<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: transparent;
      color: #1e293b;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      padding: 12px;
      user-select: none;
      min-height: 100%;
    }

    .diagram-wrap {
      width: 100%;
      max-width: 360px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 14px;
    }

    .math-var {
      font-family: "Cambria Math", "Times New Roman", Georgia, serif;
      font-style: italic;
    }

    /* Cycle Diagram Area */
    .cycle-area {
      position: relative;
      width: 280px;
      height: 280px;
    }

    .cycle-svg {
      position: absolute;
      top: 0;
      left: 0;
      width: 100%;
      height: 100%;
      overflow: visible;
    }

    .node {
      position: absolute;
      width: 80px;
      height: 80px;
      border-radius: 50%;
      background: #ffffff;
      border: 2.5px solid #cbd5e1;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      cursor: pointer;
      transform: translate(-50%, -50%);
      transition: border-color 0.25s ease, background-color 0.25s ease, box-shadow 0.25s ease;
      box-shadow: 0 2px 8px rgba(0, 0, 0, 0.06);
      z-index: 2;
    }

    .node:hover {
      border-color: #38bdf8;
      box-shadow: 0 4px 12px rgba(2, 132, 199, 0.18);
    }

    .node.active {
      border-color: #0284c7;
      background: #f0f9ff;
      box-shadow: 0 0 0 3px rgba(14, 165, 233, 0.3), 0 4px 12px rgba(2, 132, 199, 0.2);
    }

    .node .power-val {
      font-size: 17px;
      font-weight: 700;
      color: #0f172a;
      line-height: 1.15;
    }

    .node.active .power-val {
      color: #0369a1;
    }

    .node .rem-tag {
      font-size: 11px;
      font-weight: 600;
      color: #64748b;
      margin-top: 3px;
      background: #f1f5f9;
      padding: 1px 6px;
      border-radius: 10px;
    }

    .node.active .rem-tag {
      color: #0284c7;
      background: #e0f2fe;
    }

    /* Node Positions (center: 140, 140; radius: 95) */
    .node-0 { top: 45px; left: 140px; }   /* Top: Remainder 0 */
    .node-1 { top: 140px; left: 235px; }  /* Right: Remainder 1 */
    .node-2 { top: 235px; left: 140px; }  /* Bottom: Remainder 2 */
    .node-3 { top: 140px; left: 45px; }   /* Left: Remainder 3 */

    .center-label {
      position: absolute;
      top: 50%;
      left: 50%;
      transform: translate(-50%, -50%);
      font-size: 12px;
      font-weight: 800;
      color: #94a3b8;
      text-transform: uppercase;
      letter-spacing: 0.6px;
      text-align: center;
      line-height: 1.3;
      pointer-events: none;
    }

    /* Formula & Interactive Card */
    .formula-card {
      width: 100%;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 10px 14px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 8px;
    }

    .formula-main {
      font-size: 14px;
      font-weight: 600;
      color: #334155;
    }

    .formula-interactive {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 10px;
      width: 100%;
    }

    .stepper-btn {
      width: 32px;
      height: 32px;
      display: flex;
      align-items: center;
      justify-content: center;
      background: #ffffff;
      border: 1px solid #cbd5e1;
      color: #0f172a;
      font-weight: 700;
      font-size: 16px;
      border-radius: 8px;
      cursor: pointer;
      box-shadow: 0 1px 3px rgba(0,0,0,0.05);
      transition: background 0.15s, border-color 0.15s;
    }

    .stepper-btn:hover {
      background: #f1f5f9;
      border-color: #94a3b8;
    }

    .stepper-btn:active {
      background: #e2e8f0;
    }

    .calc-display {
      font-size: 15px;
      font-weight: 600;
      color: #0f172a;
      min-width: 170px;
      text-align: center;
      padding: 4px 8px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
    }

    .highlight {
      color: #0284c7;
      font-weight: 700;
    }

    .calc-note {
      font-size: 11px;
      font-weight: 500;
      color: #64748b;
    }
  </style>
</head>
<body>
  <div class="diagram-wrap">
    <div class="cycle-area">
      <svg class="cycle-svg" viewBox="0 0 280 280">
        <defs>
          <marker id="arrow" viewBox="0 0 10 10" refX="6" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse">
            <path d="M 0 1.5 L 8 5 L 0 8.5 z" fill="#0284c7" />
          </marker>
        </defs>

        
        
        <path d="M 175 54 A 95 95 0 0 1 226 105" fill="none" stroke="#38bdf8" stroke-width="2.5" marker-end="url(#arrow)" stroke-dasharray="3,1" />
        <text x="216" y="70" fill="#0284c7" font-size="12" font-weight="700" font-family="system-ui" text-anchor="middle">× <tspan class="math-var">i</tspan></text>

        
        <path d="M 226 175 A 95 95 0 0 1 175 226" fill="none" stroke="#38bdf8" stroke-width="2.5" marker-end="url(#arrow)" stroke-dasharray="3,1" />
        <text x="216" y="218" fill="#0284c7" font-size="12" font-weight="700" font-family="system-ui" text-anchor="middle">× <tspan class="math-var">i</tspan></text>

        
        <path d="M 105 226 A 95 95 0 0 1 54 175" fill="none" stroke="#38bdf8" stroke-width="2.5" marker-end="url(#arrow)" stroke-dasharray="3,1" />
        <text x="64" y="218" fill="#0284c7" font-size="12" font-weight="700" font-family="system-ui" text-anchor="middle">× <tspan class="math-var">i</tspan></text>

        
        <path d="M 54 105 A 95 95 0 0 1 105 54" fill="none" stroke="#38bdf8" stroke-width="2.5" marker-end="url(#arrow)" stroke-dasharray="3,1" />
        <text x="64" y="70" fill="#0284c7" font-size="12" font-weight="700" font-family="system-ui" text-anchor="middle">× <tspan class="math-var">i</tspan></text>
      </svg>

      <div class="center-label">mod 4<br>cycle</div>

      
      <div class="node node-0" id="node-0" onclick="setStep(0)">
        <div class="power-val"><span class="math-var">i</span><sup>4</sup> = 1</div>
        <div class="rem-tag">rem 0</div>
      </div>

      
      <div class="node node-1" id="node-1" onclick="setStep(1)">
        <div class="power-val"><span class="math-var">i</span><sup>1</sup> = <span class="math-var">i</span></div>
        <div class="rem-tag">rem 1</div>
      </div>

      
      <div class="node node-2" id="node-2" onclick="setStep(2)">
        <div class="power-val"><span class="math-var">i</span><sup>2</sup> = −1</div>
        <div class="rem-tag">rem 2</div>
      </div>

      
      <div class="node node-3" id="node-3" onclick="setStep(3)">
        <div class="power-val"><span class="math-var">i</span><sup>3</sup> = −<span class="math-var">i</span></div>
        <div class="rem-tag">rem 3</div>
      </div>
    </div>

    
    <div class="formula-card">
      <div class="formula-main">
        General Rule: <span class="math-var">i<sup>n</sup></span> = <span class="math-var">i</span><sup><span class="math-var">n</span> mod 4</sup>
      </div>
      
      <div class="formula-interactive">
        <button class="stepper-btn" onclick="adjustN(-1)" title="Decrease power" aria-label="Decrease power">−</button>
        <div class="calc-display" id="calc-display"></div>
        <button class="stepper-btn" onclick="adjustN(1)" title="Increase power" aria-label="Increase power">+</button>
      </div>

      <div class="calc-note" id="calc-note"></div>
    </div>
  </div>

  <script>
    let currentN = 1;
    const values = ['1', 'i', '−1', '−i'];
    const powerExp = ['4 (or 0)', '1', '2', '3'];

    function updateDisplay() {
      const rem = ((currentN % 4) + 4) % 4;
      const resVal = values[rem];

      // Update interactive formula calculation
      const displayEl = document.getElementById('calc-display');
      if (rem === 0) {
        displayEl.innerHTML = `<span class="math-var">i</span><sup><span class="highlight">${currentN}</span></sup> = <span class="math-var">i</span><sup><span class="highlight">0</span></sup> = <b class="highlight">${resVal}</b>`;
      } else {
        displayEl.innerHTML = `<span class="math-var">i</span><sup><span class="highlight">${currentN}</span></sup> = <span class="math-var">i</span><sup><span class="highlight">${rem}</span></sup> = <b class="highlight">${resVal === 'i' || resVal === '−i' ? (resVal === 'i' ? '<span class="math-var">i</span>' : '−<span class="math-var">i</span>') : resVal}</b>`;
      }

      // Update division note
      const noteEl = document.getElementById('calc-note');
      noteEl.innerHTML = `${currentN} ÷ 4 = ${Math.floor(currentN / 4)} with <b>remainder ${rem}</b>`;

      // Highlight active node
      for (let i = 0; i < 4; i++) {
        const el = document.getElementById('node-' + i);
        if (i === rem) {
          el.classList.add('active');
        } else {
          el.classList.remove('active');
        }
      }
    }

    function adjustN(delta) {
      currentN += delta;
      if (currentN < 0) currentN = 20;
      if (currentN > 24) currentN = 0;
      updateDisplay();
    }

    function setStep(rem) {
      const base = Math.floor(currentN / 4) * 4;
      currentN = base + rem;
      updateDisplay();
    }

    // Initial render
    updateDisplay();
  </script>
</body>
</html>

Defining the Imaginary Unit and the Complex Number System

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      width: 100%;
      min-height: 100%;
      background: #ffffff;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      padding: 16px 12px;
      user-select: none;
    }

    .math-font {
      font-family: "Cambria Math", "STIX Two Math", "Times New Roman", Georgia, serif;
      font-style: italic;
    }

    /* Anatomy Area */
    .anatomy-section {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
      margin-top: 4px;
    }

    .callout-row {
      display: flex;
      justify-content: center;
      gap: 14px;
      width: 100%;
    }

    .badge-callout {
      padding: 6px 10px;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
      text-align: center;
      line-height: 1.3;
      border: 1.5px solid transparent;
      transition: opacity 0.25s, transform 0.25s;
    }

    .badge-real {
      background: #eff6ff;
      border-color: #2563eb;
      color: #1d4ed8;
    }

    .badge-imag {
      background: #faf5ff;
      border-color: #9333ea;
      color: #7e22ce;
    }

    .badge-unit {
      background: #f0fdf4;
      border-color: #16a34a;
      color: #15803d;
      padding: 4px 10px;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 600;
      display: inline-flex;
      align-items: center;
      gap: 4px;
    }

    /* Equation Display */
    .formula-box {
      display: flex;
      align-items: center;
      justify-content: center;
      font-size: 42px;
      line-height: 1;
      padding: 8px 16px;
      border-radius: 12px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      width: 100%;
      max-width: 320px;
      gap: 4px;
    }

    .part-z { color: #64748b; }
    .part-op { color: #94a3b8; margin: 0 4px; }
    
    .part-a {
      color: #2563eb;
      font-weight: 700;
      padding: 2px 6px;
      border-radius: 6px;
      transition: all 0.25s;
    }
    
    .part-b {
      color: #9333ea;
      font-weight: 700;
      padding: 2px 6px;
      border-radius: 6px;
      transition: all 0.25s;
    }

    .part-i {
      color: #16a34a;
      font-weight: 700;
      transition: all 0.25s;
    }

    .dimmed {
      opacity: 0.2;
    }

    .highlight-a {
      background: #dbeafe;
      box-shadow: 0 0 0 2px #2563eb;
    }

    .highlight-b {
      background: #f3e8ff;
      box-shadow: 0 0 0 2px #9333ea;
    }

    /* Classification Row */
    .classes-grid {
      width: 100%;
      max-width: 540px;
      display: grid;
      grid-template-columns: repeat(3, 1fr);
      gap: 8px;
      margin-top: 14px;
    }

    .class-card {
      background: #f8fafc;
      border: 1.5px solid #e2e8f0;
      border-radius: 10px;
      padding: 10px 6px;
      display: flex;
      flex-direction: column;
      align-items: center;
      text-align: center;
      gap: 4px;
      cursor: pointer;
      transition: all 0.2s ease;
    }

    .class-card:hover {
      border-color: #cbd5e1;
      background: #f1f5f9;
    }

    .class-card.active {
      background: #ffffff;
      border-color: #0284c7;
      box-shadow: 0 4px 12px rgba(2, 132, 199, 0.15);
      transform: translateY(-2px);
    }

    .cond-tag {
      font-size: 11px;
      font-weight: 700;
      padding: 2px 6px;
      border-radius: 4px;
      background: #e2e8f0;
      color: #475569;
    }

    .class-card.active .cond-tag {
      background: #e0f2fe;
      color: #0369a1;
    }

    .type-title {
      font-size: 13px;
      font-weight: 700;
      color: #1e2945;
      margin-top: 2px;
    }

    .example-val {
      font-size: 15px;
      color: #0f172a;
      font-weight: 600;
      margin-top: 2px;
    }

    /* Helper cues */
    .interactive-cue {
      font-size: 11px;
      color: #94a3b8;
      text-align: center;
      margin-top: 6px;
    }
  </style>
</head>
<body>

  
  <div class="anatomy-section">
    <div class="callout-row">
      <div class="badge-callout badge-real" id="badgeA">
        <div>Real Part</div>
        <span class="math-font">Re(z) = a ∈ ℝ</span>
      </div>
      <div class="badge-callout badge-imag" id="badgeB">
        <div>Imaginary Part</div>
        <span class="math-font">Im(z) = b ∈ ℝ</span>
      </div>
    </div>

    
    <div class="formula-box">
      <span class="part-z math-font">z</span>
      <span class="part-op">=</span>
      <span class="part-a math-font" id="elemA">a</span>
      <span class="part-op" id="elemOp">+</span>
      <span class="part-b math-font" id="elemB">b</span>
      <span class="part-i math-font" id="elemI">i</span>
    </div>

    
    <div class="badge-unit" id="badgeI">
      <span>Imaginary Unit:</span>
      <span class="math-font">i² = −1</span>
    </div>
  </div>

  
  <div style="width: 100%; display: flex; flex-direction: column; align-items: center;">
    <div class="classes-grid">
      
      <div class="class-card" onclick="selectType(0)" id="card-0">
        <span class="cond-tag math-font">b = 0</span>
        <div class="type-title">Real</div>
        <div class="example-val math-font">z = 7</div>
      </div>

      
      <div class="class-card" onclick="selectType(1)" id="card-1">
        <span class="cond-tag math-font">a = 0, b ≠ 0</span>
        <div class="type-title">Pure Imaginary</div>
        <div class="example-val math-font">z = −3i</div>
      </div>

      
      <div class="class-card active" onclick="selectType(2)" id="card-2">
        <span class="cond-tag math-font">a ≠ 0, b ≠ 0</span>
        <div class="type-title">Complex</div>
        <div class="example-val math-font">z = 4 + 5i</div>
      </div>
    </div>
    <div class="interactive-cue">Tap a category to inspect components</div>
  </div>

  <script>
    const elA = document.getElementById('elemA');
    const elOp = document.getElementById('elemOp');
    const elB = document.getElementById('elemB');
    const elI = document.getElementById('elemI');
    const bA = document.getElementById('badgeA');
    const bB = document.getElementById('badgeB');
    const bI = document.getElementById('badgeI');

    function selectType(index) {
      for (let i = 0; i < 3; i++) {
        document.getElementById(`card-${i}`).classList.toggle('active', i === index);
      }

      if (index === 0) {
        // Real: b = 0
        elA.classList.remove('dimmed');
        elA.classList.add('highlight-a');
        elOp.classList.add('dimmed');
        elB.classList.add('dimmed');
        elB.classList.remove('highlight-b');
        elI.classList.add('dimmed');
        bA.style.opacity = '1';
        bB.style.opacity = '0.25';
        bI.style.opacity = '0.25';
      } else if (index === 1) {
        // Pure Imaginary: a = 0
        elA.classList.add('dimmed');
        elA.classList.remove('highlight-a');
        elOp.classList.add('dimmed');
        elB.classList.remove('dimmed');
        elB.classList.add('highlight-b');
        elI.classList.remove('dimmed');
        bA.style.opacity = '0.25';
        bB.style.opacity = '1';
        bI.style.opacity = '1';
      } else {
        // General Complex
        elA.classList.remove('dimmed', 'highlight-a');
        elOp.classList.remove('dimmed');
        elB.classList.remove('dimmed', 'highlight-b');
        elI.classList.remove('dimmed');
        bA.style.opacity = '1';
        bB.style.opacity = '1';
        bI.style.opacity = '1';
      }
    }
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Complex Numbers: Product of Negative Radicals</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e293b;
      background: transparent;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
      justify-content: center;
      min-height: 100%;
    }

    .math-font {
      font-family: "Cambria Math", "KaTeX_Math", Georgia, "Times New Roman", serif;
      font-style: normal;
    }

    .math-var {
      font-style: italic;
      font-family: Georgia, "Times New Roman", serif;
    }

    .rad-expr {
      display: inline-flex;
      align-items: baseline;
      white-space: nowrap;
    }

    .rad-sym {
      font-size: 1.15em;
      margin-right: 1px;
      line-height: 1;
    }

    .rad-body {
      border-top: 1.5px solid currentColor;
      padding: 0 2px 0 1px;
      margin-top: 1px;
      display: inline-block;
      line-height: 1.15;
    }

    .header-expr {
      text-align: center;
      font-size: 20px;
      font-weight: 700;
      color: #0f172a;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
    }

    .cards-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 10px;
      width: 100%;
    }

    @media (max-width: 480px) {
      .cards-grid {
        grid-template-columns: 1fr;
      }
      .header-expr {
        font-size: 18px;
      }
    }

    .eval-card {
      border-radius: 10px;
      padding: 10px;
      display: flex;
      flex-direction: column;
      gap: 8px;
      transition: opacity 0.25s ease, border-color 0.25s ease;
    }

    .card-correct {
      background: #f0fdf4;
      border: 2px solid #22c55e;
    }

    .card-incorrect {
      background: #fef2f2;
      border: 2px solid #ef4444;
    }

    .card-title {
      font-size: 13px;
      font-weight: 800;
      letter-spacing: 0.5px;
      display: flex;
      align-items: center;
      justify-content: space-between;
    }

    .card-correct .card-title { color: #15803d; }
    .card-incorrect .card-title { color: #b91c1c; }

    .step-box {
      background: #ffffff;
      border-radius: 6px;
      padding: 6px 10px;
      display: flex;
      flex-direction: column;
      gap: 2px;
      border: 1px solid rgba(0, 0, 0, 0.08);
      transition: opacity 0.25s ease, box-shadow 0.25s ease, border-color 0.25s ease;
    }

    .step-num {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      color: #64748b;
      display: flex;
      align-items: center;
      justify-content: space-between;
    }

    .step-math {
      font-size: 15px;
      min-height: 24px;
      display: flex;
      align-items: center;
      color: #1e293b;
    }

    .tag-wrong {
      color: #dc2626;
      font-weight: 700;
      font-size: 11px;
      background: #fee2e2;
      padding: 1px 6px;
      border-radius: 4px;
    }

    .tag-right {
      color: #16a34a;
      font-weight: 700;
      font-size: 11px;
      background: #dcfce7;
      padding: 1px 6px;
      border-radius: 4px;
    }

    .result-badge {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      font-size: 15px;
      font-weight: 700;
      padding: 6px;
      border-radius: 6px;
      transition: opacity 0.25s ease;
    }

    .card-correct .result-badge {
      background: #dcfce7;
      color: #166534;
      border: 1px solid #86efac;
    }

    .card-incorrect .result-badge {
      background: #fee2e2;
      color: #991b1b;
      border: 1px solid #fca5a5;
    }

    .warning-bar {
      background: #fffbeb;
      border: 1px solid #fde047;
      border-radius: 8px;
      padding: 8px 10px;
      display: flex;
      align-items: flex-start;
      gap: 8px;
      font-size: 12px;
      line-height: 1.4;
      color: #92400e;
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    .warning-icon {
      font-size: 16px;
      line-height: 1;
      flex-shrink: 0;
      margin-top: 1px;
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    .warning-content strong {
      color: #78350f;
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    .step-dimmed {
      opacity: 0.35;
    }

    .step-highlight {
      box-shadow: 0 0 0 2px #0ea5e9;
      border-color: #0ea5e9;
    }

    .controls {
      display: flex;
      justify-content: center;
      gap: 6px;
    }

    .btn-step {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      color: #334155;
      padding: 4px 12px;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 600;
      cursor: pointer;
      transition: background-color 0.15s, color 0.15s;
    }

    .btn-step:hover {
      background: #e2e8f0;
    }

    .btn-step.active {
      background: #0284c7;
      border-color: #0369a1;
      color: #ffffff;
    }
  </style>
</head>
<body>

  
  <div class="header-expr math-font">
    <span>Evaluate:</span>
    <span>
      <span class="rad-expr"><span class="rad-sym">&radic;</span><span class="rad-body">&minus;4</span></span>
      &sdot;
      <span class="rad-expr"><span class="rad-sym">&radic;</span><span class="rad-body">&minus;9</span></span>
    </span>
  </div>

  <div class="cards-grid">
    
    <div class="eval-card card-correct" id="cardCorrect">
      <div class="card-title">
        <span>CORRECT METHOD</span>
        <span>✓</span>
      </div>

      <div class="step-box" id="cStep1">
        <div class="step-num">
          <span>Step 1: Extract <span class="math-var">i</span> First</span>
          <span class="tag-right">Proper Rule</span>
        </div>
        <div class="step-math math-font">
          = (2<span class="math-var">i</span>) &sdot; (3<span class="math-var">i</span>)
        </div>
      </div>

      <div class="step-box" id="cStep2">
        <div class="step-num">
          <span>Step 2: Simplify (<span class="math-var">i</span><sup>2</sup> = &minus;1)</span>
        </div>
        <div class="step-math math-font">
          = 6<span class="math-var">i</span><sup>2</sup> = 6(&minus;1) = &minus;6
        </div>
      </div>

      <div class="result-badge math-font" id="cResult">
        <span>Answer: &minus;6</span>
        <span>✓</span>
      </div>
    </div>

    
    <div class="eval-card card-incorrect" id="cardIncorrect">
      <div class="card-title">
        <span>INCORRECT METHOD</span>
        <span>✕</span>
      </div>

      <div class="step-box" id="iStep1">
        <div class="step-num">
          <span>Step 1: Blindly Combine</span>
          <span class="tag-wrong">Invalid Step</span>
        </div>
        <div class="step-math math-font">
          = <span class="rad-expr"><span class="rad-sym">&radic;</span><span class="rad-body">(&minus;4) &sdot; (&minus;9)</span></span>
        </div>
      </div>

      <div class="step-box" id="iStep2">
        <div class="step-num">
          <span>Step 2: False Positive</span>
          <span class="tag-wrong">Sign Error!</span>
        </div>
        <div class="step-math math-font">
          = <span class="rad-expr"><span class="rad-sym">&radic;</span><span class="rad-body">36</span></span> = +6
        </div>
      </div>

      <div class="result-badge math-font" id="iResult">
        <span>Answer: +6</span>
        <span>(Wrong Sign ✕)</span>
      </div>
    </div>
  </div>

  
  <div class="warning-bar" id="warnBox">
    <div class="warning-icon">⚠️</div>
    <div class="warning-content">
      <strong>Product Rule Restriction:</strong>
      <span class="math-font"><span class="rad-expr"><span class="rad-sym">&radic;</span><span class="rad-body"><span class="math-var">a</span></span></span> &sdot; <span class="rad-expr"><span class="rad-sym">&radic;</span><span class="rad-body"><span class="math-var">b</span></span></span> = <span class="rad-expr"><span class="rad-sym">&radic;</span><span class="rad-body"><span class="math-var">ab</span></span></span></span> is <strong>only valid</strong> when at least one radicand is non-negative (<span class="math-font"><span class="math-var">a</span> &ge; 0</span> or <span class="math-font"><span class="math-var">b</span> &ge; 0</span>). Always extract <span class="math-font"><span class="math-var">i</span></span> before multiplying!
    </div>
  </div>

  
  <div class="controls">
    <button class="btn-step" id="btn1" onclick="setStep(1)">Step 1</button>
    <button class="btn-step" id="btn2" onclick="setStep(2)">Step 2</button>
    <button class="btn-step active" id="btnAll" onclick="setStep(3)">View All</button>
  </div>

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      const c2 = document.getElementById('cStep2');
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      const ir = document.getElementById('iResult');

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        cr.classList.remove('step-dimmed');
        ir.classList.remove('step-dimmed');
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          el.classList.remove('step-dimmed');
          el.classList.remove('step-highlight');
        });
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    }
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
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    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
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      padding: 12px;
      width: 100%;
      min-height: 100%;
      display: flex;
      flex-direction: column;
      justify-content: center;
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    .container {
      display: grid;
      grid-template-columns: repeat(auto-fit, minmax(290px, 1fr));
      gap: 16px;
      width: 100%;
      max-width: 800px;
      margin: 0 auto;
    }

    .panel {
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 16px 14px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      gap: 12px;
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    .badge-row {
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    .badge {
      font-size: 11px;
      font-weight: 700;
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    .badge-blue {
      background: #e0f2fe;
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    .badge-purple {
      background: #f3e8ff;
      color: #6b21a8;
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    .math-font {
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    .sqrt {
      display: inline-flex;
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      vertical-align: middle;
      white-space: nowrap;
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    .sqrt-symbol {
      font-size: 1.15em;
      line-height: 1;
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      font-weight: 400;
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    .radicand {
      border-top: 1.5px solid currentColor;
      padding: 0 2px;
      margin-left: -1.5px;
      line-height: 1.1;
      display: inline-block;
    }

    .math-flow {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 6px;
      width: 100%;
    }

    .equation-lg {
      font-size: 20px;
      font-weight: 600;
      color: #0f172a;
      text-align: center;
    }

    .equation-md {
      font-size: 17px;
      font-weight: 600;
      color: #1e293b;
      text-align: center;
    }

    .arrow-down {
      color: #64748b;
      font-size: 15px;
      font-weight: bold;
      line-height: 1;
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    .highlight-box {
      width: 100%;
      background: #ffffff;
      border: 2px solid #0284c7;
      border-radius: 10px;
      padding: 10px 8px;
      text-align: center;
      box-shadow: 0 2px 8px rgba(2, 132, 199, 0.12);
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    .highlight-box-purple {
      border-color: #7c3aed;
      box-shadow: 0 2px 8px rgba(124, 58, 237, 0.12);
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    .highlight-title {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.05em;
      margin-bottom: 4px;
      color: #0369a1;
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    .highlight-title-purple {
      color: #6d28d9;
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    .highlight-val {
      font-size: 18px;
      font-weight: 700;
      color: #0f172a;
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    .accent-i {
      color: #7c3aed;
      font-weight: 700;
    }

    .accent-blue {
      color: #0284c7;
      font-weight: 700;
    }

    .accent-teal {
      color: #0d9488;
      font-weight: 700;
    }

    /* Interactive elements */
    .interactive-section {
      width: 100%;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 10px;
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    .interactive-display {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      border-radius: 8px;
      padding: 8px 12px;
      width: 100%;
      text-align: center;
      min-height: 48px;
      display: flex;
      align-items: center;
      justify-content: center;
      font-size: 18px;
      font-weight: 600;
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      display: flex;
      gap: 6px;
      flex-wrap: wrap;
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      border: 1.5px solid #cbd5e1;
      background: #ffffff;
      color: #334155;
      font-size: 12px;
      font-weight: 700;
      font-family: inherit;
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      border-radius: 6px;
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      outline: 2px solid #7c3aed;
      outline-offset: 1px;
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      background: #7c3aed;
      border-color: #7c3aed;
      color: #ffffff;
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      font-size: 11.5px;
      color: #64748b;
      text-align: center;
      line-height: 1.3;
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      body {
        padding: 8px;
      }
      .panel {
        padding: 12px 10px;
        gap: 10px;
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      .equation-lg {
        font-size: 18px;
      }
      .equation-md {
        font-size: 15px;
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        font-size: 16px;
      }
      .interactive-display {
        font-size: 16px;
        padding: 6px 8px;
      }
      .btn-choice {
        padding: 4px 8px;
        font-size: 11px;
      }
    }
  </style>
</head>
<body>

  <div class="container">
    
    <div class="panel">
      <div class="badge-row">
        <span class="badge badge-blue">Definition of Imaginary Unit</span>
      </div>

      <div class="math-flow">
        <div class="equation-lg math-font">
          <i>x</i><sup>2</sup> = −1
        </div>
        <div class="arrow-down">↓</div>
        <div class="equation-md math-font">
          <i>x</i> = ±<span class="sqrt"><span class="sqrt-symbol">√</span><span class="radicand">−1</span></span> = ±<span class="accent-i">i</span>
        </div>
      </div>

      <div class="highlight-box">
        <div class="highlight-title">Fundamental Definition</div>
        <div class="highlight-val math-font">
          <span class="accent-i"><i>i</i></span> = <span class="sqrt"><span class="sqrt-symbol">√</span><span class="radicand">−1</span></span>
          <span style="color: #64748b; font-size: 14px; margin: 0 6px; font-weight: normal;">and</span>
          <span class="accent-i"><i>i</i></span><sup>2</sup> = −1
        </div>
      </div>

      <div class="note-text">
        The unit <b><i class="accent-i">i</i></b> allows solutions to equations with no real roots.
      </div>
    </div>

    
    <div class="panel">
      <div class="badge-row">
        <span class="badge badge-purple">Negative Radical Rule</span>
      </div>

      <div class="highlight-box highlight-box-purple">
        <div class="highlight-title highlight-title-purple">General Property (<i>k</i> &gt; 0)</div>
        <div class="highlight-val math-font" style="font-size: 17px;">
          <span class="sqrt"><span class="sqrt-symbol">√</span><span class="radicand">−<i>k</i></span></span>
          =
          <span class="sqrt"><span class="sqrt-symbol">√</span><span class="radicand"><i>k</i></span></span> · <span class="sqrt"><span class="sqrt-symbol">√</span><span class="radicand">−1</span></span>
          =
          <span class="accent-i"><i>i</i></span><span class="sqrt"><span class="sqrt-symbol">√</span><span class="radicand"><i>k</i></span></span>
        </div>
      </div>

      <div class="interactive-section">
        <div class="note-text" style="font-weight: 600; color: #475569;">
          Select an example to calculate:
        </div>

        <div class="btn-group" id="btnGroup">
          <button type="button" class="btn-choice" data-k="4">√(-4)</button>
          <button type="button" class="btn-choice" data-k="9">√(-9)</button>
          <button type="button" class="btn-choice" data-k="16">√(-16)</button>
          <button type="button" class="btn-choice active" data-k="25">√(-25)</button>
          <button type="button" class="btn-choice" data-k="49">√(-49)</button>
          <button type="button" class="btn-choice" data-k="100">√(-100)</button>
        </div>

        <div class="interactive-display math-font" id="calcDisplay">
          
        </div>
      </div>
    </div>
  </div>

  <script>
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      const root = Math.sqrt(k);
      const display = document.getElementById('calcDisplay');
      display.innerHTML = `
        <span class="sqrt"><span class="sqrt-symbol">√</span><span class="radicand">−${k}</span></span>
        &nbsp;=&nbsp;
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        &nbsp;=&nbsp;
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        renderStep(parseInt(btn.getAttribute('data-k'), 10));
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    // Initial render
    renderStep(25);
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="1:1">
  <title>Principal Argument Interval</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      width: 100%;
      height: 100%;
      background: #ffffff;
      font-family: system-ui, -apple-system, sans-serif;
      color: #1e2945;
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      align-items: center;
      justify-content: center;
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      font-weight: 600;
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      height: 82%;
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    }
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      stroke-linecap: round;
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      font-weight: 700;
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      text-anchor: middle;
      dominant-baseline: middle;
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    .neg-label { fill: #ea580c; }
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    }
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      stroke: #ffffff;
      stroke-width: 3;
      cursor: grab;
      filter: drop-shadow(0 2px 4px rgba(0,0,0,0.2));
    }
    .probe-handle:active {
      cursor: grabbing;
    }
  </style>
</head>
<body>
  <div class="wrapper">
    <div class="interval-badge">
      <span>Arg(z) &in; (&minus;&pi;, &pi;]</span>
    </div>

    <svg id="svg" viewBox="-180 -180 360 360">
      
      <circle cx="0" cy="0" r="120" class="guide-circle" />

      
      <line x1="-155" y1="0" x2="155" y2="0" class="axis-main" />
      <line x1="0" y1="-155" x2="0" y2="155" class="axis-line" />

      
      <text x="162" y="3" class="marker-label" text-anchor="start">Re</text>
      <text x="0" y="-163" class="marker-label">Im</text>

      
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      <polygon points="-114,-22 -126,-6 -110,-2" fill="#16a34a" />

      
      <path d="M 120 0 A 120 120 0 0 1 -115 10" class="neg-arc" id="arcNeg" />
      
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      <circle cx="-120" cy="-14" r="5.5" fill="#16a34a" />
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      <circle cx="-120" cy="14" r="5.5" fill="#ffffff" stroke="#ea580c" stroke-width="2.5" />
      <text x="-142" y="18" class="marker-label neg-label">&minus;&pi; (&compfn;)</text>

      
      <text x="135" y="-12" class="marker-label" style="fill:#64748b;">0</text>

      
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      <circle id="probeDot" cx="72" cy="-96" r="10" class="probe-handle" />
    </svg>

    <div class="val-readout" id="readout">Arg(z) = +0.93 rad (+53.1&deg;)</div>
  </div>

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    const probeRay = document.getElementById('probeRay');
    const probeDot = document.getElementById('probeDot');
    const angleSector = document.getElementById('angleSector');
    const readout = document.getElementById('readout');
    const R = 120;

    let currentAngle = Math.PI / 180 * 53.13; // default 53.1° (matching lesson text)
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      // Normalize to (-PI, +PI]
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      const py = -R * Math.sin(a);

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      // Draw angle sector arc
      const sweep = a >= 0 ? 0 : 1;
      const largeArc = Math.abs(a) > Math.PI ? 1 : 0;
      const sr = 38;
      const sx = sr * Math.cos(a);
      const sy = -sr * Math.sin(a);
      angleSector.setAttribute('d', `M 0 0 L ${sr} 0 A ${sr} ${sr} 0 ${largeArc} ${sweep} ${sx} ${sy} Z`);

      // Color probe based on sign
      const col = a >= 0 ? '#16a34a' : '#ea580c';
      probeRay.style.stroke = col;
      probeDot.style.fill = col;

      // Update readout
      const deg = (a * 180 / Math.PI).toFixed(1);
      const rad = a.toFixed(2);
      const sign = a > 0 ? '+' : '';
      readout.textContent = `Arg(z) = ${sign}${rad} rad (${sign}${deg}°)`;
    }

    function updateFromEvent(e) {
      const pt = svg.createSVGPoint();
      const clientX = e.touches ? e.touches[0].clientX : e.clientX;
      const clientY = e.touches ? e.touches[0].clientY : e.clientY;
      pt.x = clientX;
      pt.y = clientY;
      const cursor = pt.matrixTransform(svg.getScreenCTM().inverse());
      // svg y is down, math y is up
      const angle = Math.atan2(-cursor.y, cursor.x);
      setAngle(angle);
    }

    svg.addEventListener('pointerdown', (e) => {
      isDragging = true;
      updateFromEvent(e);
      svg.setPointerCapture(e.pointerId);
    });

    svg.addEventListener('pointermove', (e) => {
      if (isDragging) updateFromEvent(e);
    });

    svg.addEventListener('pointerup', (e) => {
      isDragging = false;
      try { svg.releasePointerCapture(e.pointerId); } catch (_) {}
    });

    svg.addEventListener('pointercancel', () => { isDragging = false; });

    // Initial render
    setAngle(currentAngle);
  </script>
</body>
</html>

Modulus and Principal Argument of Complex Numbers

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="1:1">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      -webkit-tap-highlight-color: transparent;
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    body {
      font-family: system-ui, -apple-system, sans-serif;
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      background: #ffffff;
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      transition: border-color 0.2s ease, box-shadow 0.2s ease, background-color 0.2s ease;
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    .quad-tile:hover {
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    .tile-header {
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      padding: 0 4px 2px 4px;
      font-size: 11px;
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    svg {
      width: 100%;
      height: 100%;
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      letter-spacing: 0.2px;
      transition: background 0.2s ease, color 0.2s ease;
    }
    .quad-tile.active .formula-pill {
      background: #2563eb;
      color: #ffffff;
    }

    /* Bottom Legend & Status */
    .legend-bar {
      display: flex;
      flex-wrap: wrap;
      justify-content: center;
      gap: 12px;
      margin-top: 8px;
      font-size: 11px;
      font-weight: 600;
      color: #475569;
    }
    .legend-item {
      display: flex;
      align-items: center;
      gap: 6px;
    }
    .swatch {
      width: 16px;
      height: 4px;
      border-radius: 2px;
    }
    .swatch-arg {
      background: #16a34a;
    }
    .swatch-ref {
      background: #d97706;
      border-top: 1px dashed #d97706;
      height: 3px;
    }

    .status-text {
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      font-weight: 600;
      color: #2563eb;
      text-align: center;
      height: 18px;
      line-height: 18px;
      margin-top: 4px;
    }
  </style>
</head>
<body>

  
  <div class="header-badge">
    <span>Reference Angle:</span>
    <span class="math ref-symbol">α</span>
    <span>=</span>
    <span class="math">arctan|<span class="math-normal"><sup>y</sup>/<sub>x</sub></span>|</span>
    <span style="font-weight: 400; color: #64748b; margin-left: 2px;">(0 &lt; <span class="math">α</span> &lt; <sup>π</sup>/<sub>2</sub>)</span>
  </div>

  
  <div class="grid-2x2">
    
    <div class="quad-tile" id="tile-q2" onclick="selectQuad('q2')">
      <div class="tile-header">
        <span>Quadrant II</span>
        <span class="math">x &lt; 0, y &gt; 0</span>
      </div>
      <div class="svg-wrap">
        <svg viewBox="0 0 100 100">
          <defs>
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            </marker>
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          <text x="88" y="44" font-size="7" fill="#94a3b8" font-weight="700">Re</text>
          <text x="54" y="14" font-size="7" fill="#94a3b8" font-weight="700">Im</text>

          
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          <circle cx="23" cy="23" r="3.5" fill="#1d4ed8" />
          <text x="14" y="20" font-size="8.5" fill="#1d4ed8" font-weight="700" class="math">z</text>

          
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          <path d="M 74 50 A 24 24 0 0 0 33 33" fill="none" stroke="#16a34a" stroke-width="2" marker-end="url(#arr-arg)" />
        </svg>
      </div>
      <div class="formula-pill"><span class="math">Arg(z) = π − α</span></div>
    </div>

    
    <div class="quad-tile active" id="tile-q1" onclick="selectQuad('q1')">
      <div class="tile-header">
        <span>Quadrant I</span>
        <span class="math">x &gt; 0, y &gt; 0</span>
      </div>
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        <svg viewBox="0 0 100 100">
          
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          <text x="54" y="14" font-size="7" fill="#94a3b8" font-weight="700">Im</text>

          
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          <path d="M 74 50 A 24 24 0 0 0 67 33" fill="none" stroke="#16a34a" stroke-width="2" marker-end="url(#arr-arg)" />
          <text x="73" y="41" font-size="7.5" fill="#16a34a" font-weight="700" class="math">α</text>
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      <div class="formula-pill"><span class="math">Arg(z) = α</span></div>
    </div>

    
    <div class="quad-tile" id="tile-q3" onclick="selectQuad('q3')">
      <div class="tile-header">
        <span>Quadrant III</span>
        <span class="math">x &lt; 0, y &lt; 0</span>
      </div>
      <div class="svg-wrap">
        <svg viewBox="0 0 100 100">
          
          <line x1="8" y1="50" x2="92" y2="50" stroke="#cbd5e1" stroke-width="1.5" marker-end="url(#arr-axis)" />
          <line x1="50" y1="92" x2="50" y2="8" stroke="#cbd5e1" stroke-width="1.5" marker-end="url(#arr-axis)" />
          <text x="88" y="44" font-size="7" fill="#94a3b8" font-weight="700">Re</text>
          <text x="54" y="14" font-size="7" fill="#94a3b8" font-weight="700">Im</text>

          
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          <circle cx="23" cy="77" r="3.5" fill="#1d4ed8" />
          <text x="14" y="85" font-size="8.5" fill="#1d4ed8" font-weight="700" class="math">z</text>

          
          <path d="M 32 50 A 18 18 0 0 0 37.3 62.7" fill="none" stroke="#d97706" stroke-width="1.5" stroke-dasharray="2,2" marker-end="url(#arr-ref)" />
          <text x="33" y="60" font-size="7.5" fill="#d97706" font-weight="700" class="math">α</text>

          
          <path d="M 74 50 A 24 24 0 0 1 33 67" fill="none" stroke="#16a34a" stroke-width="2" marker-end="url(#arr-arg)" />
        </svg>
      </div>
      <div class="formula-pill"><span class="math">Arg(z) = α − π</span></div>
    </div>

    
    <div class="quad-tile" id="tile-q4" onclick="selectQuad('q4')">
      <div class="tile-header">
        <span>Quadrant IV</span>
        <span class="math">x &gt; 0, y &lt; 0</span>
      </div>
      <div class="svg-wrap">
        <svg viewBox="0 0 100 100">
          
          <line x1="8" y1="50" x2="92" y2="50" stroke="#cbd5e1" stroke-width="1.5" marker-end="url(#arr-axis)" />
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          <text x="88" y="44" font-size="7" fill="#94a3b8" font-weight="700">Re</text>
          <text x="54" y="14" font-size="7" fill="#94a3b8" font-weight="700">Im</text>

          
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          <circle cx="77" cy="77" r="3.5" fill="#1d4ed8" />
          <text x="82" y="85" font-size="8.5" fill="#1d4ed8" font-weight="700" class="math">z</text>

          
          <path d="M 74 50 A 24 24 0 0 1 67 67" fill="none" stroke="#16a34a" stroke-width="2" marker-end="url(#arr-arg)" />
          <text x="73" y="63" font-size="7.5" fill="#16a34a" font-weight="700" class="math">−α</text>
        </svg>
      </div>
      <div class="formula-pill"><span class="math">Arg(z) = −α</span></div>
    </div>
  </div>

  
  <div class="status-text" id="status-line">
    Quadrant I: <span class="math">Arg(z) = α</span>
  </div>

  
  <div class="legend-bar">
    <div class="legend-item">
      <div class="swatch swatch-arg"></div>
      <span><span class="math">Arg(z)</span> ∈ (−π, π]</span>
    </div>
    <div class="legend-item">
      <div class="swatch swatch-ref"></div>
      <span>Acute Reference <span class="math">α</span></span>
    </div>
  </div>

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</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="1:1">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
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    .info-q3 { color: #d97706; }
  </style>
</head>
<body>

  <div class="controls">
    <button class="btn-step active" id="btn1">z₁ (Q2: +120°)</button>
    <button class="btn-step" id="btn2">z₂ (Q3: −135°)</button>
    <button class="btn-step" id="btn3">Compare</button>
  </div>

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    <svg viewBox="-190 -170 380 340">
      <defs>
        
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      <text x="-8" y="14" font-size="11" fill="#94a3b8" text-anchor="end">0</text>

      
      
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        <text x="-40" y="-18" font-size="11" fill="#475569" font-weight="600">α=π/3</text>

        
        
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OGM.12.NUM.3

Aligned learning resources

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