Introduction to Negative Numbers
What Is This?
Negative numbers are numbers less than zero. They are written with a minus sign in front: −1, −2, −3, and so on. Together with zero and the positive numbers, they form the complete number line that extends in both directions forever.
Why Does It Matter?
Negative numbers describe real situations every day: temperatures below freezing (−5°C), money owed (a debt of −$20), floors below ground level (basement level −1), depths below sea level (−400 m), and golf scores below par. Understanding negatives is also the gateway to algebra and coordinate graphing.
The Number Line
The number line extends in both directions from zero:
←─┼──┼──┼──┼──┼──┼──┼──┼──┼──┼──┼─→
−5 −4 −3 −2 −1 0 1 2 3 4 5
- Numbers to the right of zero are positive.
- Numbers to the left of zero are negative.
- Zero is neither positive nor negative — it is the boundary between them.
Key Insight: Direction Matters
Moving right on the number line means the numbers get larger. Moving left means the numbers get smaller.
This stays true on the negative side: −1 is larger than −5, because −1 is further right.
Comparing Negative Numbers
This is the most important — and most counterintuitive — rule:
The negative number closer to zero is the larger one.
| Comparison | Answer | Why |
|---|---|---|
| −2 vs −7 | −2 is larger | −2 is closer to zero (further right) |
| −10 vs −3 | −3 is larger | −3 is closer to zero |
| −1 vs 1 | 1 is larger | Any positive beats any negative |
| 0 vs −4 | 0 is larger | Zero beats every negative number |
⚠️ Think of temperature: −2°C is warmer than −7°C. Warmer = larger number.
Ordering Numbers with Negatives
To order from smallest to largest, work left to right along the number line.
Example: Order −3, 5, −8, 0, 2 from smallest to largest.
- Most negative first: −8, −3, 0, 2, 5 ✓
Example: Order 4, −1, −6, 3 from largest to smallest.
- Most positive first: 4, 3, −1, −6 ✓
Counting Through Zero
Counting backwards past zero continues smoothly into the negatives:
3, 2, 1, 0, −1, −2, −3, …
Counting forwards from a negative number rises through zero:
−3, −2, −1, 0, 1, 2, 3, …
Real-World Contexts
Temperature
The most familiar context. If the temperature is 4°C and it falls by 7 degrees:
- 4 − 7 = −3°C (three degrees below freezing) ✓
If the temperature is −5°C and it rises by 8 degrees:
- −5 + 8 = 3°C ✓
Money (Debt)
Owing money can be shown as a negative balance.
- You have $10 and spend $14. Balance: 10 − 14 = −$4 (you owe $4)
Elevation
Heights above sea level are positive; depths below are negative.
- A submarine at −120 m rises 50 m: −120 + 50 = −70 m ✓
Buildings
Ground floor = 0, basements are negative floors.
- From floor 2, going down 3 floors: 2 − 3 = floor −1 ✓
Finding the Difference (Distance Between)
The difference between two values is the distance between them on the number line.
Example: The temperature was −4°C at night and 6°C at midday. By how much did it rise?
- From −4 up to 0 is 4 degrees; from 0 up to 6 is 6 more.
- Total rise: 4 + 6 = 10 degrees ✓
Example: What is the difference between −8 and −3?
- From −8 to −3 is 5 (count the jumps: −8 → −7 → −6 → −5 → −4 → −3) ✓
Worked Examples
Worked Example 1: Compare
Which is larger: −6 or −2?
- −2 is closer to zero → −2 is larger ✓
Worked Example 2: Order
Order from smallest to largest: 1, −5, 3, −2, 0
- −5, −2, 0, 1, 3 ✓
Worked Example 3: Temperature drop
The temperature is 2°C and falls 9 degrees overnight. What is the new temperature?
- 2 − 9 = −7°C ✓
Worked Example 4: Temperature rise
The temperature is −6°C and rises 10 degrees. What is the new temperature?
- −6 + 10 = 4°C ✓
Worked Example 5: Difference across zero
A diver is at −15 m. A bird is flying at 25 m. What is the vertical distance between them?
- From −15 to 0: 15 m. From 0 to 25: 25 m.
- Total distance: 15 + 25 = 40 m ✓
Common Mistakes
Mistake 1: Thinking −7 is larger than −2 because 7 > 2 With negatives, the rule reverses: −7 is further from zero on the small side, so −7 < −2. Always picture the number line.
Mistake 2: Treating zero as positive or negative Zero is neither. It is the boundary point.
Mistake 3: Forgetting to pass through zero when counting Counting down from 2: 2, 1, 0, −1, −2 — don't skip zero.
Mistake 4: Subtracting the wrong way in temperature problems "Falls by 9 from 2°C" means 2 − 9 = −7, not 9 − 2 = 7. The starting value comes first.
Key Takeaways
- Negative numbers are less than zero and sit to the left of zero on the number line
- Zero is neither positive nor negative
- The negative number closer to zero is the larger one (−2 > −7)
- Any positive number is larger than any negative number
- Count smoothly through zero in both directions
- Real-world negatives: temperature, debt, elevation, building floors
Practice and Resources
Ready to practise? Try our Grade 3-5 Introduction to Negative Numbers Worksheet with number lines, comparing, ordering, and temperature problems. Test yourself with the Grade 3-5 Math Practice Test.