{
  "id": "LO1",
  "folder": "01_LO1_Number_System_Extension",
  "zip": "LO1_Number_System_Extension_SLS_xAPI.zip",
  "title": "Complex Numbers: Extending the Number System",
  "objective": "Extension of the number system from real numbers to complex numbers",
  "kind": "number_system",
  "accent": "#6d4aff",
  "stages": [
    {
      "id": "LO1-S1",
      "level": "Recognise",
      "prompt": "Is every integer also a complex number?",
      "instruction": "Choose the statement that gives the strongest reason.",
      "response": {
        "type": "choice",
        "options": [
          "No; integers have no imaginary part",
          "Yes; n = n + 0i",
          "Only positive integers are complex",
          "Only zero is complex"
        ],
        "answer": "Yes; n = n + 0i"
      },
      "model": {
        "type": "setMap",
        "focus": "−7",
        "memberships": [
          "Integer",
          "Real",
          "Complex"
        ]
      },
      "hint": [
        "Use the form a + bi.",
        "What value of b leaves an integer unchanged?"
      ],
      "worked": [
        "Write −7 as −7 + 0i.",
        "Here a = −7 and b = 0 are real.",
        "Therefore every integer lies inside the complex-number system."
      ],
      "misconception": "Treating a zero imaginary component as disqualifying a number from being complex."
    },
    {
      "id": "LO1-S2",
      "level": "Distinguish",
      "prompt": "How should zero be classified?",
      "instruction": "Use the convention that a purely imaginary number has the form bi with real b.",
      "response": {
        "type": "choice",
        "options": [
          "Real only",
          "Purely imaginary only",
          "Both real and purely imaginary",
          "Neither"
        ],
        "answer": "Both real and purely imaginary"
      },
      "model": {
        "type": "setMap",
        "focus": "0 = 0 + 0i",
        "memberships": [
          "Real",
          "Imaginary axis",
          "Complex"
        ]
      },
      "hint": [
        "Zero can be written as 0 + 0i.",
        "It also has the form bi when b = 0."
      ],
      "worked": [
        "Its real component and imaginary component are both zero.",
        "So it is real, and under the stated convention it is also purely imaginary.",
        "Some texts exclude zero from 'purely imaginary'; naming the convention removes the ambiguity."
      ],
      "misconception": "Assuming classification sets cannot overlap, or ignoring a stated convention."
    },
    {
      "id": "LO1-S3",
      "level": "Simplify",
      "prompt": "What is the real part of 2i + 5i²?",
      "instruction": "Simplify before identifying Re(z).",
      "response": {
        "type": "choice",
        "options": [
          "0",
          "2",
          "−5",
          "5"
        ],
        "answer": "−5"
      },
      "model": {
        "type": "components",
        "expression": "2i + 5i²",
        "real": "?",
        "imag": "2"
      },
      "hint": [
        "Recall i² = −1.",
        "Replace 5i² before collecting components."
      ],
      "worked": [
        "2i + 5i² = 2i + 5(−1).",
        "In standard form this is −5 + 2i.",
        "The real part is −5."
      ],
      "misconception": "Reading the unsimplified expression and missing that i² contributes a real term."
    },
    {
      "id": "LO1-S4",
      "level": "Interpret notation",
      "prompt": "Which statement is correct for z = 3 + 4i?",
      "instruction": "Distinguish an imaginary part from an imaginary term.",
      "response": {
        "type": "choice",
        "options": [
          "Im(z) = 4i",
          "Im(z) = 4",
          "Re(z) = 3i",
          "z has no real part"
        ],
        "answer": "Im(z) = 4"
      },
      "model": {
        "type": "components",
        "expression": "z = 3 + 4i",
        "real": "3",
        "imag": "4"
      },
      "hint": [
        "Im(z) returns a real coefficient.",
        "The imaginary term is bi; the imaginary part is b."
      ],
      "worked": [
        "Match z = a + bi with 3 + 4i.",
        "This gives a = 3 and b = 4.",
        "Therefore Re(z) = 3 and Im(z) = 4."
      ],
      "misconception": "Confusing the imaginary part b with the imaginary term bi."
    },
    {
      "id": "LO1-S5",
      "level": "Classify",
      "prompt": "Where does −3i belong?",
      "instruction": "Select the most complete classification.",
      "response": {
        "type": "choice",
        "options": [
          "Imaginary axis and complex numbers",
          "Real numbers only",
          "Integers only",
          "Outside the complex numbers"
        ],
        "answer": "Imaginary axis and complex numbers"
      },
      "model": {
        "type": "setMap",
        "focus": "−3i = 0 − 3i",
        "memberships": [
          "Imaginary axis",
          "Complex"
        ]
      },
      "hint": [
        "Identify a and b in a + bi.",
        "Here a = 0 and b = −3."
      ],
      "worked": [
        "−3i has real component 0.",
        "Its point lies on the imaginary axis below the origin.",
        "Every number a + bi with real a,b is complex."
      ],
      "misconception": "Thinking 'not real' means 'not complex'."
    },
    {
      "id": "LO1-S6",
      "level": "Explain",
      "prompt": "A number z = a + bi is both real and purely imaginary. What must z be?",
      "instruction": "Use both component conditions at once.",
      "response": {
        "type": "choice",
        "options": [
          "1",
          "i",
          "0",
          "Any real number"
        ],
        "answer": "0"
      },
      "model": {
        "type": "components",
        "expression": "z = a + bi",
        "real": "a = ?",
        "imag": "b = ?"
      },
      "hint": [
        "Real requires b = 0.",
        "Purely imaginary requires a = 0 under the stated form."
      ],
      "worked": [
        "Being real forces the imaginary component b to be zero.",
        "Being purely imaginary forces the real component a to be zero.",
        "Thus z = 0 + 0i = 0."
      ],
      "misconception": "Applying only one of two simultaneous classification conditions."
    }
  ]
}