Making sense of negative numbers
Use direction, distance, and opposites to calculate confidently on both sides of zero.
Interpret a negative number as a position relative to zero.
Add and subtract signed integers using movement.
Explain why subtracting a negative increases a number.
Before we begin
You’ll use: Count whole numbers and add positive integers.
Try this quick warm-up. If it feels unfamiliar, follow the hint or revisit the previous lesson.
Start at 3 and move 4 places right. Where do you finish?
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Start with the idea
Imagine a thermometer. Zero is a reference point, not the end of the scale. A temperature of −3 °C is three degrees below zero. The minus sign tells you which side of zero a number occupies.
A number line makes this visible. Positive numbers lie to the right of zero; negative numbers lie to its left. The distance from zero is the absolute value: −3 and 3 are both three units from zero.
Addition describes a change in position. Adding a positive number moves right. Adding a negative number moves left. Subtraction reverses a change: subtracting −4 undoes a move four places left, so it moves four places right.
The additive inverse of a number is , because . Subtraction is addition of the additive inverse: . In particular, . This rule follows from undoing a change; it is not a rule that every pair of minus signs can simply be removed.
Make the idea move
Move right for a positive change, left for a negative one. What value takes you exactly to zero?
Three examples, step by step
A simple move: −2 + 5
- Begin at −2, two places left of zero.
- Adding 5 means move five places right: −1, 0, 1, 2, 3.
- So . You cross zero; the starting sign does not determine the final sign.
Subtract a negative: 4 − (−3)
- Rewrite subtraction as adding the opposite: .
- The opposite of −3 is 3, so move three places right.
- The answer is 7. Check by reversing the operation: .
Temperature change
- At dawn the temperature is −6 °C. It rises by 9 °C, then falls by 5 °C.
- After the rise: . After the fall: .
- The final temperature is −2 °C. The total change is +4 °C, so is a second check.
Your turn, with a little guidance
Calculate −5 − (−8). First rewrite it as addition, then give the result.
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Independent practice
Work on paper if it helps. Give a number, decimal, or fraction such as 3/2. Correct answers to every independent exercise complete this lesson; placement mastery requires a separate assessment.
Calculate −7 + 12.
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Calculate 3 − 9.
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Calculate −4 − (−10).
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A balance is −£11. A £7 deposit arrives. What is the new balance in pounds?
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Easy mistakes. Useful lessons.
The sign depends on the final position. Twelve rightward steps outweigh seven leftward units.
That would subtract +4. Subtracting −4 reverses a leftward change and gives 7.
The knowledge check
Close the examples and explain each point below in your own words. Then solve one practice question again without a hint.
- Negative numbers describe positions below a reference point.
- Addition is a directed change; subtraction adds the opposite.
- Check a subtraction by adding the subtracted number back.
If you cannot explain a step yet, return to the visualization and change one value. What stays the same, and why?