🔬 Standard Form
Cambridge Lower Secondary · Grade 8 · Number
Standard Form Rule
A × 10n where 1 ≤ A < 10n is any integer — positive, negative or zero
Large Numbers
3,400,000 = 3.4 × 106 Move decimal left → positive power
Small Numbers
0.0052 = 5.2 × 10−3 Move decimal right → negative power
Watch a number shift into standard form! 🌌
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What you'll learn:
What standard form is: A × 10n where 1 ≤ A < 10
Converting large numbers to standard form
Converting small numbers to standard form (negative powers)
Converting standard form back to ordinary numbers
Ordering numbers given in standard form
Multiplying and dividing in standard form
Adding and subtracting in standard form
Real-world contexts: space, atoms, 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. Every number is written as:
✅ 3.4 × 106 — valid (A = 3.4, between 1 and 10)
❌ 34 × 105 — NOT valid (A = 34, must be less than 10)
❌ 0.34 × 107 — NOT valid (A = 0.34, must be at least 1)
💡 The number 1 is written as 1 × 100 since 100 = 1
Part 2: Large Numbers → Standard Form
To convert a large number to standard form, count how many places you move the decimal point to the left . That count becomes the positive power of 10.
Example: Convert 3,400,000 to standard form.
3,400,000 → place decimal after first digit: 3.400000
Decimal moved 6 places to the left → power is +6
Answer: 3.4 × 106
Example: Convert 52,000 to standard form.
52,000 → 5.2000 (decimal moved 4 places left) → 5.2 × 104
Example: Convert 807,000,000 to standard form.
807,000,000 → 8.07000000 (decimal moved 8 places left) → 8.07 × 108
📌 Large numbers: the power equals the number of digits after the first digit (including trailing zeros before the decimal point).
Part 3: Small Numbers → Standard Form
To convert a small number (between 0 and 1) to standard form, count how many places you move the decimal point to the right . That count becomes the negative power of 10.
Example: Convert 0.0052 to standard form.
0.0052 → move decimal right until A ≥ 1: 5.2
Decimal moved 3 places to the right → power is −3
Answer: 5.2 × 10−3
Example: Convert 0.000000074 to standard form.
0.000000074 → 7.4 (decimal moved 8 places right) → 7.4 × 10−8
📌 Small numbers: count the zeros after the decimal point including the first non-zero digit position. That gives the magnitude of the negative power.
⚠️ Negative power does NOT mean the number is negative — 5.2 × 10−3 = 0.0052 (a positive tiny number).
Part 4: Standard Form → Ordinary Numbers
To convert from standard form back to an ordinary number, use the power of 10 to tell you how far and which direction to move the decimal point.
Positive power → move decimal to the right (makes number bigger)
6.3 × 105 → move decimal 5 right → 630,000
Negative power → move decimal to the left (makes number smaller)
4.8 × 10−4 → move decimal 4 left → 0.00048
Standard Form Power Direction Ordinary Number 2.5 × 103 +3 Right 3 2,500 1.07 × 106 +6 Right 6 1,070,000 9.9 × 10−2 −2 Left 2 0.099 3.14 × 10−5 −5 Left 5 0.0000314
Part 5: Ordering, Multiplying, Dividing & Adding
Ordering: Compare powers first. If powers match, compare A values.
Order: 3.2 × 105 , 8.1 × 104 , 2.9 × 105
Powers: 4, 5, 5 → the 104 is smallest. Between the two 105 numbers: 2.9 < 3.2
Order (smallest first): 8.1 × 104 , 2.9 × 105 , 3.2 × 105
Multiplying: Multiply the A values, add the powers.
(3 × 104 ) × (2 × 103 ) = (3 × 2) × 104+3 = 6 × 107
(4.5 × 106 ) × (2 × 103 ) = 9 × 109
⚠️ If the product of A values ≥ 10, adjust: e.g. 15 × 107 = 1.5 × 108
Dividing: Divide the A values, subtract the powers.
(8 × 106 ) ÷ (2 × 102 ) = (8 ÷ 2) × 106−2 = 4 × 104
Adding/Subtracting: Convert to the same power first.
3.2 × 105 + 4.0 × 104 = 3.2 × 105 + 0.40 × 105 = 3.60 × 105
💡 Or convert both to ordinary numbers, add, then convert back to standard form.
💡 Worked Examples
Example 1: Converting Large Numbers to Standard Form
Write each number in standard form: (a) 47,000 (b) 360,000,000 (c) 5,090,000
(a) 47,000 → 4.7000 (decimal moved 4 places left) → 4.7 × 104
(b) 360,000,000 → 3.60000000 (decimal moved 8 places left) → 3.6 × 108
(c) 5,090,000 → 5.090000 (decimal moved 6 places left) → 5.09 × 106
📌 The Earth is about 1.5 × 1011 m from the Sun — standard form makes these distances manageable!
Example 2: Converting Small Numbers to Standard Form
Write in standard form: (a) 0.00043 (b) 0.000000009 (c) 0.071
(a) 0.00043 → 4.3 (decimal moved 4 places right) → 4.3 × 10−4
(b) 0.000000009 → 9.0 (decimal moved 9 places right) → 9 × 10−9
(c) 0.071 → 7.1 (decimal moved 2 places right) → 7.1 × 10−2
🔬 A hydrogen atom has a diameter of about 1.2 × 10−10 m — too tiny to write any other way!
Example 3: Multiplying and Dividing in Standard Form
(a) (2.5 × 104 ) × (4 × 103 ) (b) (9.6 × 108 ) ÷ (3.2 × 105 )
(a) Multiply A values: 2.5 × 4 = 10 → not valid (≥ 10), so write 10 = 1 × 101
Add powers: 4 + 3 = 7, then adjust: 10 × 107 = 1 × 108 → 1 × 108
(b) Divide A values: 9.6 ÷ 3.2 = 3
Subtract powers: 8 − 5 = 3 → answer: 3 × 103
✅ Always check A is between 1 and 10 after your calculation. Adjust if needed by changing the power.
Example 4: Adding in Standard Form + Real-World Context
The world population is approximately 8.1 × 109 . India has about 1.4 × 109 people. How many people live outside India? Give your answer in standard form.
Step 1: Same power (109 ), so subtract A values directly.
Step 2: 8.1 − 1.4 = 6.7
Step 3: Answer: 6.7 × 109 people live outside India.
🌍 Real-world check: 6.7 × 109 = 6,700,000,000 — that's 6.7 billion people!
🔭 Interactive Visualizers
📏 Place Value Slider
Type an ordinary number and drag the slider to shift the decimal point. Watch the standard form update live!
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🌌 Universe Scale Explorer
Click an object to see its size in standard form!
⚛️ Proton
🔬 Atom
🦠 Virus
🧍 Human
🌍 Earth
☀️ Sun
🌌 Milky Way
🌍
Click an object above
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Select an object to explore its size
Scale: Proton → Milky Way
✏️ Exercise 1: Convert to Standard Form
Write each ordinary number in the form A × 10n . Enter A and n separately.
✅ Check Answers
✏️ Exercise 2: Convert from Standard Form
Write each standard form number as an ordinary number.
✅ Check Answers
✏️ Exercise 3: Ordering Numbers in Standard Form
Arrange the numbers in the order requested (smallest to largest or largest to smallest). Enter each value as an ordinary number or standard form.
✅ Check Answers
✖️ Exercise 4: Multiply & Divide in Standard Form
Give your answer in standard form. Enter A and the power (n) separately.
✅ Check Answers
➕ Exercise 5: Add & Subtract in Standard Form
Give answers in standard form where possible. Enter A and power separately.
✅ Check Answers
📝 Practice: 20 Questions
Mixed standard form questions. Give answers to 2 decimal places where needed.
✅ Check All
🏆 Challenge: 8 Hard Questions
Multi-step real-world problems. Show working on paper, enter final answer.
✅ Check All