🔭 Standard Form
Cambridge Lower Secondary · Grade 7 · Number & Calculation
Large Number → Standard Form
384,400,000 = 3.844 × 108
Distance to the Moon in metres
Small Number → Standard Form
0.000000001 = 1 × 10−9
Size of an atom in metres
Multiplying in Standard Form
(3 × 104) × (2 × 103) = 6 × 107
Multiply A×A, add the powers
Watch a number shrink into Standard Form! 🔭
What you'll learn:
- Writing large numbers in standard form: A × 10n where 1 ≤ A < 10
- Writing small numbers in standard form (negative powers of 10)
- Converting standard form back to ordinary numbers
- Ordering numbers given in different forms
- Multiplying and dividing numbers in standard form
- Real-world contexts: space distances, atomic sizes, populations
📖 Learn: Standard Form
Part 1: What is Standard Form?
Standard form (also called scientific notation) is a way of writing very large or very small numbers neatly.
A number in standard form looks like this:
A × 10n
📌 A is a number where 1 ≤ A < 10 (one digit before the decimal point, not zero)
📌 n is an integer (positive, negative, or zero) — the power of 10
Examples: 3.2 × 105 • 9.07 × 10−3 • 1 × 1012
💡 NOT standard form: 32 × 104 (A is not between 1 and 10) • 0.5 × 106 (A < 1)
Part 2: Large Numbers → Standard Form
To convert a large number, count how many places you move the decimal point left to make A.
📌 Write the number out, then move the decimal point left until you have 1 digit before it
📌 Count the moves — that number becomes the positive power of 10
Example: 4,700,000 → decimal moves 6 places left → 4.7 × 106
Example: 93,000,000 → decimal moves 7 places left → 9.3 × 107
Example: 5,280 → decimal moves 3 places left → 5.28 × 103
💡 Memory tip: Large number → positive power. "Big power for big numbers!"
Part 3: Small Numbers → Standard Form
For small decimals, move the decimal point right to make A. The power becomes negative.
📌 Move the decimal right until you have 1 non-zero digit before it
📌 Count the moves — that becomes the negative power of 10
Example: 0.00045 → decimal moves 4 places right → 4.5 × 10−4
Example: 0.0000001 → decimal moves 7 places right → 1 × 10−7
Example: 0.083 → decimal moves 2 places right → 8.3 × 10−2
💡 Memory tip: Small number → negative power. "Tiny gets a minus!"
Part 4: Standard Form → Ordinary Numbers
To convert from standard form, move the decimal point in A by |n| places.
| Standard Form |
Power Means |
Ordinary Number |
| 6.2 × 104 |
move right 4 |
62,000 |
| 3.05 × 107 |
move right 7 |
30,500,000 |
| 7.1 × 10−3 |
move left 3 |
0.0071 |
| 2 × 10−5 |
move left 5 |
0.00002 |
💡 Positive power → move right (bigger). Negative power → move left (smaller).
Part 5: Multiplying & Dividing in Standard Form
Multiplying: multiply the A values, add the powers.
📌 (A × 10m) × (B × 10n) = (A × B) × 10m+n
Example: (3 × 104) × (2 × 103) = 6 × 107
Watch out: if A × B ≥ 10, adjust! (5 × 103) × (4 × 102) = 20 × 105 = 2 × 106
Dividing: divide the A values, subtract the powers.
📌 (A × 10m) ÷ (B × 10n) = (A ÷ B) × 10m−n
Example: (8 × 106) ÷ (4 × 102) = 2 × 104
💡 Always check your answer is in standard form — 1 ≤ A < 10!
💡 Worked Examples
Example 1: Converting a Large Number to Standard Form
Write 47,000,000 in standard form.
Step 1: Find A. Write 47,000,000 and move the decimal left until one digit is before it: 4.7
Step 2: Count the moves: 47000000 → 7 places to the left
Step 3: Power of 10 is +7 (positive because the original is large)
Answer: 4.7 × 107
✓ Check: 4.7 is between 1 and 10. Power is a positive integer. ✓
Example 2: Converting a Small Number to Standard Form
Write 0.0000052 in standard form.
Step 1: Find A. Move decimal right until one non-zero digit is before it: 5.2
Step 2: Count the moves: 0.0000052 → 6 places to the right
Step 3: Power of 10 is −6 (negative because the original is tiny)
Answer: 5.2 × 10−6
💡 The size of a red blood cell is about 8 × 10−6 m — very similar!
Example 3: Converting Standard Form to Ordinary Number
Write 3.06 × 105 as an ordinary number.
Step 1: Positive power means move decimal right by 5 places
Step 2: 3.06 → 3.06000 → move decimal 5 right → 306000
Answer: 306,000
Now write 4.5 × 10−4 as an ordinary number.
Step 1: Negative power means move decimal left by 4 places
Step 2: 4.5 → move decimal 4 left → 0.00045
Answer: 0.00045
Example 4: Multiplying Numbers in Standard Form
Calculate (5 × 103) × (6 × 104). Give your answer in standard form.
Step 1: Multiply A values: 5 × 6 = 30
Step 2: Add powers: 103 × 104 = 107
Step 3: 30 × 107 — but 30 ≥ 10, so adjust: 30 = 3 × 101
Step 4: 3 × 101 × 107 = 3 × 108
💡 Divide A by 10 and add 1 to the power when A ≥ 10.
🔭 Number Zoom Visualizer
Drag the slider to zoom through the scale of numbers — watch them transform into standard form!
0.0000010.00111,0001,000,000
1
= Standard Form:
1 × 100
👉 Place Value Grid — watch digits shift as the power changes!
Power: 3
Green = active digit | Peach = all significant digits | Grey = zero placeholder
Quick conversions — click to see:
✏️ Exercise 1: Write in Standard Form
Write each number in standard form A × 10n. Enter A and n separately.
✏️ Exercise 2: Standard Form → Ordinary Number
Convert each standard form number to an ordinary number.
✏️ Exercise 3: Ordering Challenge
Drag the cards from the pool into the slots in order from smallest to largest. Click a placed card to remove it.
× Exercise 4: Multiply & Divide in Standard Form
Give answers in standard form. Enter A and n.
🌍 Exercise 5: Space Explorer
Match each planet's distance from the Sun (in standard form) to its ordinary number equivalent.
📝 Practice Questions (20)
Type your answer and press Enter or click Check. Standard form: write as e.g. 3.2e5 or ordinary number.
🏆 Challenge Questions (8)
Harder problems — full worked answers included. Try each one first!