共軛、模、除法與極式。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-complex-number.md 整理；2026-10-13 脈衝 R03（使用者手動接續 2026-10-22 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}