Addition and Subtraction on Number Line
A number line is a straight line on which every point stands for a number. You have used it before with whole numbers — now you will use it with integers (positive and negative numbers). The number line is one of the best tools to understand addition and subtraction of integers because it turns arithmetic into simple movement.
The basic idea is: addition means moving to the right and subtraction means moving to the left. When you add a positive number, you step to the right. When you add a negative number (or subtract a positive number), you step to the left. Once you master this, you can solve any integer addition or subtraction problem just by counting steps.
This topic is especially useful for Class 6 students who are learning integers for the first time. Instead of memorising complicated sign rules, you can simply draw a number line and walk along it!
What is Addition and Subtraction on Number Line?
Definition: Addition and subtraction on a number line is the process of finding sums and differences by moving along a number line.
Rules for movement:
- Adding a positive number → move to the right
- Adding a negative number → move to the left
- Subtracting a positive number → move to the left
- Subtracting a negative number → move to the right
Steps to add or subtract on a number line:
- Draw a number line and mark the numbers on it.
- Locate the first number on the number line. Put your finger on it.
- Look at the operation and the second number to decide direction and number of steps.
- Count the steps and land on the answer.
Why subtraction of a negative means moving right:
Subtracting a negative is the same as adding a positive. Think of it as removing a debt — if someone cancels a Rs. 5 debt you owe, you are Rs. 5 richer. So 3 − (−5) = 3 + 5 = 8.
Types and Properties
1. Adding a Positive Number on the Number Line
Start at the first number and move right by the number of steps equal to the second number.
- Example: 3 + 4 → Start at 3, move 4 steps right → land on 7.
- Example: −2 + 5 → Start at −2, move 5 steps right → land on 3.
2. Adding a Negative Number on the Number Line
Start at the first number and move left by the number of steps equal to the absolute value of the second number.
- Example: 5 + (−3) → Start at 5, move 3 steps left → land on 2.
- Example: −1 + (−4) → Start at −1, move 4 steps left → land on −5.
3. Subtracting a Positive Number on the Number Line
Start at the first number and move left.
- Example: 6 − 4 → Start at 6, move 4 steps left → land on 2.
- Example: 2 − 7 → Start at 2, move 7 steps left → land on −5.
4. Subtracting a Negative Number on the Number Line
Start at the first number and move right (because subtracting a negative is like adding a positive).
- Example: 3 − (−2) → Start at 3, move 2 steps right → land on 5.
- Example: −4 − (−6) → Start at −4, move 6 steps right → land on 2.
Solved Examples
Example 1: Example 1: Adding two positive numbers
Problem: Find 4 + 5 using a number line.
Solution:
- Start at 4 on the number line.
- Move 5 steps to the right (adding a positive number).
- 4 → 5 → 6 → 7 → 8 → 9
Answer: 4 + 5 = 9
Example 2: Example 2: Adding a negative number to a positive number
Problem: Find 6 + (−4) using a number line.
Solution:
- Start at 6 on the number line.
- Move 4 steps to the left (adding a negative number).
- 6 → 5 → 4 → 3 → 2
Answer: 6 + (−4) = 2
Example 3: Example 3: Adding two negative numbers
Problem: Find (−3) + (−5) using a number line.
Solution:
- Start at −3 on the number line.
- Move 5 steps to the left (adding a negative number).
- −3 → −4 → −5 → −6 → −7 → −8
Answer: (−3) + (−5) = −8
Example 4: Example 4: Adding a positive number to a negative number
Problem: Find (−7) + 4 using a number line.
Solution:
- Start at −7 on the number line.
- Move 4 steps to the right (adding a positive number).
- −7 → −6 → −5 → −4 → −3
Answer: (−7) + 4 = −3
Example 5: Example 5: Subtracting a positive number
Problem: Find 3 − 8 using a number line.
Solution:
- Start at 3 on the number line.
- Move 8 steps to the left (subtracting a positive number).
- 3 → 2 → 1 → 0 → −1 → −2 → −3 → −4 → −5
Answer: 3 − 8 = −5
Example 6: Example 6: Subtracting a negative number
Problem: Find 2 − (−6) using a number line.
Solution:
- Start at 2 on the number line.
- Subtracting a negative means move right. Move 6 steps to the right.
- 2 → 3 → 4 → 5 → 6 → 7 → 8
Answer: 2 − (−6) = 2 + 6 = 8
Example 7: Example 7: Subtraction resulting in zero
Problem: Find (−4) − (−4) using a number line.
Solution:
- Start at −4 on the number line.
- Subtracting (−4) means move 4 steps right.
- −4 → −3 → −2 → −1 → 0
Answer: (−4) − (−4) = 0
Example 8: Example 8: Multiple operations on a number line
Problem: Find (−2) + 7 − 3 using a number line.
Solution:
- Start at −2.
- Add 7: move 7 steps right → −2 → −1 → 0 → 1 → 2 → 3 → 4 → 5.
- Subtract 3: from 5, move 3 steps left → 5 → 4 → 3 → 2.
Answer: (−2) + 7 − 3 = 2
Example 9: Example 9: Temperature word problem
Problem: The temperature at midnight was −6°C. By morning it rose by 9 degrees. In the evening it dropped by 5 degrees. What was the evening temperature?
Solution:
- Start at −6 on the number line.
- Rose by 9: move 9 steps right → −6 + 9 = 3°C (morning).
- Dropped by 5: move 5 steps left → 3 − 5 = −2°C (evening).
Answer: The evening temperature was −2°C.
Example 10: Example 10: Finding distance between two integers
Problem: How many steps on the number line is it from −3 to 4?
Solution:
- From −3 to 4, count the steps: −3 → −2 → −1 → 0 → 1 → 2 → 3 → 4.
- That is 7 steps.
- You can also calculate: 4 − (−3) = 4 + 3 = 7.
Answer: The distance is 7 units.
Real-World Applications
Temperature Changes: Weather reports use number lines to show temperature rises and drops. If the temperature is −3°C and it rises by 8 degrees, you move 8 steps to the right: −3 + 8 = 5°C.
Money and Spending: Think of your pocket money as a number line. If you have Rs. 100 (start at 100) and spend Rs. 60 (move 60 steps left), you have Rs. 40 left. If you spend Rs. 50 more (move left again), you go to −10, meaning you owe Rs. 10.
Elevators and Floors: A building has basement floors (negative) and upper floors (positive). Going from Basement 2 (−2) up 5 floors means −2 + 5 = Floor 3. Going down 4 floors from Floor 1 means 1 − 4 = −3 (Basement 3).
Sea Level and Altitude: Altitudes below sea level are negative. A diver at −15 metres who swims up 10 metres is at −15 + 10 = −5 metres (still underwater).
Games and Scores: In board games, moving forward is addition (right) and moving backward is subtraction (left). If you are on square 12 and draw a card saying "go back 5", you move to 12 − 5 = square 7.
Key Points to Remember
- On a number line, adding a positive number means moving to the right.
- Adding a negative number means moving to the left.
- Subtracting a positive number means moving to the left.
- Subtracting a negative number means moving to the right (double negative becomes positive).
- Always start at the first number, then move the number of steps equal to the second number.
- If you cross zero while moving left, you go into negative numbers.
- If you cross zero while moving right, you go into positive numbers.
- The number line method works for all integers — positive, negative, and zero.
- Subtraction of a negative: a − (−b) = a + b. Removing a debt makes you richer.
- Distance between two integers on a number line = larger number − smaller number.
Practice Problems
- Use a number line to find: (a) 5 + 3, (b) 5 + (−3), (c) −5 + 3, (d) −5 + (−3).
- Use a number line to find: (a) 7 − 4, (b) 4 − 7, (c) −2 − 5, (d) −2 − (−5).
- Start at −6 on the number line. Add 10. Then subtract 7. Where do you land?
- The temperature was −4°C. It rose by 11 degrees and then dropped by 8 degrees. Show on a number line and find the final temperature.
- Find the number of steps between −8 and 5 on a number line.
- A lift is at floor −3 (Basement 3). It goes up 7 floors, then down 2 floors. Which floor is it on now? Show on a number line.
- Find: (−9) + 15 − 4 + (−2) using a number line. Show step by step.
- True or false: Subtracting (−6) is the same as adding 6. Explain using a number line.
Frequently Asked Questions
Q1. Why do we move right when adding a positive number?
On a number line, numbers increase as you go right. Adding a positive number makes the total bigger, so you move towards bigger numbers — that is to the right. For example, starting at 2 and adding 3 means moving right to 5.
Q2. Why do we move left when adding a negative number?
Adding a negative number reduces the total. Since numbers decrease as you go left on the number line, you move left. For example, 5 + (−3) means starting at 5 and moving 3 steps left to land on 2.
Q3. Why does subtracting a negative number mean moving right?
Subtracting a negative is the same as adding a positive. Think of it as removing a penalty or cancelling a debt. If you owe Rs. 5 (−5) and someone cancels that debt (subtract −5), you gain Rs. 5. So 3 − (−5) = 3 + 5 = 8. On the number line, you move right.
Q4. Can the number line be used for large numbers?
Yes, but you do not need to draw every single number. You can mark only the key numbers (starting point, ending point, and zero if you cross it). The principle stays the same — move right for adding positive and left for adding negative.
Q5. What happens when you add zero on a number line?
Adding zero means you do not move at all. You stay at the same point. For example, (−3) + 0 = −3. Zero is the additive identity — it does not change the value.
Q6. How is subtraction related to addition on a number line?
Subtraction is the reverse of addition. If adding means moving right, subtracting means moving left. Also, subtracting a number is the same as adding its opposite: a − b = a + (−b). So 8 − 3 is the same as 8 + (−3) = 5.
Q7. Can results go below zero on the number line?
Yes. If you subtract a large enough number, or add a negative number with a large enough absolute value, you will cross zero and land on a negative number. For example, 3 − 8 = −5. You start at 3 and move 8 steps left, passing through zero to reach −5.
Q8. Is the number line method used in higher classes too?
The number line is mainly a learning tool for understanding integer operations. In higher classes, you will use rules (same sign add, different sign subtract) without drawing the number line every time. But the concept stays the same — the number line just helps you visualise what the rules mean.
