Advanced

Tutor model

i62 💰:9.4

i52 💰:7.9

🏆 i67 💰:62.5

i70 💰:125.0

i68 💰:125.0

🏆 i66 💰:75.0

i65 💰:125.0

i63 💰:62.5

i61 💰:75.0

i60 💰:9.4

i60 💰:37.5

i58 💰:62.5

i57 💰:22.5

i57 💰:37.5

i56 💰:45.0

⚡ i55 💰:11.25

i55 💰:37.5

i54 💰:62.5

i52 💰:35.0

i51 💰:37.5

i49 💰:15.0

i49 💰:62.5

i48 💰:11.3

i47 💰:7.5

i44 💰:25.0

i43 💰:37.5

i42 💰:187.5

i40 💰:5.0

i39 💰:100.0

i38 💰:37.5

i37 💰:11.0

i36 💰:187.5

i34 💰:12.5

i33 💰:37.5

i30 💰:1.0

i29 💰:1.5

i28 💰:150.0

i23 💰:20.0

i20 💰:4.0

i18 💰:1.0

i16 💰:25.0

i13 💰:1.5

Keep session private

What would you like to learn?

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

Lesson generation model

System default

i58 💰:18.8

i31 💰:11.0

i25 💰:31.3

Tutoring abilities

The observable universe is about $93,000,000,000$ light-years across, while a single hydrogen atom spans just $0.000000000106\text{ meters}$. Writing out dozens of trailing or leading zeros wastes time and easily leads to counting errors.

To solve this, scientists use **scientific notation**, a standard format where every number is written as a decimal between 1 and 10 multiplied by a power of 10.

In scientific notation, every value is expressed as $a \times 10^n$. The number $a$ is the **coefficient** (a decimal where $1 \le |a| < 10$), and $n$ is an integer **exponent** that indicates how many places the decimal moved.

<!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">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
      user-select: none;
    }

    .controls {
      display: flex;
      gap: 8px;
      margin-bottom: 12px;
      width: 100%;
      max-width: 460px;
    }

    .tab-btn {
      flex: 1;
      padding: 8px 12px;
      font-size: 14px;
      font-weight: 600;
      border: 1px solid #e2e8f0;
      background: #ffffff;
      color: #64748b;
      border-radius: 8px;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      transition: all 0.2s ease;
    }

    .tab-btn.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
      box-shadow: 0 2px 8px rgba(34, 183, 255, 0.25);
    }

    .stage {
      background: #ffffff;
      border-radius: 12px;
      box-shadow: 0 4px 16px rgba(0, 0, 0, 0.04);
      border: 1px solid #eef2f6;
      width: 100%;
      max-width: 460px;
      padding: 16px 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 16px;
    }

    .number-display {
      position: relative;
      height: 90px;
      width: 100%;
      display: flex;
      align-items: center;
      justify-content: center;
    }

    .digits-row {
      display: flex;
      font-size: 28px;
      font-family: ui-monospace, SFMono-Regular, Menlo, Monaco, Consolas, monospace;
      font-weight: 700;
      letter-spacing: 6px;
      position: relative;
      z-index: 2;
    }

    .digit {
      position: relative;
      width: 24px;
      text-align: center;
    }

    .digit.highlight-coeff {
      color: #0284c7;
    }

    .digit.dim {
      color: #94a3b8;
    }

    .jump-svg {
      position: absolute;
      top: 0;
      left: 0;
      width: 100%;
      height: 100%;
      pointer-events: none;
      z-index: 1;
    }

    .jump-path {
      fill: none;
      stroke: #8b5cf6;
      stroke-width: 2.5;
      stroke-linecap: round;
      stroke-dasharray: 60;
      stroke-dashoffset: 60;
      transition: stroke-dashoffset 0.35s ease;
    }

    .jump-path.drawn {
      stroke-dashoffset: 0;
    }

    .jump-label {
      font-size: 11px;
      font-weight: 700;
      fill: #8b5cf6;
      text-anchor: middle;
      opacity: 0;
      transition: opacity 0.25s ease;
    }

    .jump-label.visible {
      opacity: 1;
    }

    .decimal-pt {
      position: absolute;
      bottom: 22px;
      width: 8px;
      height: 8px;
      border-radius: 50%;
      background: #8b5cf6;
      transition: transform 0.35s cubic-bezier(0.34, 1.56, 0.64, 1);
      z-index: 3;
    }

    .result-box {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
      width: 100%;
      padding: 12px;
      background: #f8fafc;
      border-radius: 10px;
      border: 1px solid #e2e8f0;
    }

    .legend {
      display: flex;
      gap: 12px;
      font-size: 12px;
      font-weight: 600;
    }

    .legend-item {
      display: flex;
      align-items: center;
      gap: 5px;
    }

    .dot-badge {
      width: 8px;
      height: 8px;
      border-radius: 50%;
    }

    .dot-blue { background: #0284c7; }
    .dot-purple { background: #8b5cf6; }

    .play-row {
      display: flex;
      align-items: center;
      gap: 8px;
    }

    .btn-step {
      padding: 6px 14px;
      font-size: 13px;
      font-weight: 600;
      background: #ffffff;
      border: 1px solid #cbd5e1;
      border-radius: 6px;
      cursor: pointer;
      color: #334155;
    }
  </style>
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
</head>
<body>

  <div class="controls">
    <button class="tab-btn active" id="tabLarge">🔭 Large Scale</button>
    <button class="tab-btn" id="tabSmall">🔬 Small Scale</button>
  </div>

  <div class="stage">
    <div class="number-display" id="displayArea">
      <svg class="jump-svg" id="jumpSvg" viewBox="0 0 400 90"></svg>
      <div class="digits-row" id="digitsRow"></div>
      <div class="decimal-pt" id="decimalDot"></div>
    </div>

    <div class="result-box">
      <div id="mathTarget"></div>
    </div>

    <div class="legend">
      <div class="legend-item"><span class="dot-badge dot-blue"></span> 1 ≤ |a| &lt; 10</div>
      <div class="legend-item"><span class="dot-badge dot-purple"></span> 10<sup>n</sup> places</div>
    </div>

    <div class="play-row">
      <button class="btn-step" id="btnReplay">▶ Step Shift</button>
    </div>
  </div>

  <script>
    const MODES = {
      large: {
        rawDigits: ['5', '2', '0', '0', '0', '0'],
        initDotIdx: 6,
        finalDotIdx: 1,
        coeff: '5.2',
        exp: '5',
        dir: 'left'
      },
      small: {
        rawDigits: ['0', '.', '0', '0', '0', '4', '8'],
        initDotIdx: 1,
        finalDotIdx: 6,
        coeff: '4.8',
        exp: '-4',
        dir: 'right'
      }
    };

    let curMode = 'large';
    let animTimer = null;

    const digitsRow = document.getElementById('digitsRow');
    const jumpSvg = document.getElementById('jumpSvg');
    const decimalDot = document.getElementById('decimalDot');
    const mathTarget = document.getElementById('mathTarget');
    const tabLarge = document.getElementById('tabLarge');
    const tabSmall = document.getElementById('tabSmall');
    const btnReplay = document.getElementById('btnReplay');

    function renderMode(modeKey, animate = true) {
      curMode = modeKey;
      clearTimeout(animTimer);

      tabLarge.classList.toggle('active', modeKey === 'large');
      tabSmall.classList.toggle('active', modeKey === 'small');

      const config = MODES[modeKey];
      digitsRow.innerHTML = '';
      jumpSvg.innerHTML = '';

      config.rawDigits.forEach((d) => {
        const span = document.createElement('span');
        span.className = 'digit';
        span.textContent = d;
        if (modeKey === 'large') {
          if (d === '5' || d === '2') span.classList.add('highlight-coeff');
          else span.classList.add('dim');
        } else {
          if (d === '4' || d === '8') span.classList.add('highlight-coeff');
          else span.classList.add('dim');
        }
        digitsRow.appendChild(span);
      });

      katex.render(`${config.coeff} \\times 10^{${config.exp}}`, mathTarget, {
        throwOnError: false
      });

      requestAnimationFrame(() => {
        setupJumps(config, animate);
      });
    }

    function setupJumps(config, animate) {
      const charEls = digitsRow.querySelectorAll('.digit');
      if (!charEls.length) return;

      const containerRect = digitsRow.getBoundingClientRect();
      const svgRect = jumpSvg.getBoundingClientRect();

      const positions = [];
      charEls.forEach((el) => {
        const r = el.getBoundingClientRect();
        positions.push(r.left - svgRect.left);
        positions.push(r.right - svgRect.left);
      });

      const uniqueSlots = Array.from(new Set(positions)).sort((a, b) => a - b);
      const isLarge = config.dir === 'left';
      const stepsCount = Math.abs(parseInt(config.exp, 10));

      const startSlot = isLarge ? uniqueSlots[uniqueSlots.length - 1] : uniqueSlots[1];
      const endSlot = isLarge ? uniqueSlots[uniqueSlots.length - 1 - stepsCount] : uniqueSlots[1 + stepsCount];

      decimalDot.style.transition = 'none';
      decimalDot.style.left = `${startSlot - 4}px`;

      jumpSvg.innerHTML = '';

      const jumpSegments = [];
      for (let i = 0; i < stepsCount; i++) {
        let x1, x2;
        if (isLarge) {
          x1 = uniqueSlots[uniqueSlots.length - 1 - i];
          x2 = uniqueSlots[uniqueSlots.length - 2 - i];
        } else {
          x1 = uniqueSlots[1 + i];
          x2 = uniqueSlots[2 + i];
        }

        const midX = (x1 + x2) / 2;
        const topY = 25;
        const botY = 62;

        const path = document.createElementNS('http://www.w3.org/2000/svg', 'path');
        path.setAttribute('d', `M ${x1} ${botY} Q ${midX} ${topY} ${x2} ${botY}`);
        path.setAttribute('class', 'jump-path');
        jumpSvg.appendChild(path);

        const txt = document.createElementNS('http://www.w3.org/2000/svg', 'text');
        txt.setAttribute('x', midX);
        txt.setAttribute('y', topY - 4);
        txt.setAttribute('class', 'jump-label');
        txt.textContent = isLarge ? `-${i + 1}` : `+${i + 1}`;
        jumpSvg.appendChild(txt);

        jumpSegments.push({ path, txt, targetX: x2 });
      }

      if (!animate) {
        jumpSegments.forEach(s => {
          s.path.classList.add('drawn');
          s.txt.classList.add('visible');
        });
        decimalDot.style.left = `${endSlot - 4}px`;
        return;
      }

      let stepIdx = 0;
      function nextStep() {
        if (stepIdx < jumpSegments.length) {
          const seg = jumpSegments[stepIdx];
          seg.path.classList.add('drawn');
          seg.txt.classList.add('visible');
          decimalDot.style.transition = 'left 0.35s cubic-bezier(0.34, 1.56, 0.64, 1)';
          decimalDot.style.left = `${seg.targetX - 4}px`;
          stepIdx++;
          animTimer = setTimeout(nextStep, 420);
        }
      }

      animTimer = setTimeout(nextStep, 200);
    }

    tabLarge.addEventListener('click', () => renderMode('large', true));
    tabSmall.addEventListener('click', () => renderMode('small', true));
    btnReplay.addEventListener('click', () => renderMode(curMode, true));

    window.addEventListener('resize', () => renderMode(curMode, false));

    renderMode('large', true);
  </script>
</body>
</html>

In the 3rd century BCE, Archimedes first investigated powers of 10 in his treatise *The Sand Reckoner* to estimate how many grains of sand would fill the cosmos, laying the foundation for our modern exponent system.

How do we take a giant real-world measurement and rewrite it in this compact format without changing its value?

When a number is greater than 10, shift the decimal point to the left until only one non-zero digit remains in front of it. The number of places you shift becomes your **positive exponent**.

For example, the average distance from Earth to the Sun is approximately $149,600,000\text{ km}$. Moving the decimal point 8 places to the left gives $1.496 \times 10^8\text{ km}$.

<!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="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css">
  <script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.js"></script>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
      user-select: none;
    }

    .scene-header {
      display: flex;
      align-items: center;
      justify-content: space-between;
      width: 100%;
      max-width: 480px;
      padding: 6px 16px;
      background: #ffffff;
      border-radius: 12px;
      border: 1px solid #e6e6e6;
      margin-bottom: 12px;
      font-size: 15px;
      font-weight: 600;
    }

    .space-track {
      display: flex;
      align-items: center;
      gap: 8px;
    }

    .main-board {
      width: 100%;
      max-width: 480px;
      background: #ffffff;
      border: 1px solid #e6e6e6;
      border-radius: 16px;
      padding: 20px 14px 16px;
      display: flex;
      flex-direction: column;
      align-items: center;
    }

    .number-display {
      position: relative;
      width: 100%;
      max-width: 420px;
      height: 90px;
      margin-bottom: 8px;
    }

    .digits-row {
      display: flex;
      justify-content: space-between;
      align-items: center;
      position: absolute;
      bottom: 10px;
      left: 12px;
      right: 12px;
    }

    .digit-col {
      position: relative;
      width: 32px;
      text-align: center;
      font-size: 26px;
      font-weight: 700;
      font-family: "SF Mono", "Roboto Mono", monospace;
      color: #1e2945;
      transition: color 0.2s ease;
    }

    .digit-col.trailing {
      color: #94a3b8;
    }

    .digit-col.coefficient {
      color: #0284c7;
    }

    .svg-hops {
      position: absolute;
      top: 0;
      left: 0;
      width: 100%;
      height: 100%;
      pointer-events: none;
    }

    .hop-arc {
      fill: none;
      stroke: #22b7ff;
      stroke-width: 2.5;
      stroke-linecap: round;
      stroke-dasharray: 60;
      stroke-dashoffset: 60;
      transition: stroke-dashoffset 0.3s ease;
    }

    .hop-arc.active {
      stroke-dashoffset: 0;
    }

    .hop-num {
      font-size: 11px;
      font-weight: 700;
      fill: #0284c7;
      text-anchor: middle;
      opacity: 0;
      transition: opacity 0.2s ease;
    }

    .hop-num.active {
      opacity: 1;
    }

    .decimal-point {
      position: absolute;
      bottom: 12px;
      width: 9px;
      height: 9px;
      background: #ef4444;
      border-radius: 50%;
      transform: translateX(-50%);
      transition: left 0.35s cubic-bezier(0.4, 0, 0.2, 1);
      box-shadow: 0 0 0 3px rgba(239, 68, 68, 0.2);
      z-index: 5;
    }

    .formula-card {
      width: 100%;
      background: #f0f9ff;
      border: 1px solid #bae6fd;
      border-radius: 12px;
      padding: 12px 16px;
      display: flex;
      align-items: center;
      justify-content: center;
      min-height: 52px;
      margin: 10px 0 16px;
    }

    .formula-box {
      font-size: 22px;
      color: #0369a1;
      font-weight: 600;
    }

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

    .btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      border-radius: 8px;
      padding: 7px 14px;
      font-size: 14px;
      font-weight: 600;
      color: #334155;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 6px;
    }

    .btn:active {
      background: #e2e8f0;
    }

    .btn-primary {
      background: #22b7ff;
      border-color: #0ea5e9;
      color: #ffffff;
    }

    .btn-primary:active {
      background: #0284c7;
    }

    .step-pills {
      display: flex;
      gap: 4px;
      margin-top: 14px;
    }

    .pill {
      width: 24px;
      height: 6px;
      border-radius: 3px;
      background: #e2e8f0;
      cursor: pointer;
      transition: background 0.2s ease;
    }

    .pill.active {
      background: #0284c7;
    }
  </style>
</head>
<body>

  <div class="scene-header">
    <div class="space-track">🌍 <span>Earth</span></div>
    <div style="color: #64748b; font-size: 13px;">149,600,000 km</div>
    <div class="space-track"><span>Sun</span> ☀️</div>
  </div>

  <div class="main-board">
    <div class="number-display" id="displayBox">
      <svg class="svg-hops" id="hopSvg" viewBox="0 0 420 90"></svg>
      <div class="digits-row" id="digitsRow">
        <div class="digit-col" data-idx="0">1</div>
        <div class="digit-col" data-idx="1">4</div>
        <div class="digit-col" data-idx="2">9</div>
        <div class="digit-col" data-idx="3">6</div>
        <div class="digit-col" data-idx="4">0</div>
        <div class="digit-col" data-idx="5">0</div>
        <div class="digit-col" data-idx="6">0</div>
        <div class="digit-col" data-idx="7">0</div>
        <div class="digit-col" data-idx="8">0</div>
      </div>
      <div class="decimal-point" id="decDot"></div>
    </div>

    <div class="formula-card">
      <div class="formula-box" id="mathFormula"></div>
    </div>

    <div class="controls">
      <button class="btn" id="prevBtn">◀ Step</button>
      <button class="btn btn-primary" id="playBtn">▶ Auto</button>
      <button class="btn" id="nextBtn">Step ▶</button>
    </div>

    <div class="step-pills" id="stepPills"></div>
  </div>

  <script>
    const TOTAL_HOPS = 8;
    let currentStep = 0;
    let timer = null;

    const hopSvg = document.getElementById('hopSvg');
    const decDot = document.getElementById('decDot');
    const mathFormula = document.getElementById('mathFormula');
    const digitCols = document.querySelectorAll('.digit-col');
    const stepPills = document.getElementById('stepPills');
    const playBtn = document.getElementById('playBtn');

    // Create 9 pill buttons for steps 0 to 8
    for (let i = 0; i <= TOTAL_HOPS; i++) {
      const p = document.createElement('div');
      p.className = 'pill' + (i === 0 ? ' active' : '');
      p.addEventListener('click', () => { stopAuto(); setStep(i); });
      stepPills.appendChild(p);
    }

    function renderFormula(step) {
      let coeffStr = "149600000";
      if (step === 0) {
        coeffStr = "149\\,600\\,000";
      } else {
        const intPart = "149600000".slice(0, 9 - step);
        const fracPart = "149600000".slice(9 - step);
        coeffStr = intPart + "." + fracPart;
      }
      
      let latex = "";
      if (step === 0) {
        latex = "149\\,600\\,000 = 149\\,600\\,000 \\times 10^0";
      } else if (step === 8) {
        latex = "1.496 \\times 10^8 \\text{ km}";
      } else {
        latex = `${coeffStr} \\times 10^${step}`;
      }

      katex.render(latex, mathFormula, { throwOnError: false });
    }

    function updateArcs(positions) {
      hopSvg.innerHTML = '';
      for (let i = 0; i < TOTAL_HOPS; i++) {
        // hop i is from position (8 - i) to (7 - i)
        const fromX = positions[TOTAL_HOPS - i];
        const toX = positions[TOTAL_HOPS - i - 1];
        const midX = (fromX + toX) / 2;
        const topY = 28;
        const botY = 64;

        const path = document.createElementNS('http://www.w3.org/2000/svg', 'path');
        const d = `M ${fromX} ${botY} Q ${midX} ${topY} ${toX} ${botY}`;
        path.setAttribute('d', d);
        path.setAttribute('class', `hop-arc ${i < currentStep ? 'active' : ''}`);
        hopSvg.appendChild(path);

        const text = document.createElementNS('http://www.w3.org/2000/svg', 'text');
        text.setAttribute('x', midX);
        text.setAttribute('y', topY + 4);
        text.setAttribute('class', `hop-num ${i < currentStep ? 'active' : ''}`);
        text.textContent = (i + 1);
        hopSvg.appendChild(text);
      }
    }

    function getPositions() {
      const displayRect = document.getElementById('displayBox').getBoundingClientRect();
      const pos = [];
      digitCols.forEach((col, idx) => {
        const rect = col.getBoundingClientRect();
        if (idx === 0) {
          // Left edge of first digit
          pos.push(rect.left - displayRect.left + rect.width);
        } else {
          pos.push(rect.left - displayRect.left + rect.width);
        }
      });
      // pos has 9 entries corresponding to right side of each digit 0..8
      return pos;
    }

    let positions = [];

    function setStep(step) {
      currentStep = step;
      if (positions.length === 0) positions = getPositions();

      // Dot position: step 0 -> right side of digit 8 (pos[8])
      // step 8 -> right side of digit 0 (pos[0])
      const dotTargetIdx = TOTAL_HOPS - currentStep;
      decDot.style.left = `${positions[dotTargetIdx]}px`;

      // Update arcs
      updateArcs(positions);

      // Highlight digits
      digitCols.forEach((col, idx) => {
        col.className = 'digit-col';
        if (currentStep === 8) {
          if (idx <= 3) col.classList.add('coefficient');
          else col.classList.add('trailing');
        }
      });

      // Update pills
      const pills = stepPills.children;
      for (let i = 0; i <= TOTAL_HOPS; i++) {
        pills[i].className = 'pill' + (i === currentStep ? ' active' : '');
      }

      renderFormula(currentStep);
    }

    function stopAuto() {
      if (timer) {
        clearInterval(timer);
        timer = null;
        playBtn.innerHTML = '▶ Auto';
      }
    }

    function toggleAuto() {
      if (timer) {
        stopAuto();
      } else {
        playBtn.innerHTML = '⏸ Pause';
        timer = setInterval(() => {
          let next = (currentStep + 1) % (TOTAL_HOPS + 1);
          setStep(next);
        }, 1400);
      }
    }

    document.getElementById('prevBtn').addEventListener('click', () => {
      stopAuto();
      setStep(Math.max(0, currentStep - 1));
    });

    document.getElementById('nextBtn').addEventListener('click', () => {
      stopAuto();
      setStep(Math.min(TOTAL_HOPS, currentStep + 1));
    });

    playBtn.addEventListener('click', toggleAuto);

    window.addEventListener('resize', () => {
      positions = getPositions();
      setStep(currentStep);
    });

    window.addEventListener('load', () => {
      setTimeout(() => {
        positions = getPositions();
        setStep(0);
      }, 50);
    });
  </script>
</body>
</html>

Large numbers multiply by factors of 10, but what happens when we zoom into microscopic dimensions where numbers are much smaller than 1?

When a number is between 0 and 1, move the decimal point to the right until it sits immediately after the first non-zero digit. The count of jumps becomes your **negative exponent** because each jump represents dividing by 10 ($10^{-1} = \frac{1}{10}$).

For instance, a typical virus particle measures about $0.00000012\text{ meters}$ in diameter. Shifting the decimal point 7 places to the right yields $1.2 \times 10^{-7}\text{ m}$.

<!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="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css">
  <script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.js"></script>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 12px 16px;
      user-select: none;
    }

    .display-box {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 12px 16px;
      width: 100%;
      max-width: 480px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.04);
      display: flex;
      flex-direction: column;
      align-items: center;
      position: relative;
    }

    .number-row {
      display: flex;
      align-items: baseline;
      justify-content: center;
      position: relative;
      margin-top: 36px;
      margin-bottom: 8px;
    }

    .digit-slot {
      width: 24px;
      text-align: center;
      font-size: 26px;
      font-weight: 600;
      font-family: monospace;
      color: #1e2945;
      position: relative;
    }

    .digit-slot.highlight {
      color: #f59e0b;
    }

    .svg-overlay {
      position: absolute;
      top: -30px;
      left: 0;
      width: 100%;
      height: 70px;
      pointer-events: none;
      overflow: visible;
    }

    .arc-path {
      fill: none;
      stroke: #f59e0b;
      stroke-width: 2.5;
      stroke-linecap: round;
      stroke-dasharray: 60;
      stroke-dashoffset: 60;
      transition: stroke-dashoffset 0.3s ease;
    }

    .arc-path.active {
      stroke-dashoffset: 0;
    }

    .arc-label {
      font-size: 11px;
      font-weight: 700;
      fill: #d97706;
      text-anchor: middle;
      opacity: 0;
      transition: opacity 0.2s ease;
    }

    .arc-label.active {
      opacity: 1;
    }

    .decimal-dot {
      position: absolute;
      bottom: 6px;
      width: 8px;
      height: 8px;
      border-radius: 50%;
      background: #f59e0b;
      transform: translateX(-50%);
      transition: left 0.35s cubic-bezier(0.34, 1.56, 0.64, 1);
      box-shadow: 0 0 6px rgba(245, 158, 11, 0.6);
      z-index: 5;
    }

    .result-panel {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 16px;
      margin-top: 12px;
      padding: 8px 16px;
      background: #f8fafc;
      border-radius: 8px;
      width: 100%;
      min-height: 48px;
    }

    .katex-out {
      font-size: 20px;
    }

    .badge-count {
      display: inline-flex;
      align-items: center;
      gap: 4px;
      background: #fef3c7;
      color: #92400e;
      font-weight: 700;
      font-size: 13px;
      padding: 4px 10px;
      border-radius: 20px;
      border: 1px solid #fde68a;
    }

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

    .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;
      justify-content: center;
      transition: background 0.2s, border-color 0.2s;
    }

    .btn:active {
      background: #f1f5f9;
    }

    .btn-primary {
      background: #f59e0b;
      border-color: #d97706;
      color: #ffffff;
    }

    .btn-primary:active {
      background: #d97706;
    }

    .steps-row {
      display: flex;
      gap: 4px;
    }

    .step-pip {
      width: 24px;
      height: 24px;
      border-radius: 6px;
      border: 1px solid #cbd5e1;
      background: #ffffff;
      color: #64748b;
      font-size: 12px;
      font-weight: 600;
      display: flex;
      align-items: center;
      justify-content: center;
      cursor: pointer;
    }

    .step-pip.active {
      background: #f59e0b;
      border-color: #d97706;
      color: #ffffff;
    }
  </style>
</head>
<body>

  <div class="display-box">
    <div class="number-row" id="numberRow">
      <svg class="svg-overlay" id="svgOverlay"></svg>
      <div class="decimal-dot" id="dot"></div>
      
      <div class="digit-slot" data-idx="0">0</div>
      <div class="digit-slot" data-idx="1">0</div>
      <div class="digit-slot" data-idx="2">0</div>
      <div class="digit-slot" data-idx="3">0</div>
      <div class="digit-slot" data-idx="4">0</div>
      <div class="digit-slot" data-idx="5">0</div>
      <div class="digit-slot" data-idx="6">0</div>
      <div class="digit-slot" data-idx="7">1</div>
      <div class="digit-slot" data-idx="8">2</div>
    </div>

    <div class="result-panel">
      <div class="badge-count" id="badge">0 jumps</div>
      <div class="katex-out" id="mathDisplay"></div>
    </div>
  </div>

  <div class="controls">
    <button class="btn" id="prevBtn" aria-label="Previous">◀</button>
    <div class="steps-row" id="pipsContainer"></div>
    <button class="btn" id="nextBtn" aria-label="Next">▶</button>
    <button class="btn btn-primary" id="autoBtn" style="margin-left: 4px;">⏸</button>
  </div>

  <script>
    const TOTAL_JUMPS = 7;
    let currentStep = 0;
    let autoPlayTimer = null;

    const slots = document.querySelectorAll('.digit-slot');
    const dot = document.getElementById('dot');
    const svg = document.getElementById('svgOverlay');
    const badge = document.getElementById('badge');
    const mathDisplay = document.getElementById('mathDisplay');
    const pipsContainer = document.getElementById('pipsContainer');
    const prevBtn = document.getElementById('prevBtn');
    const nextBtn = document.getElementById('nextBtn');
    const autoBtn = document.getElementById('autoBtn');

    // Generate pips
    for (let i = 0; i <= TOTAL_JUMPS; i++) {
      const pip = document.createElement('div');
      pip.className = 'step-pip' + (i === 0 ? ' active' : '');
      pip.textContent = i;
      pip.onclick = () => { setStep(i); pause(); };
      pipsContainer.appendChild(pip);
    }

    function initArcs() {
      svg.innerHTML = '';
      const w = 24; // slot width
      const originX = 24; // after first 0

      for (let i = 0; i < TOTAL_JUMPS; i++) {
        const x1 = originX + i * w;
        const x2 = x1 + w;
        const midX = (x1 + x2) / 2;
        const yBase = 58;
        const yTop = 32;

        const path = document.createElementNS('http://www.w3.org/2000/svg', 'path');
        path.setAttribute('d', `M ${x1} ${yBase} Q ${midX} ${yTop} ${x2} ${yBase}`);
        path.setAttribute('class', 'arc-path');
        path.setAttribute('id', `arc-${i}`);
        svg.appendChild(path);

        const text = document.createElementNS('http://www.w3.org/2000/svg', 'text');
        text.setAttribute('x', midX);
        text.setAttribute('y', yTop - 4);
        text.setAttribute('class', 'arc-label');
        text.setAttribute('id', `label-${i}`);
        text.textContent = (i + 1);
        svg.appendChild(text);
      }
    }

    function updateView() {
      const slotWidth = 24;
      const baseLeft = 24;
      dot.style.left = `${baseLeft + currentStep * slotWidth}px`;

      // Update arcs
      for (let i = 0; i < TOTAL_JUMPS; i++) {
        const arc = document.getElementById(`arc-${i}`);
        const lbl = document.getElementById(`label-${i}`);
        if (i < currentStep) {
          arc.classList.add('active');
          lbl.classList.add('active');
        } else {
          arc.classList.remove('active');
          lbl.classList.remove('active');
        }
      }

      // Highlight digit 1 when jumped over
      slots.forEach((s, idx) => {
        if (idx === 7 && currentStep >= 7) {
          s.classList.add('highlight');
        } else {
          s.classList.remove('highlight');
        }
      });

      // Update pips
      document.querySelectorAll('.step-pip').forEach((p, idx) => {
        p.classList.toggle('active', idx === currentStep);
      });

      // Badge
      badge.textContent = `${currentStep} ${currentStep === 1 ? 'jump' : 'jumps'}`;

      // Math
      let latex = '';
      if (currentStep === 0) {
        latex = '0.00000012\\text{ m}';
      } else if (currentStep === 7) {
        latex = '1.2 \\times 10^{-7}\\text{ m}';
      } else {
        latex = `\\rightarrow 10^{-${currentStep}}`;
      }
      try {
        katex.render(latex, mathDisplay, { throwOnError: false });
      } catch (e) {
        mathDisplay.textContent = latex;
      }
    }

    function setStep(step) {
      currentStep = Math.max(0, Math.min(TOTAL_JUMPS, step));
      updateView();
    }

    function next() {
      if (currentStep < TOTAL_JUMPS) {
        setStep(currentStep + 1);
      } else {
        setStep(0);
      }
    }

    function prev() {
      if (currentStep > 0) {
        setStep(currentStep - 1);
      } else {
        setStep(TOTAL_JUMPS);
      }
    }

    function startAuto() {
      if (autoPlayTimer) clearInterval(autoPlayTimer);
      autoBtn.textContent = '⏸';
      autoPlayTimer = setInterval(() => {
        next();
      }, 1400);
    }

    function pause() {
      if (autoPlayTimer) {
        clearInterval(autoPlayTimer);
        autoPlayTimer = null;
      }
      autoBtn.textContent = '▶';
    }

    function toggleAuto() {
      if (autoPlayTimer) {
        pause();
      } else {
        startAuto();
      }
    }

    prevBtn.onclick = () => { prev(); pause(); };
    nextBtn.onclick = () => { next(); pause(); };
    autoBtn.onclick = toggleAuto;

    // Init
    initArcs();
    updateView();
    startAuto();
  </script>
</body>
</html>

A common mistake is simply counting the zeros in the decimal. You must always count the **total decimal jumps**, not the zeros, because jumping past non-zero digits like $1$ also counts as a place shift.

What if you are given a scientific measurement and need to convert it back into everyday decimal form?

To convert back to **standard form**, reverse the decimal movement: a positive exponent shifts the decimal point to the right (making the number bigger), while a negative exponent shifts it to the left (making the number smaller).

For example, $4.05 \times 10^5$ becomes $405,000$ after moving the decimal 5 places right, while $8.3 \times 10^{-4}$ expands to $0.00083$ after moving 4 places left and filling empty spaces with placeholder zeros.

<!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>Scientific Notation Decimal Shifts</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      user-select: none;
    }

    body {
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 16px 12px;
      gap: 12px;
    }

    /* Mode selector tabs */
    .mode-switch {
      display: flex;
      gap: 8px;
      width: 100%;
      max-width: 440px;
    }

    .tab-btn {
      flex: 1;
      padding: 10px 14px;
      font-size: 15px;
      font-weight: 600;
      border: 2px solid #e2e8f0;
      background: #ffffff;
      color: #64748b;
      border-radius: 10px;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      transition: all 0.2s ease;
    }

    .tab-btn.active {
      border-color: #22b7ff;
      background: #e8f7ff;
      color: #0077b6;
      box-shadow: 0 2px 8px rgba(34, 183, 255, 0.15);
    }

    /* Main Stage */
    .stage {
      width: 100%;
      max-width: 520px;
      background: #ffffff;
      border-radius: 12px;
      padding: 20px 16px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 16px;
      box-shadow: 0 2px 10px rgba(0,0,0,0.04);
      position: relative;
    }

    .expression-header {
      font-size: 22px;
      font-weight: 700;
      letter-spacing: 0.5px;
      color: #1e2945;
      display: flex;
      align-items: center;
      gap: 4px;
    }

    .exp-badge {
      background: #e8f7ff;
      color: #0088cc;
      padding: 2px 6px;
      border-radius: 6px;
      font-size: 16px;
    }

    /* Digit Slots Display */
    .display-row {
      position: relative;
      display: flex;
      align-items: center;
      justify-content: center;
      height: 70px;
      width: 100%;
    }

    .slots-container {
      display: flex;
      align-items: center;
      justify-content: center;
      position: relative;
    }

    .slot {
      width: 32px;
      height: 48px;
      display: flex;
      align-items: center;
      justify-content: center;
      font-size: 26px;
      font-weight: 700;
      color: #1e2945;
      position: relative;
    }

    .slot.zero-pad {
      color: #22b7ff;
    }

    .slot.dim {
      opacity: 0.2;
    }

    /* SVG Hops overlay */
    .hops-svg {
      position: absolute;
      top: -12px;
      left: 0;
      width: 100%;
      height: 60px;
      pointer-events: none;
      overflow: visible;
    }

    .hop-arc {
      fill: none;
      stroke: #22b7ff;
      stroke-width: 2.5;
      stroke-linecap: round;
      stroke-dasharray: 60;
      stroke-dashoffset: 60;
      transition: stroke-dashoffset 0.35s ease;
    }

    .hop-arc.drawn {
      stroke-dashoffset: 0;
    }

    .hop-arrow {
      fill: #22b7ff;
      opacity: 0;
      transition: opacity 0.2s ease;
    }

    .hop-arrow.drawn {
      opacity: 1;
    }

    /* Decimal Point Dot */
    .dot-marker {
      position: absolute;
      bottom: 6px;
      width: 8px;
      height: 8px;
      background: #ef4444;
      border-radius: 50%;
      transform: translateX(-50%);
      transition: left 0.35s cubic-bezier(0.34, 1.3, 0.64, 1);
      box-shadow: 0 0 6px rgba(239, 68, 68, 0.5);
      z-index: 5;
    }

    /* Outcome badge */
    .result-view {
      display: flex;
      align-items: center;
      gap: 10px;
      font-size: 20px;
      font-weight: 700;
      color: #0f172a;
      background: #f1f5f9;
      padding: 8px 18px;
      border-radius: 20px;
    }

    .result-view span.res-val {
      color: #0077b6;
    }

    /* Controls */
    .controls {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
      width: 100%;
      max-width: 440px;
    }

    .btn {
      padding: 8px 16px;
      font-size: 14px;
      font-weight: 600;
      border-radius: 8px;
      border: 1px solid #cbd5e1;
      background: #ffffff;
      color: #334155;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 6px;
    }

    .btn:active {
      background: #f1f5f9;
    }

    .btn-primary {
      background: #22b7ff;
      border-color: #22b7ff;
      color: #ffffff;
    }

    .btn-primary:active {
      background: #009ee3;
    }

    .step-indicator {
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      min-width: 70px;
      text-align: center;
    }

    /* Scale context indicator */
    .scale-tag {
      font-size: 12px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.8px;
      padding: 3px 8px;
      border-radius: 4px;
    }

    .scale-tag.pos {
      background: #dcfce7;
      color: #15803d;
    }

    .scale-tag.neg {
      background: #fef3c7;
      color: #b45309;
    }
  </style>
</head>
<body>

  
  <div class="mode-switch">
    <button class="tab-btn active" id="tabPos" onclick="setMode('pos')">
      <span>4.05 × 10⁵</span>
      <span class="scale-tag pos">≥ 10</span>
    </button>
    <button class="tab-btn" id="tabNeg" onclick="setMode('neg')">
      <span>8.3 × 10⁻⁴</span>
      <span class="scale-tag neg">&lt; 1</span>
    </button>
  </div>

  
  <div class="stage">
    <div class="expression-header" id="expHeader">
      4.05 × 10<span class="exp-badge">5</span>
    </div>

    
    <div class="display-row" id="displayRow">
      <svg class="hops-svg" id="hopsSvg" viewBox="0 0 300 60"></svg>
      <div class="slots-container" id="slotsContainer"></div>
      <div class="dot-marker" id="dotMarker"></div>
    </div>

    
    <div class="result-view">
      <span>=</span>
      <span class="res-val" id="resultVal">4.05</span>
    </div>
  </div>

  
  <div class="controls">
    <button class="btn" onclick="prevStep()">◀</button>
    <button class="btn btn-primary" id="playBtn" onclick="togglePlay()">▶ Auto</button>
    <button class="btn" onclick="nextStep()">▶</button>
    <div class="step-indicator" id="stepCounter">Hop 0 / 5</div>
  </div>

  <script>
    const CONFIGS = {
      pos: {
        raw: '4.05 × 10⁵',
        headerHtml: '4.05 × 10<sup style="color:#0088cc;font-size:16px;">+5</sup>',
        maxSteps: 5,
        direction: 1, // right
        // Slots layout: [4, 0, 5, 0, 0, 0]
        digits: ['4', '0', '5', '0', '0', '0'],
        initialDotIndex: 0, // after '4' (between slot 0 and slot 1)
        activeAfterStep: [1, 2, 2, 3, 4, 5], // which slots are un-dimmed at each step
        results: ['4.05', '40.5', '405', '4,050', '40,500', '405,000']
      },
      neg: {
        raw: '8.3 × 10⁻⁴',
        headerHtml: '8.3 × 10<sup style="color:#b45309;font-size:16px;">-4</sup>',
        maxSteps: 4,
        direction: -1, // left
        // Slots layout: [0, 0, 0, 0, 8, 3]
        digits: ['0', '0', '0', '0', '8', '3'],
        initialDotIndex: 4, // after '8' (between slot 4 and slot 5)
        activeAfterStep: [2, 2, 2, 2, 2], // activation logic
        results: ['8.3', '0.83', '0.083', '0.0083', '0.00083']
      }
    };

    let currentMode = 'pos';
    let currentStep = 0;
    let timer = null;
    const SLOT_WIDTH = 32;

    function renderBoard() {
      const cfg = CONFIGS[currentMode];
      const container = document.getElementById('slotsContainer');
      container.innerHTML = '';

      cfg.digits.forEach((d, idx) => {
        const slot = document.createElement('div');
        slot.className = 'slot';
        slot.id = `slot-${idx}`;
        slot.textContent = d;
        container.appendChild(slot);
      });

      updateView();
    }

    function updateView() {
      const cfg = CONFIGS[currentMode];
      document.getElementById('expHeader').innerHTML = cfg.headerHtml;
      document.getElementById('stepCounter').textContent = `Hop ${currentStep} / ${cfg.maxSteps}`;
      document.getElementById('resultVal').textContent = cfg.results[currentStep];

      // Update digit styling (dim / pad)
      cfg.digits.forEach((d, idx) => {
        const el = document.getElementById(`slot-${idx}`);
        if (!el) return;

        if (currentMode === 'pos') {
          if (idx <= 2) {
            el.className = 'slot';
          } else {
            el.className = idx <= (currentStep + 1) ? 'slot zero-pad' : 'slot dim';
          }
        } else {
          // neg mode
          const numZerosNeeded = currentStep;
          const zeroStart = 4 - numZerosNeeded;
          if (idx >= 4) {
            el.className = 'slot';
          } else if (idx >= zeroStart) {
            el.className = 'slot zero-pad';
          } else {
            el.className = 'slot dim';
          }
        }
      });

      // Update Decimal Dot Position
      // Decimal sits on the boundary between slots
      const dot = document.getElementById('dotMarker');
      const dotSlotIdx = cfg.initialDotIndex + (cfg.direction * currentStep);
      const dotX = (dotSlotIdx + 1) * SLOT_WIDTH;
      dot.style.left = `${dotX}px`;

      // Render SVG hop arcs
      renderHops();
    }

    function renderHops() {
      const cfg = CONFIGS[currentMode];
      const svg = document.getElementById('hopsSvg');
      const totalWidth = cfg.digits.length * SLOT_WIDTH;
      svg.setAttribute('viewBox', `0 0 ${totalWidth} 60`);
      svg.innerHTML = '';

      for (let i = 0; i < currentStep; i++) {
        let fromX, toX;
        if (cfg.direction === 1) {
          fromX = (cfg.initialDotIndex + 1 + i) * SLOT_WIDTH;
          toX = fromX + SLOT_WIDTH;
        } else {
          fromX = (cfg.initialDotIndex + 1 - i) * SLOT_WIDTH;
          toX = fromX - SLOT_WIDTH;
        }

        const midX = (fromX + toX) / 2;
        const midY = 15;
        const base = 50;

        const path = document.createElementNS('http://www.w3.org/2000/svg', 'path');
        path.setAttribute('d', `M ${fromX} ${base} Q ${midX} ${midY} ${toX} ${base}`);
        path.setAttribute('class', 'hop-arc drawn');
        svg.appendChild(path);

        // Hop direction arrow head
        const arrow = document.createElementNS('http://www.w3.org/2000/svg', 'polygon');
        if (cfg.direction === 1) {
          arrow.setAttribute('points', `${toX-3},${base-6} ${toX+2},${base} ${toX-3},${base+4}`);
        } else {
          arrow.setAttribute('points', `${toX+3},${base-6} ${toX-2},${base} ${toX+3},${base+4}`);
        }
        arrow.setAttribute('class', 'hop-arrow drawn');
        svg.appendChild(arrow);
      }
    }

    function setMode(mode) {
      if (timer) togglePlay();
      currentMode = mode;
      currentStep = 0;
      document.getElementById('tabPos').classList.toggle('active', mode === 'pos');
      document.getElementById('tabNeg').classList.toggle('active', mode === 'neg');
      renderBoard();
    }

    function nextStep() {
      const cfg = CONFIGS[currentMode];
      if (currentStep < cfg.maxSteps) {
        currentStep++;
        updateView();
      } else if (timer) {
        currentStep = 0;
        updateView();
      }
    }

    function prevStep() {
      if (currentStep > 0) {
        currentStep--;
        updateView();
      }
    }

    function togglePlay() {
      const btn = document.getElementById('playBtn');
      if (timer) {
        clearInterval(timer);
        timer = null;
        btn.textContent = '▶ Auto';
      } else {
        btn.textContent = '⏸ Pause';
        timer = setInterval(() => {
          const cfg = CONFIGS[currentMode];
          if (currentStep >= cfg.maxSteps) {
            currentStep = 0;
            updateView();
          } else {
            nextStep();
          }
        }, 1100);
      }
    }

    // Initialize on load
    renderBoard();
  </script>
</body>
</html>

You can always sanity-check your answer: positive exponents must produce standard values of 10 or greater, while negative exponents must always produce fractional values strictly less than 1.

math

Using Scientific Notation for Large and Small Scales. Write very large and very small numbers in scientific notation and convert back to standard form.

Time and Duration: Year 9 Planning Tool (AC9M9M02)

Introduction to Scientific Notation and Large/Small Numbers

Convert Between Ordinary Decimal and Scientific Notation (Year 9 Curriculum)

When teaching AC9M9M02 to Year 9 students, ground abstract index notation in real-world contexts of measurement and duration (e.g., geological eras, deep space distances, nano-second computer cycles). Emphasize the place-value reasoning behind powers of ten rather than rote 'zero counting'. Explicitly model calculator entry using the 'EXP' or 'EE' key to prevent syntax errors, and regularly contrast scientific notation with standard SI metric prefixes.

Using Scientific Notation for Large and Small Scales

Anatomy of Scientific Notation

Converting Large Numbers

Converting Small Numbers