<!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">
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  </style>
</head>
<body>

  <div class="header-bar">
    <span>Rule:</span>
    <span class="rule-math">z = a <span class="flip-highlight">+</span> bi &nbsp;➔&nbsp; <span class="z-bar">z</span> = a <span class="flip-highlight">−</span> bi</span>
  </div>

  <div class="grid">
    
    <div class="item-card">
      <div class="card-top">
        <span class="tag">Standard</span>
      </div>
      <div class="math-row">
        <div class="expr">z = 5 <span class="sign-box flip-orig">+</span> 2i</div>
        <div class="arrow">➔</div>
        <div class="expr"><span class="z-bar">z</span> = 5 <span class="sign-box flip-target">−</span> 2i</div>
      </div>
      <div class="subtext">Imaginary sign flips</div>
    </div>

    
    <div class="item-card">
      <div class="card-top">
        <span class="tag">Negative Real</span>
      </div>
      <div class="math-row">
        <div class="expr">z = −4 <span class="sign-box flip-orig">−</span> 9i</div>
        <div class="arrow">➔</div>
        <div class="expr"><span class="z-bar">z</span> = −4 <span class="sign-box flip-target">+</span> 9i</div>
      </div>
      <div class="subtext">Real part (−4) unchanged</div>
    </div>

    
    <div class="item-card">
      <div class="card-top">
        <span class="tag real">Pure Real (b = 0)</span>
      </div>
      <div class="math-row">
        <div class="expr">z = 8</div>
        <div class="arrow">➔</div>
        <div class="expr"><span class="z-bar">z</span> = 8 <span class="sign-box same">✓</span></div>
      </div>
      <div class="subtext">Real numbers equal their conjugate</div>
    </div>

    
    <div class="item-card">
      <div class="card-top">
        <span class="tag imag">Pure Imaginary (a = 0)</span>
      </div>
      <div class="math-row">
        <div class="expr">z = <span class="sign-box flip-orig">−</span>3i</div>
        <div class="arrow">➔</div>
        <div class="expr"><span class="z-bar">z</span> = <span class="sign-box flip-target">+</span>3i</div>
      </div>
      <div class="subtext">Entire term sign flips</div>
    </div>
  </div>

</body>
</html>

Finding the Conjugate of a Complex Number

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<head>
  <meta charset="UTF-8">
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  <title>Conjugate Multiplication</title>
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      opacity: 1;
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      font-family: "SF Pro Text", "Cambria Math", "Times New Roman", serif;
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    sup {
      font-size: 12px;
      line-height: 0;
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      color: #15803d;
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      color: #475569;
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  </style>
</head>
<body>

  
  <div class="stepper-nav">
    <button class="step-btn active" onclick="setStep(0)">1. FOIL</button>
    <button class="step-btn" onclick="setStep(1)">2. Cancel</button>
    <button class="step-btn" onclick="setStep(2)">3. i² = −1</button>
    <button class="step-btn" onclick="setStep(3)">4. Result</button>
  </div>

  
  <div class="main-stage">
    
    <div class="expr-row" id="row-0">
      <span class="step-tag">Step 1</span>
      <div class="expr-math">
        (a + bi)(a − bi) = a² − abi + abi − b²i²
      </div>
    </div>

    
    <div class="expr-row" id="row-1">
      <span class="step-tag">Step 2</span>
      <div class="expr-math">
        = a² <span class="strikethrough">− abi</span> <span class="strikethrough">+ abi</span> − b²i²
      </div>
    </div>

    
    <div class="expr-row" id="row-2">
      <span class="step-tag">Step 3</span>
      <div class="expr-math">
        = a² − b²<span class="sub-box">(−1)</span>
      </div>
    </div>

    
    <div class="expr-row highlight-final" id="row-3">
      <span class="step-tag">Step 4</span>
      <div class="final-badge">
        <span>a² + b²</span>
        <span class="real-pill">Real</span>
      </div>
    </div>
  </div>

  
  <div class="example-panel">
    <span class="example-tag">Example</span>
    <span class="example-math">(3 + 4i)(3 − 4i) = 3² + 4² = 9 + 16 = <strong>25</strong></span>
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HSN.CN.A.3

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