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Override for the tutor's lesson content generation calls. Auto uses the system default (Claude Sonnet 5 for lesson text, Opus 5 for diagrams).

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Tutoring abilities

If you punch $1 \div 4$ into a calculator, it stops cleanly at $0.25$, but punch in $1 \div 3$ and you get an endless march of $0.3333\dots$

Why do some fractions stop while others loop forever, and is every repeating decimal actually a hidden fraction?

A **rational number** is any number that can be written as a ratio $\frac{a}{b}$, where $a$ and $b$ are integers (whole numbers or their negatives) and $b \neq 0$.

The word "rational" comes from the word *ratio* — if you can write it as one integer divided by another, it belongs to this family.

<!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;
      -webkit-tap-highlight-color: transparent;
    }

    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: #ffffff;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
    }

    /* Header & Badge */
    .header-row {
      display: flex;
      flex-wrap: wrap;
      align-items: center;
      justify-content: center;
      gap: 8px;
      margin-bottom: 8px;
    }

    .badge {
      display: inline-flex;
      align-items: center;
      gap: 4px;
      padding: 4px 10px;
      border-radius: 999px;
      font-size: 13px;
      font-weight: 600;
      background: #eff6ff;
      color: #1d4ed8;
      border: 1px solid #bfdbfe;
    }

    /* Preset selector pills */
    .preset-bar {
      display: flex;
      gap: 6px;
      flex-wrap: wrap;
      justify-content: center;
      margin-bottom: 10px;
    }

    .pill-btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      border-radius: 6px;
      padding: 4px 10px;
      font-size: 13px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
      transition: background 0.2s, color 0.2s, box-shadow 0.2s, border-color 0.2s;
    }

    .pill-btn.active {
      background: #2563eb;
      color: #ffffff;
      border-color: #2563eb;
      box-shadow: 0 2px 6px rgba(37, 99, 235, 0.3);
    }

    /* Main Fraction Structure Card */
    .main-fraction-card {
      width: 100%;
      max-width: 480px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 10px 14px;
      display: grid;
      grid-template-columns: 80px 1fr;
      align-items: center;
      gap: 12px;
      margin-bottom: 8px;
    }

    .fraction-display {
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      font-size: 26px;
      font-weight: 700;
      color: #0f172a;
      line-height: 1;
      padding: 4px 0;
    }

    .frac-num {
      color: #2563eb;
      padding-bottom: 3px;
      transition: color 0.2s;
    }

    .frac-line {
      width: 100%;
      height: 3px;
      background: #334155;
      border-radius: 2px;
      margin: 2px 0;
    }

    .frac-den {
      color: #0891b2;
      padding-top: 3px;
      transition: color 0.2s;
    }

    .labels-col {
      display: flex;
      flex-direction: column;
      gap: 6px;
    }

    .label-row {
      display: flex;
      align-items: center;
      gap: 6px;
      font-size: 13px;
    }

    .label-tag {
      font-weight: 700;
      font-size: 11px;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      padding: 2px 6px;
      border-radius: 4px;
    }

    .tag-num {
      background: #dbeafe;
      color: #1e40af;
    }

    .tag-den {
      background: #cffafe;
      color: #0e7490;
    }

    .label-desc {
      color: #334155;
      font-size: 12px;
    }

    .label-desc span {
      font-weight: 600;
    }

    /* Branching Section */
    .branches-container {
      width: 100%;
      max-width: 480px;
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 8px;
    }

    .branch-card {
      background: #ffffff;
      border: 2px solid #e2e8f0;
      border-radius: 10px;
      padding: 10px 8px;
      text-align: center;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      transition: border-color 0.25s, box-shadow 0.25s, background 0.25s, opacity 0.25s;
      opacity: 0.5;
    }

    .branch-card.active {
      opacity: 1;
      border-color: #2563eb;
      background: #f8fafc;
      box-shadow: 0 4px 12px rgba(37, 99, 235, 0.12);
    }

    .branch-title {
      font-size: 13px;
      font-weight: 700;
      margin-bottom: 4px;
      color: #0f172a;
    }

    .branch-card.active .branch-title.term-title {
      color: #059669;
    }

    .branch-card.active .branch-title.rep-title {
      color: #7c3aed;
    }

    .branch-formula {
      font-size: 15px;
      font-weight: 700;
      color: #1e2945;
      margin: 4px 0;
      background: #ffffff;
      padding: 3px 6px;
      border-radius: 4px;
      border: 1px solid #e2e8f0;
      width: 100%;
    }

    .bar-notation {
      text-decoration: overline;
    }

    .branch-sub {
      font-size: 11px;
      color: #64748b;
      font-weight: 500;
      margin-top: 2px;
    }

    /* Small Screen Tuning */
    @media (max-width: 380px) {
      .main-fraction-card {
        grid-template-columns: 65px 1fr;
        gap: 8px;
        padding: 8px 10px;
      }
      .fraction-display {
        font-size: 22px;
      }
      .branch-formula {
        font-size: 13px;
      }
      .branch-title {
        font-size: 12px;
      }
    }
  </style>
</head>
<body>

  
  <div class="header-row">
    <div class="badge">
      <span>✦</span> Ratio of Integers (Rational)
    </div>
  </div>

  
  <div class="preset-bar" id="presetBar">
    <button class="pill-btn active" data-num="3" data-den="4" data-type="term" data-res="0.75" data-alt="">3/4</button>
    <button class="pill-btn" data-num="2" data-den="3" data-type="rep" data-res="0.666..." data-alt="0.6̄">2/3</button>
    <button class="pill-btn" data-num="-5" data-den="8" data-type="term" data-res="-0.625" data-alt="">-5/8</button>
    <button class="pill-btn" data-num="1" data-den="6" data-type="rep" data-res="0.166..." data-alt="0.16̄">1/6</button>
    <button class="pill-btn" data-num="7" data-den="9" data-type="rep" data-res="0.777..." data-alt="0.7̄">7/9</button>
  </div>

  
  <div class="main-fraction-card">
    <div class="fraction-display">
      <span class="frac-num" id="fracNum">3</span>
      <div class="frac-line"></div>
      <span class="frac-den" id="fracDen">4</span>
    </div>
    <div class="labels-col">
      <div class="label-row">
        <span class="label-tag tag-num">Numerator</span>
        <span class="label-desc">Integer <b id="valA">a = 3</b></span>
      </div>
      <div class="label-row">
        <span class="label-tag tag-den">Denominator</span>
        <span class="label-desc">Non-zero <b id="valB">b = 4</b></span>
      </div>
    </div>
  </div>

  
  <div class="branches-container">
    
    <div class="branch-card active" id="cardTerm">
      <div class="branch-title term-title">Terminating Decimal</div>
      <div class="branch-formula" id="formulaTerm">3 ÷ 4 = 0.75</div>
      <div class="branch-sub">Stops cleanly</div>
    </div>

    
    <div class="branch-card" id="cardRep">
      <div class="branch-title rep-title">Repeating Decimal</div>
      <div class="branch-formula" id="formulaRep">2 ÷ 3 = 0.6̄</div>
      <div class="branch-sub">Periodic pattern</div>
    </div>
  </div>

  <script>
    const buttons = document.querySelectorAll('.pill-btn');
    const fracNum = document.getElementById('fracNum');
    const fracDen = document.getElementById('fracDen');
    const valA = document.getElementById('valA');
    const valB = document.getElementById('valB');
    const cardTerm = document.getElementById('cardTerm');
    const cardRep = document.getElementById('cardRep');
    const formulaTerm = document.getElementById('formulaTerm');
    const formulaRep = document.getElementById('formulaRep');

    buttons.forEach(btn => {
      btn.addEventListener('click', () => {
        buttons.forEach(b => b.classList.remove('active'));
        btn.classList.add('active');

        const num = btn.dataset.num;
        const den = btn.dataset.den;
        const type = btn.dataset.type;
        const res = btn.dataset.res;
        const alt = btn.dataset.alt;

        fracNum.textContent = num;
        fracDen.textContent = den;
        valA.textContent = `a = ${num}`;
        valB.textContent = `b = ${den}`;

        if (type === 'term') {
          cardTerm.classList.add('active');
          cardRep.classList.remove('active');
          formulaTerm.textContent = `${num} ÷ ${den} = ${res}`;
          formulaRep.textContent = '2 ÷ 3 = 0.6̄';
        } else {
          cardRep.classList.add('active');
          cardTerm.classList.remove('active');
          formulaRep.textContent = `${num} ÷ ${den} = ${alt || res}`;
          formulaTerm.textContent = '3 ÷ 4 = 0.75';
        }
      });
    });
  </script>
</body>
</html>

How do we know whether a fraction will stop or repeat forever when we divide it out?

To turn any fraction $\frac{a}{b}$ into a decimal, divide the top number $a$ by the bottom number $b$ using long division.

A **terminating decimal** ends because the division hits a remainder of zero, like $\frac{3}{8} = 0.375$.

A **repeating decimal** never ends because the remainders start looping, creating a repeating block of digits indicated with a top bar (a **vinculum**), like $\frac{4}{11} = 0.3636\dots = 0.\overline{36}$.

<!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; }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      color: #1e2945;
      background: transparent;
      padding: 8px;
    }
    .wrapper {
      display: flex;
      flex-direction: column;
      gap: 8px;
      max-width: 600px;
      margin: 0 auto;
    }
    .controls {
      display: flex;
      justify-content: center;
      align-items: center;
      gap: 6px;
    }
    .step-btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      border-radius: 6px;
      padding: 4px 10px;
      font-size: 13px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
    }
    .step-btn.active {
      background: #2563eb;
      border-color: #2563eb;
      color: #ffffff;
    }
    .play-btn {
      background: #2563eb;
      color: white;
      border: none;
      border-radius: 6px;
      padding: 4px 12px;
      font-size: 13px;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 4px;
    }
    .panels {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 8px;
    }
    @media (max-width: 440px) {
      .panels { gap: 4px; }
    }
    .panel {
      background: #ffffff;
      border: 1.5px solid #e2e8f0;
      border-radius: 10px;
      padding: 10px 8px;
      display: flex;
      flex-direction: column;
      align-items: center;
      min-height: 290px;
      position: relative;
    }
    .panel-header {
      font-size: 15px;
      font-weight: 700;
      margin-bottom: 8px;
      color: #0f172a;
    }
    .long-division {
      font-family: ui-monospace, SFMono-Regular, Menlo, Monaco, Consolas, monospace;
      font-size: 14px;
      line-height: 1.25;
      position: relative;
      width: 100%;
      max-width: 150px;
      margin: 0 auto;
    }
    .row {
      display: flex;
      justify-content: flex-end;
      min-height: 18px;
    }
    .char {
      width: 11px;
      text-align: center;
      display: inline-block;
      transition: opacity 0.3s ease;
    }
    .bracket-row {
      display: flex;
      align-items: center;
      justify-content: flex-end;
    }
    .divisor {
      font-weight: 600;
      padding-right: 4px;
    }
    .dividend-box {
      border-top: 2px solid #1e2945;
      border-left: 2px solid #1e2945;
      border-top-left-radius: 4px;
      padding-left: 3px;
      display: flex;
    }
    .sub-line {
      border-bottom: 1.5px solid #64748b;
      margin-top: 1px;
      margin-bottom: 1px;
    }
    .rem-highlight-green {
      background: #dcfce7;
      color: #15803d;
      font-weight: 700;
      border-radius: 3px;
      padding: 0 1px;
    }
    .rem-highlight-purple {
      background: #f3e8ff;
      color: #7e22ce;
      font-weight: 700;
      border-radius: 3px;
      padding: 0 1px;
    }
    .badge {
      margin-top: auto;
      padding: 6px 8px;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 700;
      text-align: center;
      width: 100%;
      transition: opacity 0.3s ease;
    }
    .badge-terminating {
      background: #dcfce7;
      border: 1px solid #86efac;
      color: #166534;
    }
    .badge-repeating {
      background: #f3e8ff;
      border: 1px solid #d8b4fe;
      color: #6b21a8;
    }
    .loop-svg {
      position: absolute;
      right: 0px;
      top: 112px;
      width: 32px;
      height: 90px;
      pointer-events: none;
      transition: opacity 0.3s ease;
    }
    .hidden {
      opacity: 0;
      visibility: hidden;
    }
  </style>
</head>
<body>

<div class="wrapper">
  <div class="controls">
    <button class="step-btn active" onclick="setStep(0)">Step 1</button>
    <button class="step-btn" onclick="setStep(1)">Step 2</button>
    <button class="step-btn" onclick="setStep(2)">Step 3</button>
    <button class="step-btn" onclick="setStep(3)">Step 4</button>
    <button class="play-btn" id="playBtn" onclick="togglePlay()">⏸</button>
  </div>

  <div class="panels">
    
    <div class="panel">
      <div class="panel-header">3 ÷ 8</div>
      
      <div class="long-division">
        
        <div class="row">
          <span class="char"></span>
          <span class="char"></span>
          <span class="char">0</span>
          <span class="char">.</span>
          <span class="char s1">3</span>
          <span class="char s2">7</span>
          <span class="char s3">5</span>
        </div>
        
        
        <div class="bracket-row">
          <span class="divisor">8</span>
          <div class="dividend-box">
            <span class="char">3</span>
            <span class="char">.</span>
            <span class="char">0</span>
            <span class="char s2">0</span>
            <span class="char s3">0</span>
          </div>
        </div>

        
        <div class="row s1">
          <span class="char">-</span>
          <span class="char">2</span>
          <span class="char">4</span>
          <span class="char"></span>
          <span class="char"></span>
        </div>
        <div class="sub-line s1"></div>
        <div class="row s1">
          <span class="char"></span>
          <span class="char">6</span>
          <span class="char s2">0</span>
          <span class="char"></span>
        </div>

        
        <div class="row s2">
          <span class="char">-</span>
          <span class="char">5</span>
          <span class="char">6</span>
          <span class="char"></span>
        </div>
        <div class="sub-line s2"></div>
        <div class="row s2">
          <span class="char"></span>
          <span class="char"></span>
          <span class="char">4</span>
          <span class="char s3">0</span>
        </div>

        
        <div class="row s3">
          <span class="char"></span>
          <span class="char">-</span>
          <span class="char">4</span>
          <span class="char">0</span>
        </div>
        <div class="sub-line s3"></div>
        <div class="row s3">
          <span class="char"></span>
          <span class="char"></span>
          <span class="char"></span>
          <span class="char rem-highlight-green s3">0</span>
        </div>
      </div>

      <div class="badge badge-terminating s3">
        Remainder = 0<br>Terminating
      </div>
    </div>

    
    <div class="panel">
      <div class="panel-header">4 ÷ 11</div>

      <div class="long-division">
        
        <div class="row">
          <span class="char"></span>
          <span class="char"></span>
          <span class="char"></span>
          <span class="char">0</span>
          <span class="char">.</span>
          <span class="char s1">3</span>
          <span class="char s2">6</span>
          <span class="char s3">3</span>
          <span class="char s4">6</span>
        </div>

        
        <div class="bracket-row">
          <span class="divisor">11</span>
          <div class="dividend-box">
            <span class="char">4</span>
            <span class="char">.</span>
            <span class="char">0</span>
            <span class="char s2">0</span>
            <span class="char s3">0</span>
            <span class="char s4">0</span>
          </div>
        </div>

        
        <div class="row s1">
          <span class="char">-</span>
          <span class="char">3</span>
          <span class="char">3</span>
          <span class="char"></span>
          <span class="char"></span>
          <span class="char"></span>
        </div>
        <div class="sub-line s1"></div>
        <div class="row s1">
          <span class="char"></span>
          <span class="char rem-highlight-purple">7</span>
          <span class="char s2">0</span>
          <span class="char"></span>
          <span class="char"></span>
        </div>

        
        <div class="row s2">
          <span class="char">-</span>
          <span class="char">6</span>
          <span class="char">6</span>
          <span class="char"></span>
          <span class="char"></span>
        </div>
        <div class="sub-line s2"></div>
        <div class="row s2">
          <span class="char"></span>
          <span class="char"></span>
          <span class="char rem-highlight-purple">4</span>
          <span class="char s3">0</span>
          <span class="char"></span>
        </div>

        
        <div class="row s3">
          <span class="char"></span>
          <span class="char">-</span>
          <span class="char">3</span>
          <span class="char">3</span>
          <span class="char"></span>
        </div>
        <div class="sub-line s3"></div>
        <div class="row s3">
          <span class="char"></span>
          <span class="char"></span>
          <span class="char rem-highlight-purple">7</span>
          <span class="char s4">0</span>
        </div>

        
        <div class="row s4">
          <span class="char"></span>
          <span class="char">-</span>
          <span class="char">6</span>
          <span class="char">6</span>
        </div>
        <div class="sub-line s4"></div>
        <div class="row s4">
          <span class="char"></span>
          <span class="char"></span>
          <span class="char"></span>
          <span class="char rem-highlight-purple">4</span>
        </div>
      </div>

      
      <svg class="loop-svg s3" viewBox="0 0 32 90">
        <path d="M 4,10 C 28,15 28,65 6,75" fill="none" stroke="#9333ea" stroke-width="2" stroke-dasharray="3,3" />
        <polyline points="10,70 4,76 12,80" fill="none" stroke="#9333ea" stroke-width="2" />
      </svg>

      <div class="badge badge-repeating s4">
        Repeats 7, 4...<br>0.3̄6̄
      </div>
    </div>
  </div>
</div>

<script>
  let currentStep = 0;
  let isPlaying = true;
  let timer = null;

  function updateView() {
    // Update step buttons
    document.querySelectorAll('.step-btn').forEach((btn, idx) => {
      btn.classList.toggle('active', idx === currentStep);
    });

    // Elements visibility for step 1, 2, 3, 4
    const s1 = currentStep >= 0;
    const s2 = currentStep >= 1;
    const s3 = currentStep >= 2;
    const s4 = currentStep >= 3;

    document.querySelectorAll('.s1').forEach(el => el.classList.toggle('hidden', !s1));
    document.querySelectorAll('.s2').forEach(el => el.classList.toggle('hidden', !s2));
    document.querySelectorAll('.s3').forEach(el => el.classList.toggle('hidden', !s3));
    document.querySelectorAll('.s4').forEach(el => el.classList.toggle('hidden', !s4));
  }

  function setStep(step) {
    currentStep = step;
    updateView();
  }

  function togglePlay() {
    isPlaying = !isPlaying;
    document.getElementById('playBtn').textContent = isPlaying ? '⏸' : '▶';
  }

  // Auto-play loop
  timer = setInterval(() => {
    if (isPlaying) {
      currentStep = (currentStep + 1) % 4;
      updateView();
    }
  }, 3200);

  updateView();
</script>

</body>
</html>

Converting fractions to decimals is straightforward, but what if you are handed a decimal and need to work backwards to find its fraction?

For a terminating decimal, write the digits over their corresponding **place value** power of 10 ($10, 100, 1000$) and simplify.

For example, in $0.375$, the last digit is in the thousandths place, so you write $\frac{375}{1000}$ and divide the top and bottom by $125$ to get $\frac{3}{8}$.

What happens when the decimal goes on forever — how can we put infinitely many digits over a power of 10?

We can eliminate infinite repeating tails using an algebraic subtraction trick: multiply by a power of 10 to shift the repeating block, then subtract the original number.

Let's convert $0.\overline{7} = 0.777\dots$ into a fraction step by step.

<!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;
      color: #1e2945;
      background: #ffffff;
      padding: 16px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      user-select: none;
    }

    .stepper-nav {
      display: flex;
      gap: 8px;
      margin-bottom: 12px;
      width: 100%;
      max-width: 420px;
      justify-content: center;
    }

    .step-btn {
      flex: 1;
      padding: 8px 4px;
      border: 1px solid #e2e8f0;
      background: #f8fafc;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      text-align: center;
      transition: all 0.2s ease;
    }

    .step-btn.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
      box-shadow: 0 2px 8px rgba(34, 183, 255, 0.3);
    }

    .stage-container {
      width: 100%;
      max-width: 420px;
      min-height: 230px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      padding: 16px;
      position: relative;
    }

    .math-row {
      font-size: 20px;
      font-family: "KaTeX_Main", "Times New Roman", serif;
      display: flex;
      align-items: center;
      justify-content: center;
      margin: 4px 0;
      transition: opacity 0.3s ease, transform 0.3s ease;
    }

    /* Stacked subtraction layout */
    .calc-table {
      font-size: 20px;
      font-family: "KaTeX_Main", "Times New Roman", serif;
      display: inline-grid;
      grid-template-columns: auto auto auto;
      column-gap: 8px;
      row-gap: 6px;
      align-items: center;
      text-align: right;
    }

    .op-sign {
      color: #64748b;
      font-weight: 600;
      text-align: left;
    }

    .dec-part {
      position: relative;
      display: inline-block;
      text-align: left;
    }

    .strike-line {
      position: absolute;
      top: 50%;
      left: -4px;
      right: -4px;
      height: 3px;
      background: #ef4444;
      border-radius: 2px;
      transform: translateY(-50%) rotate(-6deg) scaleX(0);
      transform-origin: left center;
      transition: transform 0.4s cubic-bezier(0.34, 1.56, 0.64, 1);
    }

    .strike-line.active {
      transform: translateY(-50%) rotate(-6deg) scaleX(1);
    }

    .calc-divider {
      grid-column: 1 / -1;
      height: 2px;
      background: #1e2945;
      margin: 2px 0;
      border-radius: 1px;
    }

    .final-box {
      margin-top: 10px;
      padding: 8px 18px;
      background: #ffffff;
      border: 2px solid #22b7ff;
      border-radius: 10px;
      box-shadow: 0 4px 12px rgba(34, 183, 255, 0.18);
      transform: scale(0.95);
      opacity: 0;
      transition: all 0.3s ease;
    }

    .final-box.show {
      transform: scale(1);
      opacity: 1;
    }

    .controls {
      display: flex;
      align-items: center;
      gap: 12px;
      margin-top: 12px;
    }

    .ctrl-btn {
      padding: 6px 14px;
      border-radius: 20px;
      border: 1px solid #cbd5e1;
      background: #ffffff;
      font-size: 13px;
      font-weight: 600;
      color: #334155;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 4px;
    }
  </style>
</head>
<body>

  <div class="stepper-nav">
    <button class="step-btn active" onclick="goToStep(0)">1. Let x</button>
    <button class="step-btn" onclick="goToStep(1)">2. &times;10</button>
    <button class="step-btn" onclick="goToStep(2)">3. Subtract</button>
    <button class="step-btn" onclick="goToStep(3)">4. Solve</button>
  </div>

  <div class="stage-container" id="stage">
    
  </div>

  <div class="controls">
    <button class="ctrl-btn" id="playBtn" onclick="togglePlay()">⏸ Pause</button>
  </div>

  <script>
    let currentStep = 0;
    let timer = null;
    let isPlaying = true;
    const TOTAL_STEPS = 4;
    const STEP_DURATION = 3500;

    function renderStep(idx) {
      const stage = document.getElementById('stage');
      const btns = document.querySelectorAll('.step-btn');
      btns.forEach((b, i) => b.classList.toggle('active', i === idx));

      if (idx === 0) {
        stage.innerHTML = `
          <div class="math-row" style="font-size: 24px;">
            <span id="eq1">x = 0.777...</span>
          </div>
        `;
      } else if (idx === 1) {
        stage.innerHTML = `
          <div class="math-row" style="color: #64748b; font-size: 18px; margin-bottom: 8px;">
            10 &times; (x = 0.777...)
          </div>
          <div class="math-row" style="font-size: 24px; color: #0284c7; font-weight: bold;">
            10x = 7.777...
          </div>
        `;
      } else if (idx === 2) {
        stage.innerHTML = `
          <div class="calc-table">
            <span class="op-sign"></span>
            <span>10x</span>
            <span>= 7<span class="dec-part">.777...<span class="strike-line" id="slash1"></span></span></span>

            <span class="op-sign">&minus;</span>
            <span>x</span>
            <span>= 0<span class="dec-part">.777...<span class="strike-line" id="slash2"></span></span></span>

            <div class="calc-divider"></div>

            <span></span>
            <span style="font-weight:bold; color:#0284c7;">9x</span>
            <span style="font-weight:bold; color:#0284c7; text-align: left;">= 7</span>
          </div>
        `;
        setTimeout(() => {
          const s1 = document.getElementById('slash1');
          const s2 = document.getElementById('slash2');
          if (s1 && s2) {
            s1.classList.add('active');
            s2.classList.add('active');
          }
        }, 300);
      } else if (idx === 3) {
        stage.innerHTML = `
          <div class="math-row" style="color: #64748b; font-size: 18px; margin-bottom: 6px;">
            9x = 7
          </div>
          <div class="final-box" id="finalBox">
            <div class="math-row" style="font-size: 26px; color: #0284c7; margin: 0;">
              x = <span style="display:inline-flex; flex-direction:column; vertical-align:middle; text-align:center; margin-left:6px; font-size: 0.9em; line-height: 1.1;">
                <span>7</span>
                <span style="border-top:2px solid #0284c7;">9</span>
              </span>
            </div>
          </div>
        `;
        setTimeout(() => {
          const box = document.getElementById('finalBox');
          if (box) box.classList.add('show');
        }, 100);
      }
    }

    function goToStep(idx) {
      currentStep = idx;
      renderStep(currentStep);
    }

    function nextStep() {
      currentStep = (currentStep + 1) % TOTAL_STEPS;
      renderStep(currentStep);
    }

    function startAutoPlay() {
      if (timer) clearInterval(timer);
      timer = setInterval(nextStep, STEP_DURATION);
    }

    function togglePlay() {
      isPlaying = !isPlaying;
      const btn = document.getElementById('playBtn');
      if (isPlaying) {
        btn.textContent = '⏸ Pause';
        startAutoPlay();
      } else {
        btn.textContent = '▶ Play';
        clearInterval(timer);
      }
    }

    // Initialize
    renderStep(0);
    startAutoPlay();
  </script>
</body>
</html>

If two digits repeat, like $x = 0.\overline{45}$, multiply by $100$ so the repeating blocks line up: $100x - x = 45.\overline{45} - 0.\overline{45}$, giving $99x = 45$, or $x = \frac{45}{99} = \frac{5}{11}$.

What if the repeating pattern doesn't start immediately after the decimal point, like in $0.1\overline{6}$?

When non-repeating digits sit in front of the pattern, multiply by powers of 10 to isolate both the start and the end of the repeating block.

Let $x = 0.1\overline{6}$. Multiply by $100$ to get $100x = 16.\overline{6}$ and by $10$ to get $10x = 1.\overline{6}$.

Subtracting gives $100x - 10x = 16.\overline{6} - 1.\overline{6}$, which simplifies to $90x = 15$, giving $x = \frac{15}{90} = \frac{1}{6}$.

A common mistake here is subtracting $100x - x$, which leaves $99x = 16.5$ — always choose powers of 10 that make the decimal tails match completely so they cancel to whole numbers.

Is it pure luck that every fraction turns into a repeating or terminating decimal, or is there a mathematical guarantee?

## Why Fractions Must Repeat or Terminate

When you divide an integer $a$ by a non-zero integer $b$, the only possible remainders are $0, 1, 2, \dots, b-1$.

This relies on the **Pigeonhole Principle** — first formalized by mathematician Peter Gustav Lejeune Dirichlet in 1834 — which states that if you place more items into containers than there are containers, at least one container must hold more than one item.

<!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>Pigeonhole Principle - Decimal Cycles</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      background: #ffffff;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      user-select: none;
    }

    .header {
      font-size: 16px;
      font-weight: 700;
      color: #1e2945;
      text-align: center;
      margin-bottom: 6px;
    }

    .formula-tag {
      display: inline-block;
      background: #f1f5f9;
      padding: 2px 8px;
      border-radius: 6px;
      font-family: monospace;
      font-size: 15px;
      color: #0284c7;
      border: 1px solid #e2e8f0;
    }

    /* Remainder Slots (Pigeonholes) */
    .slots-container {
      width: 100%;
      max-width: 480px;
      display: grid;
      grid-template-columns: 1.1fr repeat(6, 1fr);
      gap: 5px;
      margin: 8px 0;
    }

    .slot {
      background: #f8fafc;
      border: 2px dashed #cbd5e1;
      border-radius: 8px;
      padding: 6px 2px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 68px;
      transition: all 0.3s ease;
      position: relative;
    }

    .slot.term-slot {
      border-color: #86efac;
      background: #f0fdf4;
    }

    .slot-label {
      font-size: 12px;
      font-weight: 700;
      color: #64748b;
    }

    .slot.term-slot .slot-label {
      color: #16a34a;
    }

    .slot-content {
      font-size: 18px;
      min-height: 24px;
      display: flex;
      align-items: center;
      justify-content: center;
      transition: transform 0.3s ease;
    }

    .slot.occupied {
      background: #eff6ff;
      border-color: #38bdf8;
      border-style: solid;
    }

    .slot.collision {
      background: #fef2f2;
      border-color: #f87171;
      border-style: solid;
      box-shadow: 0 0 0 2px rgba(248, 113, 113, 0.4);
    }

    /* Main Visual Stage */
    .stage-area {
      width: 100%;
      max-width: 480px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 10px 14px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 125px;
      position: relative;
    }

    .step-view {
      display: none;
      width: 100%;
      text-align: center;
      animation: fadeIn 0.3s ease;
    }

    .step-view.active {
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      gap: 6px;
    }

    @keyframes fadeIn {
      from { opacity: 0; transform: translateY(4px); }
      to { opacity: 1; transform: translateY(0); }
    }

    .badge-pill {
      font-size: 13px;
      font-weight: 700;
      padding: 3px 10px;
      border-radius: 20px;
      display: inline-flex;
      align-items: center;
      gap: 5px;
    }
    .badge-green { background: #dcfce7; color: #15803d; }
    .badge-blue { background: #e0f2fe; color: #0369a1; }
    .badge-orange { background: #ffedd5; color: #c2410c; }
    .badge-purple { background: #f3e8ff; color: #7e22ce; }

    .step-desc {
      font-size: 14px;
      font-weight: 600;
      color: #334155;
    }

    .cycle-circle {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      font-family: monospace;
      font-size: 16px;
      font-weight: 700;
      color: #0284c7;
      background: #ffffff;
      padding: 6px 14px;
      border-radius: 30px;
      border: 1px solid #bae6fd;
    }

    .loop-arrow {
      font-size: 18px;
      color: #0284c7;
      animation: spinSlow 6s linear infinite;
    }
    @keyframes spinSlow {
      100% { transform: rotate(360deg); }
    }

    /* Stepper Controls */
    .controls {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 10px;
      margin-top: 8px;
      width: 100%;
      max-width: 480px;
    }

    .btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      color: #334155;
      padding: 6px 14px;
      border-radius: 8px;
      font-size: 14px;
      font-weight: 600;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 4px;
      transition: background 0.2s, border-color 0.2s;
    }
    .btn:active { background: #f1f5f9; }
    .btn-play {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
    }
    .btn-play:active { background: #0369a1; }

    .dots-nav {
      display: flex;
      gap: 6px;
    }

    .dot-btn {
      width: 10px;
      height: 10px;
      border-radius: 50%;
      background: #cbd5e1;
      border: none;
      cursor: pointer;
      padding: 0;
      transition: background 0.2s;
    }

    .dot-btn.active {
      background: #0284c7;
      transform: scale(1.15);
    }
  </style>
</head>
<body>

  <div class="header">
    Divide by <span class="formula-tag">b = 7</span>: Exactly 7 Remainders
  </div>

  
  <div class="slots-container">
    <div class="slot term-slot" id="slot-0">
      <span class="slot-label">R = 0</span>
      <span class="slot-content" id="pigeon-0">🛑</span>
    </div>
    <div class="slot" id="slot-1">
      <span class="slot-label">R = 1</span>
      <span class="slot-content" id="pigeon-1"></span>
    </div>
    <div class="slot" id="slot-2">
      <span class="slot-label">R = 2</span>
      <span class="slot-content" id="pigeon-2"></span>
    </div>
    <div class="slot" id="slot-3">
      <span class="slot-label">R = 3</span>
      <span class="slot-content" id="pigeon-3"></span>
    </div>
    <div class="slot" id="slot-4">
      <span class="slot-label">R = 4</span>
      <span class="slot-content" id="pigeon-4"></span>
    </div>
    <div class="slot" id="slot-5">
      <span class="slot-label">R = 5</span>
      <span class="slot-content" id="pigeon-5"></span>
    </div>
    <div class="slot" id="slot-6">
      <span class="slot-label">R = 6</span>
      <span class="slot-content" id="pigeon-6"></span>
    </div>
  </div>

  
  <div class="stage-area">
    
    <div class="step-view active" id="step-view-0">
      <div class="badge-pill badge-green">✓ Case 1: Remainder = 0</div>
      <div class="step-desc">Decimal terminates cleanly</div>
      <div style="font-family: monospace; font-size: 15px; color: #16a34a; font-weight: 700;">
        1 ÷ 4 = 0.25 (R = 0)
      </div>
    </div>

    
    <div class="step-view" id="step-view-1">
      <div class="badge-pill badge-blue">Case 2: 6 Non-Zero Slots</div>
      <div class="step-desc">Remainders {1, 2, 3, 4, 5, 6} available</div>
      <div style="font-size: 13px; color: #64748b; font-weight: 600;">
        🕊️ Steps 1–3 fill empty holes
      </div>
    </div>

    
    <div class="step-view" id="step-view-2">
      <div class="badge-pill badge-blue">Steps 1–6: All Slots Filled</div>
      <div class="step-desc">At most 6 steps without zero</div>
      <div style="font-size: 13px; color: #64748b; font-weight: 600;">
        6 holes completely occupied!
      </div>
    </div>

    
    <div class="step-view" id="step-view-3">
      <div class="badge-pill badge-orange">✦ 7th Step: Pigeonhole Principle!</div>
      <div class="step-desc">7th remainder MUST repeat an occupied slot</div>
      <div style="font-size: 13px; color: #c2410c; font-weight: 700;">
        Overlap forces duplicate remainder
      </div>
    </div>

    
    <div class="step-view" id="step-view-4">
      <div class="badge-pill badge-purple">🔁 Endless Decimal Loop</div>
      <div class="cycle-circle">
        <span class="loop-arrow">↻</span>
        <span>1 ÷ 7 = 0.142857142857...</span>
      </div>
    </div>
  </div>

  
  <div class="controls">
    <button class="btn" id="prevBtn" aria-label="Previous step">←</button>
    <div class="dots-nav">
      <button class="dot-btn active" data-idx="0"></button>
      <button class="dot-btn" data-idx="1"></button>
      <button class="dot-btn" data-idx="2"></button>
      <button class="dot-btn" data-idx="3"></button>
      <button class="dot-btn" data-idx="4"></button>
    </div>
    <button class="btn" id="nextBtn" aria-label="Next step">→</button>
    <button class="btn btn-play" id="playBtn">⏸</button>
  </div>

  <script>
    const totalSteps = 5;
    let currentStep = 0;
    let isPlaying = true;
    let timer = null;

    const views = [0, 1, 2, 3, 4].map(i => document.getElementById(`step-view-${i}`));
    const dots = document.querySelectorAll('.dot-btn');
    const playBtn = document.getElementById('playBtn');

    function updateSlots(step) {
      // Clear all non-zero slots
      for (let i = 1; i <= 6; i++) {
        const slot = document.getElementById(`slot-${i}`);
        const pigeon = document.getElementById(`pigeon-${i}`);
        slot.className = 'slot';
        pigeon.textContent = '';
      }

      const slot0 = document.getElementById('slot-0');
      slot0.className = 'slot term-slot';

      if (step === 0) {
        slot0.className = 'slot term-slot collision';
      } else if (step === 1) {
        // First few remainders for 1/7: 3, 2, 6
        ['3', '2', '6'].forEach(id => {
          document.getElementById(`slot-${id}`).className = 'slot occupied';
          document.getElementById(`pigeon-${id}`).textContent = '🕊️';
        });
      } else if (step === 2) {
        // All 6 non-zero slots filled
        for (let i = 1; i <= 6; i++) {
          document.getElementById(`slot-${i}`).className = 'slot occupied';
          document.getElementById(`pigeon-${i}`).textContent = '🕊️';
        }
      } else if (step === 3 || step === 4) {
        // Collision on remainder 3
        for (let i = 1; i <= 6; i++) {
          document.getElementById(`slot-${i}`).className = 'slot occupied';
          document.getElementById(`pigeon-${i}`).textContent = '🕊️';
        }
        const collSlot = document.getElementById('slot-3');
        collSlot.className = 'slot collision';
        document.getElementById('pigeon-3').textContent = '🕊️🕊️';
      }
    }

    function render(step) {
      views.forEach((v, idx) => {
        v.classList.toggle('active', idx === step);
      });
      dots.forEach((d, idx) => {
        d.classList.toggle('active', idx === step);
      });
      updateSlots(step);
    }

    function setStep(newStep) {
      currentStep = (newStep + totalSteps) % totalSteps;
      render(currentStep);
    }

    function nextStep() {
      setStep(currentStep + 1);
    }

    function prevStep() {
      setStep(currentStep - 1);
    }

    function startTimer() {
      if (timer) clearInterval(timer);
      timer = setInterval(nextStep, 3500);
      playBtn.textContent = '⏸';
      isPlaying = true;
    }

    function pauseTimer() {
      if (timer) clearInterval(timer);
      playBtn.textContent = '▶';
      isPlaying = false;
    }

    document.getElementById('nextBtn').addEventListener('click', () => {
      pauseTimer();
      nextStep();
    });

    document.getElementById('prevBtn').addEventListener('click', () => {
      pauseTimer();
      prevStep();
    });

    playBtn.addEventListener('click', () => {
      if (isPlaying) pauseTimer();
      else startTimer();
    });

    dots.forEach((dot, idx) => {
      dot.addEventListener('click', () => {
        pauseTimer();
        setStep(idx);
      });
    });

    // Initialize
    render(0);
    startTimer();
  </script>
</body>
</html>

Because there are only $b-1$ possible non-zero remainders, long division must either hit $0$ or repeat an earlier remainder within $b-1$ steps.

This proves that every single rational number is guaranteed to be either a terminating decimal or a repeating decimal — no exceptions.

math

Rational Numbers as Fractions and Terminating or Repeating Decimals. Express rational numbers as fractions and as terminating or repeating decimals.

Sums and Products of Rational and Irrational Numbers (HSN.RN.B.3)

Converting Repeating Decimals to Fractions & Repeating or Terminating?

Writing Repeating Decimals as Fractions Review & Practice

In high school (9th-10th grade), teach this standard by transitioning students from mechanical long division to algebraic and structural reasoning. Connect decimal representations directly to denominator prime factorizations ($2^m 5^n$) and the Pigeonhole Principle. Emphasize multi-step algebraic elimination ($10^k x - 10^j x$) for mixed repeating decimals, and reinforce CCSS HSN.RN.B.3 by using these fractional representations in formal proofs by contradiction showing the irrationality of sums and products.

Rational Numbers as Fractions and Terminating or Repeating Decimals

What Makes a Number Rational?

Fractions to Decimals