IGCSE Standard Form: Complete Guide for Cambridge IGCSE Mathematics

IGCSE standard form (also called scientific notation) is an essential topic in Cambridge IGCSE Mathematics that appears in both Paper 2 and Paper 4. Mastering converting to and from standard form and performing calculations is crucial for working with very large or very small numbers.

This comprehensive IGCSE standard form guide covers everything you need to know, including writing numbers in standard form, converting from standard form to ordinary numbers, adding, subtracting, multiplying, and dividing numbers in standard form, step-by-step worked examples, common exam questions, and expert tips from Tutopiya’s IGCSE maths tutors. We’ll also show you how to avoid the most common mistakes that cost students valuable marks.

🎯 What you’ll learn: By the end of this guide, you’ll know how to write numbers in standard form, convert between standard form and ordinary numbers, perform all operations with standard form, and apply these skills to solve problems in IGCSE exams.

Already studying with Tutopiya? Practice these skills with our dedicated IGCSE Number practice deck featuring exam-style questions and instant feedback.

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Why IGCSE Standard Form Matters

IGCSE standard form is essential for working with very large or very small numbers. Here’s why it’s so important:

• High frequency topic: Standard form questions appear regularly in IGCSE maths papers
• Real-world applications: Used in science, engineering, astronomy, and measurements
• Exam weight: Typically worth 4-8 marks per paper
• Calculator skills: Essential for using scientific calculators effectively
• Foundation for advanced topics: Required for understanding exponential functions and scientific calculations

Key insight from examiners: Students often make errors with the power of 10 or forget to adjust the coefficient. This guide will help you master these systematically.

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Understanding Standard Form

Standard form (or scientific notation) is a way of writing very large or very small numbers in the form:

Format: a × 10ⁿ

Where:

• a is a number between 1 and 10 (1 ≤ a < 10)
• n is an integer (positive, negative, or zero)
• × 10ⁿ represents the power of 10

Examples

• 3,000,000 = 3 × 10⁶
• 0.0004 = 4 × 10⁻⁴
• 450,000 = 4.5 × 10⁵
• 0.0072 = 7.2 × 10⁻³

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Converting Ordinary Numbers to Standard Form

For Large Numbers (Greater than 1)

Method:

1. Write the number with a decimal point after the first non-zero digit
2. Count how many places you moved the decimal point to the left
3. This count is the positive power of 10

Example 1: Convert 5,400,000 to standard form

Solution:

1. Place decimal after first digit: 5.4
2. Count places moved: 6 places to the left
3. Answer: 5.4 × 10⁶

Example 2: Convert 123,000 to standard form

Solution:

1. Place decimal after first digit: 1.23
2. Count places moved: 5 places to the left
3. Answer: 1.23 × 10⁵

For Small Numbers (Less than 1)

Method:

1. Write the number with a decimal point after the first non-zero digit
2. Count how many places you moved the decimal point to the right
3. This count is the negative power of 10

Example 1: Convert 0.0008 to standard form

Solution:

1. Place decimal after first non-zero digit: 8
2. Count places moved: 4 places to the right
3. Answer: 8 × 10⁻⁴

Example 2: Convert 0.0000456 to standard form

Solution:

1. Place decimal after first non-zero digit: 4.56
2. Count places moved: 5 places to the right
3. Answer: 4.56 × 10⁻⁵

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Converting Standard Form to Ordinary Numbers

For Positive Powers of 10

Method: Move the decimal point to the right by the number in the power.

Example 1: Convert 3.2 × 10⁴ to an ordinary number

Solution:

1. Power is 4, so move decimal 4 places right
2. 3.2 × 10⁴ = 32,000

Example 2: Convert 7.85 × 10⁶ to an ordinary number

Solution:

1. Power is 6, so move decimal 6 places right
2. 7.85 × 10⁶ = 7,850,000

For Negative Powers of 10

Method: Move the decimal point to the left by the number in the power.

Example 1: Convert 4.5 × 10⁻³ to an ordinary number

Solution:

1. Power is -3, so move decimal 3 places left
2. 4.5 × 10⁻³ = 0.0045

Example 2: Convert 2.3 × 10⁻⁵ to an ordinary number

Solution:

1. Power is -5, so move decimal 5 places left
2. 2.3 × 10⁻⁵ = 0.000023

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Calculations with Standard Form

Multiplication

Method:

1. Multiply the coefficients (numbers in front)
2. Add the powers of 10
3. Adjust if the coefficient is not between 1 and 10

Formula: (a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ

Example: Calculate (3 × 10⁴) × (2 × 10⁵)

Solution:

1. Multiply coefficients: 3 × 2 = 6
2. Add powers: 4 + 5 = 9
3. Answer: 6 × 10⁹

Example: Calculate (4.2 × 10³) × (5 × 10⁶)

Solution:

1. Multiply coefficients: 4.2 × 5 = 21
2. Add powers: 3 + 6 = 9
3. Adjust: 21 × 10⁹ = 2.1 × 10¹⁰ (21 is not between 1 and 10)

Answer: 2.1 × 10¹⁰

Division

Method:

1. Divide the coefficients
2. Subtract the powers of 10
3. Adjust if the coefficient is not between 1 and 10

Formula: (a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ

Example: Calculate (8 × 10⁷) ÷ (2 × 10³)

Solution:

1. Divide coefficients: 8 ÷ 2 = 4
2. Subtract powers: 7 - 3 = 4
3. Answer: 4 × 10⁴

Example: Calculate (6 × 10⁸) ÷ (4 × 10²)

Solution:

1. Divide coefficients: 6 ÷ 4 = 1.5
2. Subtract powers: 8 - 2 = 6
3. Answer: 1.5 × 10⁶

Addition and Subtraction

Method:

1. Convert both numbers to have the same power of 10
2. Add or subtract the coefficients
3. Keep the same power of 10
4. Adjust if needed

Example: Calculate (3 × 10⁴) + (2 × 10⁴)

Solution:

1. Same power, so add coefficients: 3 + 2 = 5
2. Answer: 5 × 10⁴

Example: Calculate (5 × 10⁵) + (3 × 10⁴)

Solution:

1. Convert to same power: 3 × 10⁴ = 0.3 × 10⁵
2. Add: 5 × 10⁵ + 0.3 × 10⁵ = 5.3 × 10⁵

Answer: 5.3 × 10⁵

Example: Calculate (7 × 10³) - (2 × 10²)

Solution:

1. Convert to same power: 2 × 10² = 0.2 × 10³
2. Subtract: 7 × 10³ - 0.2 × 10³ = 6.8 × 10³

Answer: 6.8 × 10³

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Step-by-Step Method for Standard Form Problems

1. Identify the operation - Conversion, multiplication, division, addition, or subtraction?
2. For calculations: Ensure coefficients are between 1 and 10
3. Apply the rules:
   • Multiplication: Multiply coefficients, add powers
   • Division: Divide coefficients, subtract powers
   • Addition/Subtraction: Same power, then add/subtract coefficients
4. Adjust if needed - Coefficient must be 1 ≤ a < 10
5. Check your answer - Does it make sense?

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Worked Examples

Example 1: Converting to Standard Form

Convert 0.000072 to standard form.

Solution:

1. Place decimal after first non-zero: 7.2
2. Count places moved right: 5 places
3. Answer: 7.2 × 10⁻⁵

Example 2: Converting from Standard Form

Convert 4.5 × 10⁶ to an ordinary number.

Solution:

1. Power is 6, so move decimal 6 places right
2. Answer: 4,500,000

Example 3: Multiplication

Calculate (2.5 × 10³) × (4 × 10⁵)

Solution:

1. Multiply: 2.5 × 4 = 10
2. Add powers: 3 + 5 = 8
3. Adjust: 10 × 10⁸ = 1 × 10⁹

Answer: 1 × 10⁹

Example 4: Division

Calculate (8.4 × 10⁷) ÷ (2 × 10²)

Solution:

1. Divide: 8.4 ÷ 2 = 4.2
2. Subtract powers: 7 - 2 = 5
3. Answer: 4.2 × 10⁵

Example 5: Addition

Calculate (3 × 10⁴) + (5 × 10³)

Solution:

1. Convert: 5 × 10³ = 0.5 × 10⁴
2. Add: 3 × 10⁴ + 0.5 × 10⁴ = 3.5 × 10⁴

Answer: 3.5 × 10⁴

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Common Examiner Traps (and How to Dodge Them)

• Coefficient not between 1 and 10 - Always adjust after calculations
• Wrong power sign - Positive for large numbers, negative for small numbers
• Addition/subtraction errors - Must have same power of 10 first
• Decimal point placement - Count carefully when converting
• Forgetting to adjust - After multiplication, check if coefficient needs adjustment
• Calculator display - Some calculators show standard form differently

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IGCSE Standard Form Practice Questions

Question 1: Conversions

a) Convert 6,500,000 to standard form b) Convert 0.00034 to standard form c) Convert 3.2 × 10⁵ to an ordinary number

Solution: a) 6.5 × 10⁶ b) 3.4 × 10⁻⁴ c) 320,000

Question 2: Multiplication

Calculate (4 × 10³) × (3 × 10⁶)

Solution: 4 × 3 = 12, 3 + 6 = 9 12 × 10⁹ = 1.2 × 10¹⁰

Answer: 1.2 × 10¹⁰

Question 3: Division

Calculate (9 × 10⁸) ÷ (3 × 10²)

Solution: 9 ÷ 3 = 3, 8 - 2 = 6 3 × 10⁶

Answer: 3 × 10⁶

Question 4: Addition

Calculate (7 × 10⁴) + (2 × 10³)

Solution: 2 × 10³ = 0.2 × 10⁴ 7 × 10⁴ + 0.2 × 10⁴ = 7.2 × 10⁴

Answer: 7.2 × 10⁴

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Frequently Asked Questions About IGCSE Standard Form

What is standard form?

Standard form (scientific notation) writes numbers as a × 10ⁿ where 1 ≤ a < 10 and n is an integer.

How do I convert a large number to standard form?

Place decimal after first digit, count places moved left, use positive power. Example: 5,400,000 = 5.4 × 10⁶.

How do I convert a small number to standard form?

Place decimal after first non-zero digit, count places moved right, use negative power. Example: 0.0008 = 8 × 10⁻⁴.

How do I multiply numbers in standard form?

Multiply coefficients, add powers, then adjust if coefficient is not between 1 and 10.

How do I add numbers in standard form?

Convert to same power of 10, then add coefficients.

Why must the coefficient be between 1 and 10?

This is the standard convention - it ensures each number has a unique representation in standard form.

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Related IGCSE Maths Resources

Strengthen your IGCSE Mathematics preparation with these comprehensive guides:

• IGCSE Exponents and Surds: Complete Guide - Master index laws and surd operations
• IGCSE Estimation and Rounding: Complete Guide - Master rounding and significant figures
• IGCSE Maths Revision Notes, Syllabus and Preparation Tips - Complete syllabus overview, topic breakdown, and revision strategies
• IGCSE Past Papers Guide - Access free IGCSE past papers and exam resources

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