Find: \(5 + (-3)\).
Adding a negative means subtract.
5 - 3 = 2.
Integers and Directed Numbers
Directed numbers include both positive and negative values and are used to represent change, temperature and movement. They are often used alongside order of operations, and in GCSE Maths you need to apply rules of operations with negative numbers accurately.
Overview
Integers are whole numbers, including negative numbers, zero and positive numbers.
Directed numbers are numbers that have a direction, shown by a plus or minus sign.
\( \dots, -3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,\ \dots \)
Negative numbers appear in contexts such as temperature, money, coordinates and graphs, so it is important to work confidently with signs in calculations.
What you should understand after this topic
- Understand what integers are
- Understand what directed numbers mean
- Add and subtract positive and negative numbers
- Multiply and divide signed numbers
- Avoid common mistakes with signs
Key Definitions
Integer
A whole number that can be negative, zero or positive.
Directed Number
A number with a positive or negative sign showing direction or value.
Positive Number
A number greater than zero.
Negative Number
A number less than zero.
Zero
The number that is neither positive nor negative.
Opposite
A number the same distance from zero on the other side of the number line.
Key Rules
How to Solve
Step 1: Understand integers and directed numbers
Integers are whole numbers, both positive and negative, including zero.
Directed numbers include positive and negative values.
Integers
\(-5,\ -1,\ 0,\ 3,\ 12\)
Not integers
\(2.5,\ \frac{1}{3},\ -4.7\)
Step 2: Use a number line
Numbers increase to the right and decrease to the left.
Exam tip: Right = positive, left = negative.
Step 3: Adding directed numbers
Think of movement on a number line.
\( 3 + (-5) = -2 \)
\( -4 + 7 = 3 \)
Positive → move right
Negative → move left
Quick rule: Same signs → add, different signs → subtract.
Step 4: Subtracting directed numbers
Change subtraction into addition of the opposite.
\( 5 - (-3) = 5 + 3 = 8 \)
\( -2 - 6 = -8 \)
Key rule: Subtracting a negative = adding a positive.
Step 5: Multiplying and dividing
Use sign rules after calculating the numbers.
Exam tip: Same signs → positive, different signs → negative.
+ × +
Positive
+ × −
Negative
− × +
Negative
− × −
Positive
Step 6: Comparing numbers
Numbers further to the right are greater.
\( -2 > -5 \)
Negative numbers closer to zero are greater.
Step 7: Apply to real situations
See order of operations for combining calculations.
Temperature
\(-3^\circ C\) means below zero.
Money
\(-£20\) means owing money.
Floors
\(-1\) means below ground level.
Coordinates
Negative values show direction.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics.
Edexcel
Work out \( -7 + 12 \).
Edexcel
Work out \( -9 - 5 \).
Edexcel
Work out \( -6 \times 4 \).
Edexcel
Work out \( -20 \div (-5) \).
Edexcel
Work out \( 5 - (-8) + 3 \).
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on reasoning and real-life applications of directed numbers.
AQA
The temperature in a city is −3°C at midnight. By midday, it has risen by 11°C. What is the temperature at midday?
AQA
A submarine is at a depth of −45 metres. It rises 18 metres and then descends 12 metres. What is its final depth?
AQA
Evaluate \( -4^2 \).
AQA
Evaluate \( (-4)^2 \).
AQA
Explain why \( -4^2 \) and \( (-4)^2 \) give different answers.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising order of operations and algebraic reasoning with directed numbers.
OCR
Arrange the following numbers in order of size, starting with the smallest: \( -5,\; 3,\; -2,\; 0,\; 7 \).
OCR
Work out \( -3 \times (-7) + 5 \).
OCR
Work out \( 12 - 3(-4) \).
OCR
Find the value of \( x \) if \( x - 7 = -12 \).
OCR
A bank account is overdrawn by £35. A deposit of £60 is made, followed by a withdrawal of £18. Calculate the final balance.
Exam Checklist
Step 1
Read the signs carefully before starting.
Step 2
Use a number line if the question feels confusing.
Step 3
For subtraction, think about adding the opposite.
Step 4
For multiplication and division, use the sign rules separately from the number part.
Most common exam mistakes
Add / subtract confusion
Forgetting that \( - - \) becomes \( + \).
Sign rule confusion
Writing a negative answer for negative times negative.
Ordering negatives
Thinking the number with the bigger digits is always greater.
Bracket confusion
Ignoring brackets around a negative number.
Common Mistakes
These are common mistakes students make when working with integers and directed numbers in GCSE Maths.
Assuming subtraction always makes numbers smaller
Incorrect
A student thinks subtracting any number reduces the value.
Correct
Subtracting a negative increases the value. For example, \(5 - (-3) = 5 + 3 = 8\).
Forgetting that subtracting a negative means adding
Incorrect
A student writes \(7 - (-2) = 5\).
Correct
Subtracting a negative is the same as adding. \(7 - (-2) = 7 + 2 = 9\).
Mixing up sign rules in multiplication and division
Incorrect
A student gets the sign wrong when multiplying or dividing negatives.
Correct
Remember the rules: positive × positive = positive, negative × negative = positive, and positive × negative = negative.
Misunderstanding negative number size
Incorrect
A student says \(-8\) is greater than \(-3\).
Correct
On a number line, numbers further to the left are smaller. So \(-8 < -3\).
Ignoring brackets with negative numbers
Incorrect
A student calculates \(-3^2\) as 9 instead of \(-9\).
Correct
Without brackets, the square applies only to the number: \(-3^2 = -9\). To square the negative, write \((-3)^2 = 9\).
Try It Yourself
Practise calculating with positive and negative integers.
Foundation
Foundation Practice
Understand addition and subtraction with positive and negative numbers.
Find: \(7 + (-2)\).
Subtract 2 from 7.
7 - 2 = 5.
Find: \(-4 + 6\).
Move 6 steps right from -4.
-4 + 6 = 2.
Find: \(-8 + 3\).
Move towards zero.
-8 + 3 = -5.
Find: \(6 - 9\).
Subtracting a larger number gives negative.
6 - 9 = -3.
Find: \(4 - 7\).
Think number line.
4 - 7 = -3.
Find: \(-3 - 4\).
Move further left.
-3 - 4 = -7.
Find: \(-6 - 2\).
Subtract means go left.
-6 - 2 = -8.
A student says \(-5 + 2 = 7\). What is wrong?
Check direction on number line.
-5 + 2 = -3, not 7.
Find: \(-10 + 6\).
Move toward zero.
-10 + 6 = -4.
Higher
Higher Practice
Work with multiplication, division and mixed operations.
Find: \((-3) × 4\).
Negative × positive = negative.
-3 × 4 = -12.
Find: \((-5) × 3\).
Multiply normally, keep sign.
-15.
Find: \((-4) × (-2)\).
Negative × negative = positive.
(-4)(-2) = 8.
Find: \((-6) × (-3)\).
Two negatives make positive.
18.
Find: \(12 ÷ (-3)\).
Positive ÷ negative = negative.
12 ÷ -3 = -4.
Find: \((-15) ÷ 5\).
Divide normally, keep sign.
-3.
Find: \((-20) ÷ (-4)\).
Negative ÷ negative = positive.
5.
Find: \((-24) ÷ (-6)\).
Divide signs first.
4.
Find: \(-3 + 5 × (-2)\).
Do multiplication first.
5 × -2 = -10 → -3 + (-10) = -13.
Find: \(-4 × (-2) + 3\).
Multiply first.
-4 × -2 = 8 → 8 + 3 = 11.
Games
Practise this topic with interactive games.
Games coming soon.
Integers and Directed Numbers Video Tutorial
Frequently Asked Questions
What are directed numbers?
Directed numbers are numbers with a sign, either positive or negative, used to show direction or change.
How do I add negative numbers?
Adding a negative is the same as subtracting. For example, 5 + (-3) = 2.
Why do two negatives make a positive when multiplying?
Multiplying two negatives gives a positive because the direction reverses twice.