{
  "id": "LO3",
  "folder": "03_LO3_Equality_of_Complex_Numbers",
  "zip": "LO3_Equality_of_Complex_Numbers_SLS_xAPI.zip",
  "title": "Complex Equality Component Balance",
  "objective": "Equality of complex numbers",
  "kind": "equality",
  "accent": "#a64b00",
  "stages": [
    {
      "id": "LO3-S1",
      "level": "Match components",
      "prompt": "When does a + bi = c + di?",
      "instruction": "Choose the complete condition.",
      "response": {
        "type": "choice",
        "options": [
          "a = c only",
          "b = d only",
          "a = c and b = d",
          "a + b = c + d"
        ],
        "answer": "a = c and b = d"
      },
      "model": {
        "type": "components",
        "expression": "a + bi = c + di",
        "real": "a ↔ c",
        "imag": "b ↔ d"
      },
      "hint": [
        "Equate the real parts and equate the imaginary parts.",
        "Match a with c and b with d."
      ],
      "worked": [
        "Equality means the two Argand points coincide.",
        "Their horizontal coordinates must match: a = c.",
        "Their vertical coordinates must match: b = d."
      ],
      "misconception": "Using only a total or matching just one component."
    },
    {
      "id": "LO3-S2",
      "level": "Use conjugacy",
      "prompt": "If z = z*, what can you conclude?",
      "instruction": "Compare the imaginary components.",
      "response": {
        "type": "choice",
        "options": [
          "z is real",
          "z is purely imaginary and nonzero",
          "z = i",
          "Nothing"
        ],
        "answer": "z is real"
      },
      "model": {
        "type": "mirror",
        "z": {
          "re": 2,
          "im": 2
        },
        "showConjugate": true
      },
      "hint": [
        "Write z = a + bi and z* = a − bi.",
        "Equality forces b = −b."
      ],
      "worked": [
        "a + bi = a − bi requires b = −b.",
        "Hence 2b = 0, so b = 0.",
        "Therefore z lies on the real axis."
      ],
      "misconception": "Thinking conjugation changes the real component."
    },
    {
      "id": "LO3-S3",
      "level": "Recover coefficients",
      "prompt": "1 + 2i is a root of 2z² + pz + q = 0, where p,q are real. Find p and q.",
      "instruction": "Enter p as the real field and q as the imaginary-labelled second field.",
      "response": {
        "type": "complex",
        "labels": [
          "p",
          "q"
        ],
        "answer": {
          "re": -4,
          "im": 10
        }
      },
      "model": {
        "type": "roots",
        "roots": [
          {
            "re": 1,
            "im": 2
          },
          {
            "re": 1,
            "im": -2
          }
        ],
        "labels": [
          "given root",
          "required conjugate"
        ]
      },
      "hint": [
        "A real-coefficient polynomial also has root 1 − 2i.",
        "Use sum = −p/2 and product = q/2."
      ],
      "worked": [
        "The root pair is 1 ± 2i.",
        "Their sum is 2, so −p/2 = 2 and p = −4.",
        "Their product is 5, so q/2 = 5 and q = 10."
      ],
      "misconception": "Using only the given root and ignoring the real-coefficient conjugate pair."
    },
    {
      "id": "LO3-S4",
      "level": "Square roots",
      "prompt": "Solve z² = 3 + 4i.",
      "instruction": "Choose the exact pair of values.",
      "response": {
        "type": "choice",
        "options": [
          "z = ±(2 + i)",
          "z = ±(2 − i)",
          "z = 3 ± 4i",
          "z = ±(1 + 2i)"
        ],
        "answer": "z = ±(2 + i)"
      },
      "model": {
        "type": "components",
        "expression": "(a + bi)² = 3 + 4i",
        "real": "a² − b² = 3",
        "imag": "2ab = 4"
      },
      "hint": [
        "Equate real and imaginary components after squaring.",
        "Test small integers satisfying 2ab = 4."
      ],
      "worked": [
        "(2 + i)² = 4 + 4i + i² = 3 + 4i.",
        "A square equation has the opposite root as well.",
        "Thus z = ±(2 + i)."
      ],
      "misconception": "Taking square roots of the real and imaginary parts separately."
    },
    {
      "id": "LO3-S5",
      "level": "Cube roots",
      "prompt": "Which set contains all roots of z³ = −8?",
      "instruction": "Use modulus and equally spaced arguments.",
      "response": {
        "type": "choice",
        "options": [
          "{−2, 1 + √3i, 1 − √3i}",
          "{−2, 2i, −2i}",
          "{2, −1 + √3i, −1 − √3i}",
          "{−8, 8i, −8i}"
        ],
        "answer": "{−2, 1 + √3i, 1 − √3i}"
      },
      "model": {
        "type": "roots",
        "radius": 2,
        "angles": [
          60,
          180,
          300
        ]
      },
      "hint": [
        "The cube-root modulus is 2.",
        "The arguments differ by 120°."
      ],
      "worked": [
        "Write −8 with argument π and modulus 8.",
        "Cube-root arguments are π/3, π, and 5π/3.",
        "The roots are 1 + √3i, −2, and 1 − √3i."
      ],
      "misconception": "Returning only the real cube root or spacing roots by 180°."
    },
    {
      "id": "LO3-S6",
      "level": "Solve a system",
      "prompt": "Solve z + iw − 1 + i = 0 and 3z + 2w* − 4i = 0.",
      "instruction": "Enter z and w as four components.",
      "response": {
        "type": "complexPair",
        "answer": {
          "z": {
            "re": 2,
            "im": 2
          },
          "w": {
            "re": -3,
            "im": 1
          }
        }
      },
      "model": {
        "type": "components",
        "expression": "z=a+bi, w=c+di",
        "real": "a−d=1; 3a+2c=0",
        "imag": "b+c=−1; 3b−2d=4"
      },
      "hint": [
        "Expand iw = −d + ci and w* = c − di.",
        "Equate real and imaginary components in each equation."
      ],
      "worked": [
        "The four real equations are a−d=1, b+c=−1, 3a+2c=0, and 3b−2d=4.",
        "Solving gives a=2, b=2, c=−3, d=1.",
        "Therefore z=2+2i and w=−3+i."
      ],
      "misconception": "Conjugating w without changing the sign of its imaginary component, or expanding iw incorrectly."
    }
  ]
}