Step-by-step worked examples and graded practice questions on negative numbers — ordering positive and negative numbers, and the four operations with negative numbers.
📚 Foundation & Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
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A negative number is a number less than zero, written with a minus sign — for example −5. Negative numbers are used to represent values below a starting point, such as temperatures below freezing, depths below sea level, or a bank balance that's overdrawn.
On a number line, negative numbers sit to the left of zero. The further left a number is, the smaller its value — so −10 is smaller than −2, even though 10 is bigger than 2.
Numbers increase in value from left to right. −5 is smaller than −3, and −3 is smaller than 0.
Ordering positive and negative numbers
Worked Example 1
Write these numbers in order, smallest first: 3, −7, 0, −2, 5, −1
1
Picture the number line: the most negative number is smallest, and the most positive number is largest
2
Order from furthest left to furthest right: −7, −2, −1, 0, 3, 5
Answer−7, −2, −1, 0, 3, 5
Adding and subtracting negative numbers
Two signs written next to each other combine into a single sign: adding a negative is the same as subtracting, and subtracting a negative is the same as adding.
Written as
Simplifies to
Example
+ followed by −
−
5 + (−3) = 5 − 3 = 2
− followed by −
+
5 − (−3) = 5 + 3 = 8
− followed by +
−
5 − (+3) = 5 − 3 = 2
Worked Example 2
Work out (a) −4 + 9 and (b) −4 − (−9)
1
(a) Start at −4 and move 9 places to the right (adding moves right): −4 + 9 = 5
2
(b) Two minus signs together become a plus: −4 − (−9) = −4 + 9 = 5
Answer(a) 5 (b) 5
Multiplying and dividing negative numbers
For multiplying and dividing, count how many negative signs are involved: an even number of negatives gives a positive answer; an odd number of negatives gives a negative answer.
Signs
Result
positive × positive
positive
positive × negative
negative
negative × negative
positive
Worked Example 3
Work out (a) −6 × 3, (b) −6 × −3, and (c) −20 ÷ −4
1
(a) One negative sign (odd) → negative answer: 6 × 3 = 18, so −6 × 3 = −18
2
(b) Two negative signs (even) → positive answer: 6 × 3 = 18, so −6 × −3 = 18
3
(c) Two negative signs (even) → positive answer: 20 ÷ 4 = 5, so −20 ÷ −4 = 5
Answer(a) −18 (b) 18 (c) 5
Negative numbers in context
Worked Example 4
The temperature at 6am was −4°C. By midday it had risen by 11°C. What was the temperature at midday?
1
A rise in temperature means adding: −4 + 11
2
Start at −4 on the number line and move 11 places right: −4 + 11 = 7
Answer7°C
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1Write these numbers in order, smallest first: 4, −6, −1, 2, −8Foundation
Q2Work out −3 + 8Foundation
Q3Work out 6 − 10Foundation
Q4Work out −5 × 4Foundation
Q5The temperature was −2°C. It fell by 5°C. What is the new temperature?Foundation
Higher (Grade 5–7)
Q6Work out −7 − (−12)Higher
Q7Work out −8 × −6Higher
Q8Work out −36 ÷ 9Higher
Q9Work out −45 ÷ −5Higher
Q10The temperature at midnight was −6°C. By 3pm it had risen to 9°C. By how many degrees did it rise?Higher
Higher — Hard (Grade 8–9)
Q11Work out (−3)² and −3², explaining why the two answers are different.Grade 8–9
Q12Work out (−2) × (−3) × (−4)Grade 8–9
Q13The temperature in London is −3°C. The temperature in Moscow is 4 times as cold. Work out the temperature in Moscow.Grade 8–9
Q14Work out −2 − 3 × (−4)Grade 8–9
Q15Two numbers multiply to give a positive answer and add to give a negative answer. What can you say about the signs of the two numbers?Grade 8–9
Answers
Foundation (Q1–Q5)
Q1−8, −6, −1, 2, 4
Q25
Q3−4
Q4−20
Q5−7°C
Higher (Q6–Q10)
Q65(−7 + 12)
Q748
Q8−4
Q99
Q1015°C(9 − (−6))
Higher — Hard (Q11–Q15)
Q11(−3)² = 9; −3² = −9(the brackets mean "square −3"; without brackets, only the 3 is squared, then the minus is applied)
Q12−24(three negatives — odd — gives a negative answer)
Q15Both numbers must be negative — two negatives multiply to a positive, and two negatives add to a negative.
Common mistakes
Common Mistake 1
Thinking −8 is bigger than −3
Because 8 is bigger than 3, it's tempting to think −8 > −3. In fact −8 is smaller — it's further to the left on the number line, and further below zero.
Common Mistake 2
Not combining double signs before calculating
5 − (−3) is not 5 − 3. The two minus signs combine into a plus first: 5 − (−3) = 5 + 3 = 8.
Common Mistake 3
Confusing (−3)² with −3²
(−3)² means −3 multiplied by itself: (−3) × (−3) = 9. But −3² means "the negative of 3 squared": −(3 × 3) = −9. The brackets change the answer completely.
Common Mistake 4
Losing track of the sign in multi-step calculations
In questions like −2 − 3 × (−4), always do multiplication before subtraction, and carry the negative sign through every step — writing each stage down avoids losing track.
Exam tips
💡 Exam Tip 1
Sketch a number line for tricky questions
If you're unsure whether to add or subtract, a quick number line sketch showing your starting point and direction of movement removes the guesswork.
💡 Exam Tip 2
Simplify double signs first
Before doing any arithmetic, rewrite "+ −", "− −" and "− +" as a single sign. This turns a negative-number question into an ordinary addition or subtraction.
💡 Exam Tip 3
Count the negative signs for multiplication and division
Even count of negatives → positive answer. Odd count of negatives → negative answer. This works no matter how many numbers are being multiplied or divided together.
💡 Exam Tip 4
Watch out for missing brackets
Always check whether a "squared" negative number has brackets around it. (−4)² and −4² give different answers — read the question exactly as written.
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