<!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">
  <title>Simplifying Imaginary Numbers</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #ffffff;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px 16px;
    }
    .flow-container {
      width: 100%;
      max-width: 440px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 6px;
    }
    .step-box {
      width: 100%;
      background: #f8fafc;
      border: 2px solid #e2e8f0;
      border-radius: 10px;
      padding: 8px 14px;
      display: flex;
      align-items: center;
      justify-content: space-between;
      transition: all 0.25s ease;
      cursor: pointer;
      user-select: none;
    }
    .step-box:hover {
      border-color: #94a3b8;
    }
    .step-box.active {
      border-color: #0284c7;
      background: #ffffff;
      box-shadow: 0 4px 12px rgba(2, 132, 199, 0.12);
    }
    .step-box.final {
      background: #f0f9ff;
      border-color: #bae6fd;
    }
    .step-box.final.active {
      background: #e0f2fe;
      border-color: #0284c7;
      box-shadow: 0 4px 14px rgba(2, 132, 199, 0.2);
    }
    .step-left {
      display: flex;
      align-items: center;
      gap: 10px;
    }
    .step-num {
      width: 24px;
      height: 24px;
      border-radius: 50%;
      background: #e2e8f0;
      color: #1e2945;
      font-size: 13px;
      font-weight: 700;
      display: flex;
      align-items: center;
      justify-content: center;
      flex-shrink: 0;
      transition: all 0.25s ease;
    }
    .active .step-num {
      background: #0284c7;
      color: #ffffff;
    }
    .step-label {
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
    }
    .active .step-label {
      color: #1e2945;
    }
    .step-math {
      font-family: "Cambria Math", "Times New Roman", serif;
      font-size: 19px;
      font-weight: 600;
      color: #1e2945;
      letter-spacing: 0.5px;
    }
    .color-sq { color: #d97706; font-weight: 700; }
    .color-neg { color: #7c3aed; font-weight: 700; }
    .color-rem { color: #0284c7; font-weight: 700; }

    .arrow {
      color: #94a3b8;
      font-size: 14px;
      line-height: 1;
      height: 14px;
      display: flex;
      align-items: center;
      justify-content: center;
      transition: color 0.25s ease;
    }
    .arrow.active {
      color: #0284c7;
      font-weight: 700;
    }

    /* Controls */
    .controls {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
      margin-top: 10px;
      width: 100%;
    }
    .btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      color: #1e2945;
      padding: 5px 12px;
      border-radius: 6px;
      font-size: 13px;
      font-weight: 600;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 4px;
      transition: background 0.15s ease;
    }
    .btn:hover {
      background: #f1f5f9;
    }
    .dot-nav {
      display: flex;
      gap: 6px;
      align-items: center;
    }
    .dot {
      width: 9px;
      height: 9px;
      border-radius: 50%;
      background: #cbd5e1;
      cursor: pointer;
      border: none;
      padding: 0;
      transition: all 0.25s ease;
    }
    .dot.active {
      background: #0284c7;
      transform: scale(1.25);
    }
  </style>
</head>
<body>

  <div class="flow-container">
    
    <div class="step-box active" id="step0" onclick="goToStep(0)">
      <div class="step-left">
        <span class="step-num">1</span>
        <span class="step-label">Original radical</span>
      </div>
      <div class="step-math">&radic;<span style="text-decoration:overline;">&minus;72</span></div>
    </div>

    <div class="arrow" id="arrow0">&darr;</div>

    
    <div class="step-box" id="step1" onclick="goToStep(1)">
      <div class="step-left">
        <span class="step-num">2</span>
        <span class="step-label">Factor out &minus;1</span>
      </div>
      <div class="step-math">&radic;<span style="text-decoration:overline;"><span class="color-sq">36</span> &middot; <span class="color-rem">2</span> &middot; <span class="color-neg">(&minus;1)</span></span></div>
    </div>

    <div class="arrow" id="arrow1">&darr;</div>

    
    <div class="step-box" id="step2" onclick="goToStep(2)">
      <div class="step-left">
        <span class="step-num">3</span>
        <span class="step-label">Evaluate &radic;36 &amp; &radic;&minus;1</span>
      </div>
      <div class="step-math"><span class="color-sq">6</span> &middot; &radic;<span style="text-decoration:overline;"><span class="color-rem">2</span></span> &middot; <span class="color-neg">i</span></div>
    </div>

    <div class="arrow" id="arrow2">&darr;</div>

    
    <div class="step-box final" id="step3" onclick="goToStep(3)">
      <div class="step-left">
        <span class="step-num">4</span>
        <span class="step-label">Final form</span>
      </div>
      <div class="step-math"><span class="color-sq">6</span><span class="color-neg">i</span>&radic;<span style="text-decoration:overline;"><span class="color-rem">2</span></span></div>
    </div>

    
    <div class="controls">
      <button class="btn" id="prevBtn" onclick="prevStep()">← Prev</button>
      
      <div class="dot-nav">
        <button class="dot active" id="dot0" onclick="goToStep(0)" aria-label="Step 1"></button>
        <button class="dot" id="dot1" onclick="goToStep(1)" aria-label="Step 2"></button>
        <button class="dot" id="dot2" onclick="goToStep(2)" aria-label="Step 3"></button>
        <button class="dot" id="dot3" onclick="goToStep(3)" aria-label="Step 4"></button>
      </div>

      <button class="btn" id="nextBtn" onclick="nextStep()">Next →</button>
    </div>
  </div>

  <script>
    let currentStep = 0;
    const totalSteps = 4;
    let autoPlayTimer = null;

    function renderState() {
      for (let i = 0; i < totalSteps; i++) {
        const stepEl = document.getElementById('step' + i);
        const dotEl = document.getElementById('dot' + i);
        
        if (i === currentStep) {
          stepEl.classList.add('active');
          dotEl.classList.add('active');
        } else {
          stepEl.classList.remove('active');
          dotEl.classList.remove('active');
        }

        if (i < 3) {
          const arrowEl = document.getElementById('arrow' + i);
          if (i === currentStep - 1) {
            arrowEl.classList.add('active');
          } else {
            arrowEl.classList.remove('active');
          }
        }
      }
    }

    function goToStep(index) {
      currentStep = index;
      renderState();
      resetAutoPlay();
    }

    function nextStep() {
      currentStep = (currentStep + 1) % totalSteps;
      renderState();
      resetAutoPlay();
    }

    function prevStep() {
      currentStep = (currentStep - 1 + totalSteps) % totalSteps;
      renderState();
      resetAutoPlay();
    }

    function startAutoPlay() {
      autoPlayTimer = setInterval(() => {
        currentStep = (currentStep + 1) % totalSteps;
        renderState();
      }, 3500);
    }

    function resetAutoPlay() {
      clearInterval(autoPlayTimer);
      startAutoPlay();
    }

    startAutoPlay();
  </script>
</body>
</html>

Simplifying Expressions with Imaginary 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="4:3">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <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: #1e2945;
      padding: 12px 8px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
    }
    .comparison-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 12px;
      width: 100%;
      max-width: 600px;
    }
    .panel-card {
      border-radius: 12px;
      padding: 12px 10px;
      display: flex;
      flex-direction: column;
      align-items: center;
      text-align: center;
    }
    .panel-wrong {
      background: #fef2f2;
      border: 2px solid #ef4444;
    }
    .panel-right {
      background: #f0fdf4;
      border: 2px solid #10b981;
    }
    .header-badge {
      display: inline-flex;
      align-items: center;
      gap: 6px;
      font-weight: 700;
      font-size: 15px;
      padding: 4px 12px;
      border-radius: 20px;
      margin-bottom: 12px;
    }
    .badge-wrong {
      background: #fee2e2;
      color: #dc2626;
      border: 1px solid #fca5a5;
    }
    .badge-right {
      background: #dcfce7;
      color: #16a34a;
      border: 1px solid #86efac;
    }
    .math-flow {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 8px;
      width: 100%;
    }
    .math-step {
      font-size: 16px;
      line-height: 1.2;
      display: flex;
      align-items: center;
      justify-content: center;
    }
    .arrow {
      color: #9ca3af;
      font-size: 14px;
      user-select: none;
    }
    .final-row {
      display: inline-flex;
      align-items: center;
      gap: 6px;
      font-weight: 700;
      font-size: 18px;
      padding: 4px 12px;
      border-radius: 8px;
      margin-top: 4px;
    }
    .final-wrong {
      background: #fee2e2;
      color: #dc2626;
    }
    .final-right {
      background: #dcfce7;
      color: #16a34a;
    }
    .mark {
      font-size: 18px;
    }
    @media (max-width: 380px) {
      .comparison-grid {
        gap: 8px;
      }
      .panel-card {
        padding: 10px 6px;
      }
      .header-badge {
        font-size: 13px;
        padding: 3px 8px;
        margin-bottom: 8px;
      }
      .math-step {
        font-size: 14px;
      }
      .final-row {
        font-size: 16px;
        padding: 3px 8px;
      }
    }
  </style>
</head>
<body>

  <div class="comparison-grid">
    
    <div class="panel-card panel-wrong">
      <div class="header-badge badge-wrong">
        <span>✗</span>
        <span>Wrong Way</span>
      </div>
      <div class="math-flow">
        <div class="math-step" id="w1"></div>
        <div class="arrow">↓</div>
        <div class="math-step" id="w2"></div>
        <div class="arrow">↓</div>
        <div class="math-step" id="w3"></div>
        <div class="arrow">↓</div>
        <div class="final-row final-wrong">
          <span id="w4"></span>
          <span class="mark">✗</span>
        </div>
      </div>
    </div>

    
    <div class="panel-card panel-right">
      <div class="header-badge badge-right">
        <span>✓</span>
        <span>Right Way</span>
      </div>
      <div class="math-flow">
        <div class="math-step" id="r1"></div>
        <div class="arrow">↓</div>
        <div class="math-step" id="r2"></div>
        <div class="arrow">↓</div>
        <div class="math-step" id="r3"></div>
        <div class="arrow">↓</div>
        <div class="final-row final-right">
          <span id="r4"></span>
          <span class="mark">✓</span>
        </div>
      </div>
    </div>
  </div>

  <script>
    function renderMath() {
      const equations = [
        { id: 'w1', tex: '\\sqrt{-4} \\cdot \\sqrt{-9}' },
        { id: 'w2', tex: '\\sqrt{(-4)(-9)}' },
        { id: 'w3', tex: '\\sqrt{36}' },
        { id: 'w4', tex: '6' },
        { id: 'r1', tex: '\\sqrt{-4} \\cdot \\sqrt{-9}' },
        { id: 'r2', tex: '(2i)(3i)' },
        { id: 'r3', tex: '6i^2 = 6(-1)' },
        { id: 'r4', tex: '-6' }
      ];

      if (window.katex) {
        equations.forEach(eq => {
          const el = document.getElementById(eq.id);
          if (el) {
            katex.render(eq.tex, el, { throwOnError: false, displayMode: false });
          }
        });
      } else {
        // Fallback plain text representation if KaTeX CDN is unreachable
        document.getElementById('w1').textContent = '√(-4) · √(-9)';
        document.getElementById('w2').textContent = '√((-4)(-9))';
        document.getElementById('w3').textContent = '√36';
        document.getElementById('w4').textContent = '6';
        document.getElementById('r1').textContent = '√(-4) · √(-9)';
        document.getElementById('r2').textContent = '(2i)(3i)';
        document.getElementById('r3').textContent = '6i² = 6(-1)';
        document.getElementById('r4').textContent = '-6';
      }
    }

    window.addEventListener('DOMContentLoaded', renderMath);
    if (document.readyState !== 'loading') {
      renderMath();
    }
  </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;
    }

    body {
      width: 100%;
      min-height: 100%;
      background: transparent;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      padding: 16px;
      gap: 20px;
    }

    .container {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 20px;
    }

    /* Math typography styling */
    .math-var {
      font-family: "Cambria Math", "Times New Roman", Times, serif;
      font-style: italic;
      font-weight: 700;
    }

    .radicand {
      border-top: 2px solid currentColor;
      padding-top: 1px;
      margin-left: 1px;
      display: inline-block;
      line-height: 1;
    }

    /* Badges Row */
    .badges-row {
      display: flex;
      flex-wrap: wrap;
      justify-content: center;
      align-items: center;
      gap: 12px;
      width: 100%;
    }

    .formula-badge {
      flex: 1 1 180px;
      background: #e0f2fe;
      border: 2px solid #0284c7;
      border-radius: 12px;
      padding: 12px 16px;
      text-align: center;
      font-size: 22px;
      font-weight: 700;
      color: #1e2945;
      box-shadow: 0 2px 6px rgba(2, 132, 199, 0.12);
      display: flex;
      align-items: center;
      justify-content: center;
      min-height: 56px;
      letter-spacing: 0.5px;
    }

    /* Squaring Flow Area (No outer border wrapper) */
    .flow-section {
      width: 100%;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 16px;
      padding: 8px 0;
    }

    .flow-row {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
      width: 100%;
    }

    .node-box {
      width: 76px;
      height: 76px;
      background: #f8fafc;
      border: 2px solid #cbd5e1;
      border-radius: 12px;
      display: flex;
      align-items: center;
      justify-content: center;
      font-size: 26px;
      font-weight: 700;
      color: #1e2945;
      transition: all 0.3s ease;
      user-select: none;
    }

    .node-box.active-in {
      border-color: #0284c7;
      background: #e0f2fe;
      box-shadow: 0 0 0 3px rgba(2, 132, 199, 0.25);
    }

    .node-box.active-out {
      border-color: #059669;
      background: #d1fae5;
      color: #065f46;
      box-shadow: 0 0 0 3px rgba(5, 150, 105, 0.25);
    }

    .flow-arrow-container {
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      position: relative;
      flex: 1;
      max-width: 130px;
    }

    .op-pill {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      border-radius: 20px;
      padding: 3px 12px;
      font-size: 15px;
      font-weight: 700;
      color: #0284c7;
      margin-bottom: 6px;
      white-space: nowrap;
      transition: all 0.3s ease;
    }

    .op-pill.active-pill {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
    }

    .arrow-svg {
      width: 100%;
      height: 24px;
      overflow: visible;
    }

    .arrow-line {
      transition: stroke 0.3s ease;
    }

    /* Explanation message */
    .flow-desc {
      font-size: 14px;
      color: #475569;
      text-align: center;
      min-height: 20px;
      line-height: 1.4;
    }

    /* Interactive Control */
    .controls {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
    }

    .action-btn {
      background: #0284c7;
      color: #ffffff;
      border: none;
      border-radius: 8px;
      padding: 10px 20px;
      font-size: 15px;
      font-weight: 600;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 8px;
      transition: opacity 0.2s ease, background-color 0.2s ease;
    }

    .action-btn:hover {
      opacity: 0.9;
    }

    .action-btn:active {
      opacity: 0.8;
    }

    @media (max-width: 380px) {
      .formula-badge {
        font-size: 19px;
        min-height: 50px;
        padding: 10px 12px;
      }
      .node-box {
        width: 64px;
        height: 64px;
        font-size: 22px;
      }
      .op-pill {
        font-size: 13px;
        padding: 2px 8px;
      }
    }
  </style>
</head>
<body>

  <div class="container">
    
    <div class="badges-row">
      <div class="formula-badge">
        <span class="math-var">i</span>&nbsp;=&nbsp;√<span class="radicand">&minus;1</span>
      </div>
      <div class="formula-badge">
        <span class="math-var">i</span>²&nbsp;=&nbsp;&minus;1
      </div>
    </div>

    
    <div class="flow-section">
      <div class="flow-row">
        <div class="node-box" id="nodeIn">
          <span class="math-var">i</span>
        </div>
        
        <div class="flow-arrow-container">
          <span class="op-pill" id="opLabel">( )²</span>
          <svg class="arrow-svg" viewBox="0 0 100 24" preserveAspectRatio="none">
            <defs>
              <marker id="arrowhead" markerWidth="8" markerHeight="8" refX="6" refY="4" orient="auto">
                <path d="M 0 0 L 8 4 L 0 8 Z" fill="#0284c7" id="arrowheadPath" />
              </marker>
            </defs>
            <line class="arrow-line" x1="4" y1="12" x2="90" y2="12" stroke="#0284c7" stroke-width="2.5" marker-end="url(#arrowhead)" />
          </svg>
        </div>

        <div class="node-box" id="nodeOut">
          &minus;1
        </div>
      </div>

      <div class="flow-desc" id="flowDesc">
        Start with imaginary unit <em>i</em>
      </div>

      <div class="controls">
        <button class="action-btn" id="stepBtn" type="button">
          <span>Apply Power</span> ( )²
        </button>
      </div>
    </div>
  </div>

  <script>
    let step = 0;
    const nodeIn = document.getElementById('nodeIn');
    const nodeOut = document.getElementById('nodeOut');
    const opLabel = document.getElementById('opLabel');
    const stepBtn = document.getElementById('stepBtn');
    const flowDesc = document.getElementById('flowDesc');

    function updateState() {
      if (step === 0) {
        nodeIn.classList.add('active-in');
        nodeOut.classList.remove('active-out');
        opLabel.classList.remove('active-pill');
        flowDesc.innerHTML = 'Start with the imaginary unit <span class="math-var">i</span>';
        stepBtn.innerHTML = '<span>Apply Power</span> ( )²';
      } else {
        nodeIn.classList.remove('active-in');
        nodeOut.classList.add('active-out');
        opLabel.classList.add('active-pill');
        flowDesc.innerHTML = 'Squaring gives: (<span class="math-var">i</span>)² = (√&minus;1)² = <strong>&minus;1</strong>';
        stepBtn.innerHTML = '<span>Reset</span> ↺';
      }
    }

    stepBtn.addEventListener('click', () => {
      step = step === 0 ? 1 : 0;
      updateState();
    });

    // Initialize state
    updateState();
  </script>
</body>
</html>

OGM.10.RNQ.4

Aligned learning resources

Related diagrams