IGCSE Directed Numbers: Complete Guide for Cambridge IGCSE Mathematics

IGCSE directed numbers (positive and negative numbers) are fundamental concepts in Cambridge IGCSE Mathematics that appear throughout the curriculum. Mastering operations with positive and negative numbers is essential for solving algebraic problems, working with coordinates, and understanding real-world scenarios like temperature, elevation, and financial transactions.

This comprehensive IGCSE directed numbers guide covers everything you need to know, including number line representation, addition and subtraction rules, multiplication and division rules, order of operations, 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 add, subtract, multiply, and divide positive and negative numbers confidently, use the number line effectively, 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 Directed Numbers Matter

IGCSE directed numbers are essential building blocks that appear in almost every topic. Here’s why they’re so important:

• Foundation topic: Required for understanding algebra, coordinates, and all number operations
• High frequency: Questions involving negative numbers appear in every IGCSE maths paper
• Real-world applications: Used in temperature, elevation, financial transactions, and scientific calculations
• Algebra foundation: Essential for solving equations and working with algebraic expressions
• Common errors: Many students lose marks due to sign errors - this guide will help you avoid them

Key insight from examiners: Students often make mistakes with double negatives, subtraction of negative numbers, and the order of operations. This guide will help you master these systematically.

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Understanding Directed Numbers: The Basics

Directed numbers are numbers that have both size (magnitude) and direction. They include:

• Positive numbers: +1, +2, +3, … (or simply 1, 2, 3, …)
• Negative numbers: -1, -2, -3, …
• Zero: 0 (neither positive nor negative)

The Number Line

The number line is a visual representation of numbers:

<---|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|---->
   -5   -4   -3   -2   -1    0    1    2    3    4    5

Key points:

• Numbers to the right are greater than numbers to the left
• Zero is the reference point
• Negative numbers are to the left of zero
• Positive numbers are to the right of zero

Real-World Examples

• Temperature: -5°C (5 degrees below zero), +20°C (20 degrees above zero)
• Elevation: -50 m (50 meters below sea level), +100 m (100 meters above sea level)
• Money: -$50 (a debt of $50), +$100 (a credit of $100)
• Time: -3 hours (3 hours before), +2 hours (2 hours after)

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Addition of Directed Numbers

Rule 1: Adding Two Positive Numbers

Rule: (+a) + (+b) = +(a + b)

Simply add the numbers and keep the positive sign.

Examples:

• (+5) + (+3) = +8 or simply 5 + 3 = 8
• (+12) + (+7) = +19 or 12 + 7 = 19

Rule 2: Adding Two Negative Numbers

Rule: (-a) + (-b) = -(a + b)

Add the numbers and keep the negative sign.

Examples:

• (-5) + (-3) = -8
• (-12) + (-7) = -19

Memory tip: “Two negatives make a more negative”

Rule 3: Adding Positive and Negative Numbers

Rule: Subtract the smaller absolute value from the larger, and keep the sign of the number with the larger absolute value.

Method:

1. Find the difference between the numbers (ignore signs)
2. Keep the sign of the number with the larger absolute value

Examples:

• (+8) + (-3) = +5 (8 - 3 = 5, and 8 > 3, so positive)
• (-8) + (+3) = -5 (8 - 3 = 5, and 8 > 3, so negative)
• (+5) + (-7) = -2 (7 - 5 = 2, and 7 > 5, so negative)

Alternative method (number line):

• Start at the first number
• Move right for positive, left for negative
• Where you end up is your answer

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Subtraction of Directed Numbers

Key Principle

Subtracting is the same as adding the opposite: a - b = a + (-b)

Rule 1: Subtracting a Positive Number

Rule: a - (+b) = a + (-b)

Change subtraction to addition of the negative.

Examples:

• (+8) - (+3) = (+8) + (-3) = +5
• (-5) - (+2) = (-5) + (-2) = -7

Rule 2: Subtracting a Negative Number

Rule: a - (-b) = a + (+b)

Subtracting a negative is the same as adding a positive.

Examples:

• (+8) - (-3) = (+8) + (+3) = +11
• (-5) - (-2) = (-5) + (+2) = -3

Memory tip: “Two negatives make a positive” (when subtracting)

Rule 3: Double Negatives

Rule: -(-a) = +a

A negative sign in front of a negative number makes it positive.

Examples:

• -(-5) = +5
• -(-12) = +12

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Multiplication of Directed Numbers

The Sign Rules for Multiplication

First Number	Second Number	Result
+	+	+
+	-	-
-	+	-
-	-	+

Memory tip:

• Same signs → positive result
• Different signs → negative result

Examples

Same signs (positive result):

• (+5) × (+3) = +15 or 5 × 3 = 15
• (-5) × (-3) = +15

Different signs (negative result):

• (+5) × (-3) = -15
• (-5) × (+3) = -15

Multiplying More Than Two Numbers

Method: Count the number of negative signs

• Even number of negatives → positive result
• Odd number of negatives → negative result

Examples:

• (-2) × (-3) × (-4) = ?

  • 3 negative signs (odd) → negative result
  • 2 × 3 × 4 = 24
  • Answer: -24

• (-1) × (-2) × (-3) × (-4) = ?

  • 4 negative signs (even) → positive result
  • 1 × 2 × 3 × 4 = 24
  • Answer: +24

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Division of Directed Numbers

The Sign Rules for Division

The rules are the same as multiplication:

First Number	Second Number	Result
+	+	+
+	-	-
-	+	-
-	-	+

Memory tip: Same as multiplication - same signs positive, different signs negative.

Examples

Same signs (positive result):

• (+15) ÷ (+3) = +5 or 15 ÷ 3 = 5
• (-15) ÷ (-3) = +5

Different signs (negative result):

• (+15) ÷ (-3) = -5
• (-15) ÷ (+3) = -5

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Order of Operations (BODMAS/BIDMAS)

When working with directed numbers in expressions, follow the order of operations:

1. Brackets
2. Orders (powers, roots)
3. Division and Multiplication (left to right)
4. Addition and Subtraction (left to right)

Example: Calculate: -3 + 4 × (-2)

Solution:

1. Multiplication first: 4 × (-2) = -8
2. Then addition: -3 + (-8) = -11

Answer: -11

Example: Calculate: (-5)² - 3 × (-2)

Solution:

1. Orders first: (-5)² = 25 (negative squared is positive)
2. Multiplication: 3 × (-2) = -6
3. Subtraction: 25 - (-6) = 25 + 6 = 31

Answer: 31

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Step-by-Step Method for Directed Number Problems

1. Identify the operation(s) - Addition, subtraction, multiplication, or division?
2. Apply the sign rules - Use the rules for that operation
3. Follow BODMAS - If there are multiple operations
4. Check your answer - Does it make sense? Use the number line to verify

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Worked Examples

Example 1: Addition

Calculate: (-7) + (+12)

Solution:

1. Different signs, so subtract: 12 - 7 = 5
2. Larger number is positive, so answer is positive

Answer: +5

Example 2: Subtraction

Calculate: (-8) - (-5)

Solution:

1. Subtracting a negative = adding a positive: (-8) - (-5) = (-8) + (+5)
2. Different signs, so subtract: 8 - 5 = 3
3. Larger number is negative, so answer is negative

Answer: -3

Example 3: Multiplication

Calculate: (-4) × (-6) × (-2)

Solution:

1. Count negatives: 3 (odd number) → negative result
2. Multiply: 4 × 6 × 2 = 48

Answer: -48

Example 4: Division

Calculate: (-24) ÷ (+6)

Solution:

1. Different signs → negative result
2. Divide: 24 ÷ 6 = 4

Answer: -4

Example 5: Mixed Operations

Calculate: -5 + 3 × (-2) - (-4)

Solution:

1. Multiplication first: 3 × (-2) = -6
2. Expression becomes: -5 + (-6) - (-4)
3. Simplify: -5 - 6 + 4 (subtracting negative = adding positive)
4. Add: -11 + 4 = -7

Answer: -7

Example 6: Powers

Calculate: (-3)² and (-3)³

Solution:

• (-3)² = (-3) × (-3) = +9 (even power → positive)
• (-3)³ = (-3) × (-3) × (-3) = -27 (odd power → negative)

Answers: +9 and -27

Key rule:

• Even powers of negative numbers are positive
• Odd powers of negative numbers are negative

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Common Examiner Traps (and How to Dodge Them)

• Double negatives: -(-5) = +5, not -5
• Subtracting negatives: 5 - (-3) = 5 + 3 = 8, not 5 - 3 = 2
• Order of operations: Always do multiplication/division before addition/subtraction
• Powers of negatives: (-2)² = +4, but -2² = -4 (brackets matter!)
• Sign errors in multiplication: Remember: same signs = positive, different signs = negative
• Forgetting zero: 0 - 5 = -5, not 5

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IGCSE Directed Numbers Practice Questions

Question 1: Addition and Subtraction

Calculate: a) (-12) + (+8) b) (+15) - (-7) c) (-9) - (+4)

Solution: a) (-12) + (+8) = -4 (12 - 8 = 4, negative is larger) b) (+15) - (-7) = 15 + 7 = 22 c) (-9) - (+4) = -9 - 4 = -13

Answers: a) -4 b) +22 c) -13

Question 2: Multiplication and Division

Calculate: a) (-6) × (-4) b) (+18) ÷ (-3) c) (-2) × (+5) × (-3)

Solution: a) (-6) × (-4) = +24 (same signs) b) (+18) ÷ (-3) = -6 (different signs) c) (-2) × (+5) × (-3) = +30 (2 negatives = even = positive)

Answers: a) +24 b) -6 c) +30

Question 3: Mixed Operations

Calculate: -8 + 4 × (-3) - (-5)

Solution:

1. Multiplication: 4 × (-3) = -12
2. Expression: -8 + (-12) - (-5) = -8 - 12 + 5
3. Calculate: -20 + 5 = -15

Answer: -15

Question 4: Powers

Calculate: a) (-4)² b) (-4)³ c) -4²

Solution: a) (-4)² = 16 (even power) b) (-4)³ = -64 (odd power) c) -4² = -16 (no brackets, so only 4 is squared)

Answers: a) 16 b) -64 c) -16

Question 5: Word Problem

The temperature at midnight was -3°C. By noon, it had risen by 8°C. Then it dropped by 5°C. What was the final temperature?

Solution:

1. Starting temperature: -3°C
2. Rises by 8°C: -3 + 8 = +5°C
3. Drops by 5°C: +5 - 5 = 0°C

Answer: 0°C

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Frequently Asked Questions About IGCSE Directed Numbers

What are directed numbers?

Directed numbers are numbers with both size and direction - they include positive numbers, negative numbers, and zero. Examples: +5, -3, 0.

How do I add a positive and negative number?

Subtract the smaller absolute value from the larger, and keep the sign of the number with the larger absolute value. Example: (+8) + (-3) = +5.

What happens when I subtract a negative number?

Subtracting a negative is the same as adding a positive: a - (-b) = a + b. Example: 5 - (-3) = 5 + 3 = 8.

How do I multiply negative numbers?

• Same signs (both positive or both negative) → positive result
• Different signs (one positive, one negative) → negative result

What’s the difference between (-2)² and -2²?

• (-2)² = 4 (the negative 2 is squared, so the whole thing is positive)
• -2² = -4 (only the 2 is squared, then the negative is applied)

Brackets matter!

How do I remember the rules?

Addition/Subtraction:

• Two positives or two negatives: add and keep the sign
• One positive, one negative: subtract and keep the sign of the larger

Multiplication/Division:

• Same signs → positive
• Different signs → negative

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Related IGCSE Maths Resources

Strengthen your IGCSE Mathematics preparation with these comprehensive guides:

• IGCSE Factors and Multiples: Complete Guide - Master prime factorization, HCF, and LCM
• IGCSE Set Language and Absolute Value: Complete Guide - Master set notation and absolute value equations
• 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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