Subtraction of Integers
You have learned how to add integers — both positive and negative. Now it is time to learn subtraction. The great news is that if you understand integer addition well, subtraction is just one small step further. Here is the secret: subtracting an integer is the same as adding its additive inverse (its opposite). That means every subtraction problem can be converted into an addition problem! For example, 5 - 3 is the same as 5 + (-3). And 5 - (-3) is the same as 5 + (+3) = 8. This one rule — change the sign and add — makes integer subtraction simple and powerful. But why does this rule work? Think of it with temperature. If the temperature is 5°C and it drops by 3 degrees, you get 2°C. That is 5 - 3 = 2. But what if the temperature is 5°C and the cold "drops" are removed? Removing a 3-degree drop means it actually gets warmer by 3 degrees: 5 - (-3) = 5 + 3 = 8°C. Subtracting a negative is like removing a debt — it makes things better! In this chapter, you will master the rule for integer subtraction, practice it with number lines, and apply it to real-world problems involving temperature, altitude, money and more. Let us get started.
What is Subtraction of Integers?
Subtraction of Integers follows one fundamental rule:
To subtract an integer, add its additive inverse (opposite).
In symbols: a - b = a + (-b)
This means:
- Subtracting a positive number: a - (+b) = a + (-b) — you move to the left on the number line.
- Subtracting a negative number: a - (-b) = a + (+b) = a + b — you move to the right on the number line.
Why does this work?
Subtraction asks: "What must I add to b to get a?" If a = 5 and b = 3, the answer is 2 because 3 + 2 = 5. If a = 5 and b = -3, the answer is 8 because -3 + 8 = 5. In both cases, a - b = a + (-b) gives the correct result.
The Sign Change Rule (Step-by-Step):
Step 1: Write the subtraction problem: a - b.
Step 2: Change the subtraction sign to addition.
Step 3: Change the sign of the number being subtracted (b becomes -b, or -b becomes +b).
Step 4: Now solve using the rules of integer addition that you already know.
Examples of the Rule:
7 - (+3) = 7 + (-3) = 4 (change +3 to -3)
7 - (-3) = 7 + (+3) = 10 (change -3 to +3)
(-7) - (+3) = (-7) + (-3) = -10 (change +3 to -3)
(-7) - (-3) = (-7) + (+3) = -4 (change -3 to +3)
On the Number Line:
Subtraction on a number line is the reverse of addition:
- Subtracting a positive number: move to the left.
- Subtracting a negative number: move to the right (since removing a negative is adding a positive).
Types and Properties
Let us look at all the cases of integer subtraction systematically:
Case 1: Positive minus Positive (both positive)
a - b where both a and b are positive.
Convert: a - b = a + (-b). This is now adding a positive and a negative.
Example: 8 - 5 = 8 + (-5) = 3. (You already knew this from primary school!)
Example: 3 - 8 = 3 + (-8) = -5. (The result can be negative if you subtract a bigger number.)
Case 2: Positive minus Negative
a - (-b) where a is positive and b is negative.
Convert: a - (-b) = a + b. Subtracting a negative becomes adding a positive!
Example: 6 - (-4) = 6 + 4 = 10.
Think of it as: removing a debt of Rs. 4 means you gain Rs. 4.
This is the most important new concept — two negatives make a positive when subtracting.
Case 3: Negative minus Positive
(-a) - b where the first number is negative and you subtract a positive.
Convert: (-a) - b = (-a) + (-b) = -(a + b).
Example: (-6) - 4 = (-6) + (-4) = -10.
Think of it as: you owe Rs. 6 and spend Rs. 4 more — now you owe Rs. 10.
Case 4: Negative minus Negative
(-a) - (-b) where both numbers are negative.
Convert: (-a) - (-b) = (-a) + b.
Example: (-8) - (-3) = (-8) + 3 = -5.
Example: (-3) - (-8) = (-3) + 8 = 5.
Think of it as: you owe Rs. 8 and someone forgives Rs. 3 of your debt — now you owe only Rs. 5.
Properties of Integer Subtraction:
Not Commutative: a - b is NOT equal to b - a (in general).
Example: 5 - 3 = 2, but 3 - 5 = -2. They are different.
Not Associative: (a - b) - c is NOT equal to a - (b - c) (in general).
Example: (10 - 4) - 2 = 6 - 2 = 4, but 10 - (4 - 2) = 10 - 2 = 8. Different.
Identity Element: a - 0 = a (subtracting zero changes nothing). But 0 - a = -a (not the same as a).
Finding the Difference: The difference between two integers on a number line is found by subtraction. The distance (always positive) is the absolute value of the difference: |a - b|.
Solved Examples
Example 1: Example 1: Positive minus positive (result positive)
Problem: Find 15 - 8.
Solution:
Convert to addition: 15 - 8 = 15 + (-8).
Different signs: subtract absolute values: 15 - 8 = 7. Larger absolute value (15) is positive, so result is positive.
Answer: 15 - 8 = 7
Example 2: Example 2: Positive minus positive (result negative)
Problem: Find 6 - 14.
Solution:
Convert to addition: 6 - 14 = 6 + (-14).
Different signs: subtract absolute values: 14 - 6 = 8. Larger absolute value (14) is negative, so result is negative.
Number line: Start at 6. Move 14 steps to the left. 6 → 5 → 4 → 3 → 2 → 1 → 0 → -1 → -2 → -3 → -4 → -5 → -6 → -7 → -8.
Answer: 6 - 14 = -8
Example 3: Example 3: Subtracting a negative number
Problem: Find 5 - (-7).
Solution:
Apply the rule: subtracting a negative means adding a positive.
5 - (-7) = 5 + 7 = 12.
Think of it as: You have Rs. 5 and someone cancels a Rs. 7 debt. That is like gaining Rs. 7, so you now have Rs. 12.
Number line: Start at 5. Since we are subtracting a negative, move 7 steps to the right. 5 → 6 → 7 → 8 → 9 → 10 → 11 → 12.
Answer: 5 - (-7) = 12
Example 4: Example 4: Negative minus positive
Problem: Find (-9) - 4.
Solution:
Convert to addition: (-9) - 4 = (-9) + (-4).
Same signs (both negative): add absolute values: 9 + 4 = 13. Both are negative, so result is negative.
Think of it as: You owe Rs. 9 and spend Rs. 4 more. Now you owe Rs. 13.
Answer: (-9) - 4 = -13
Example 5: Example 5: Negative minus negative
Problem: Find (-12) - (-5).
Solution:
Apply the rule: subtracting a negative means adding a positive.
(-12) - (-5) = (-12) + 5.
Different signs: subtract absolute values: 12 - 5 = 7. Larger absolute value (12) is negative, so result is negative.
Answer: (-12) - (-5) = -7
Example 6: Example 6: Negative minus negative (result positive)
Problem: Find (-3) - (-10).
Solution:
(-3) - (-10) = (-3) + 10.
Different signs: subtract absolute values: 10 - 3 = 7. Larger absolute value (10) is positive, so result is positive.
Think of it as: You owe Rs. 3, and Rs. 10 of debt is forgiven. After clearing your Rs. 3 debt, you have Rs. 7 to spare.
Answer: (-3) - (-10) = 7
Example 7: Example 7: Temperature word problem
Problem: The temperature in Moscow is -18°C and in Delhi it is 25°C. What is the difference in temperature between Delhi and Moscow?
Solution:
Difference = Temperature of Delhi - Temperature of Moscow
= 25 - (-18)
= 25 + 18 (subtracting a negative = adding a positive)
= 43°C
Answer: Delhi is 43°C warmer than Moscow.
Example 8: Example 8: Altitude difference problem
Problem: The summit of a hill is 850 metres above sea level. A point at the bottom of a nearby lake is 45 metres below sea level. What is the difference in altitude between the summit and the lake bottom?
Solution:
Summit altitude = +850 m. Lake bottom altitude = -45 m.
Difference = 850 - (-45) = 850 + 45 = 895 metres.
Answer: The difference in altitude is 895 metres.
Example 9: Example 9: Mixed operations with integers
Problem: Simplify: 10 - 15 + 7 - (-3) - 8.
Solution:
Step 1: Convert all subtractions to additions:
= 10 + (-15) + 7 + (+3) + (-8)
Step 2: Group positives and negatives:
Positives: 10 + 7 + 3 = 20
Negatives: (-15) + (-8) = -23
Step 3: Add the groups:
20 + (-23) = -3
Answer: -3
Example 10: Example 10: Showing subtraction is not commutative
Problem: Show that subtraction is not commutative using a = 7 and b = -3.
Solution:
Calculate a - b:
7 - (-3) = 7 + 3 = 10
Calculate b - a:
(-3) - 7 = (-3) + (-7) = -10
Since 10 is not equal to -10, a - b is not equal to b - a.
In fact, a - b and b - a are always additive inverses of each other: if one is +10, the other is -10. They have the same absolute value but opposite signs.
Conclusion: Subtraction is NOT commutative.
Real-World Applications
Integer subtraction is used in many real-world situations:
Temperature Differences: Finding how much warmer one place is than another requires subtraction. If Mumbai is 30°C and Leh is -10°C, the difference is 30 - (-10) = 40°C. Weather forecasters calculate temperature changes by subtracting the earlier reading from the later one.
Altitude and Depth Differences: Geographers calculate altitude differences using subtraction. The difference between Mount Everest (+8849 m) and the Dead Sea (-430 m) is 8849 - (-430) = 9279 metres. This is the greatest land elevation range on Earth.
Banking and Finance: When you check how much your balance changed, you subtract the old balance from the new one. If your balance went from Rs. 5000 to Rs. 3200, the change is 3200 - 5000 = -1800 (a decrease of Rs. 1800). Comparing profits and losses also requires integer subtraction.
Time Zones: India is UTC +5:30 and New York is UTC -5. The time difference is calculated by subtraction: 5.5 - (-5) = 10.5 hours. This is why India is 10.5 hours ahead of New York.
Sports: In cricket, the Net Run Rate (NRR) involves subtraction of run rates, which can be positive or negative. In football, the goal difference is (goals scored) minus (goals conceded), which uses subtraction and can be negative.
Elevation Profiles: Road engineers and hikers look at elevation profiles — graphs showing height along a path. The steepness is calculated by subtracting altitudes at different points. Tunnels and bridges require precise altitude difference calculations.
Key Points to Remember
- The fundamental rule: to subtract an integer, add its opposite. a - b = a + (-b).
- Subtracting a positive number makes the result smaller (move left on number line).
- Subtracting a negative number makes the result larger (move right on number line): a - (-b) = a + b.
- Two negatives in subtraction make a positive: 5 - (-3) = 5 + 3 = 8.
- Subtraction is NOT commutative: a - b does not always equal b - a.
- Subtraction is NOT associative: (a - b) - c does not always equal a - (b - c).
- Converting all subtractions to additions makes complex problems easier to solve.
- The distance between two integers on the number line is |a - b| (always positive).
- a - 0 = a, but 0 - a = -a.
- Integer subtraction is used for temperature differences, altitude differences, time zone calculations, and financial comparisons.
Practice Problems
- Find: (a) 10 - 16, (b) (-8) - 3, (c) 7 - (-5), (d) (-4) - (-9).
- Use a number line to show (-3) - 5. What is the result?
- The temperature in Leh is -12°C and in Jaipur it is 35°C. Find the temperature difference.
- Simplify: 20 - 35 + 10 - (-5) - 8.
- Show that (8 - 3) - 2 is not equal to 8 - (3 - 2). What property does this demonstrate?
- A submarine at -200 m rises to -50 m. How many metres did it rise? Use subtraction.
- Find two integers whose difference is -7. Can you find three different pairs?
- If a - b = 15 and a = 8, find b.
Frequently Asked Questions
Q1. Why does subtracting a negative give a positive result?
Subtracting a negative means removing something negative, which has a positive effect. Think of money: if you owe Rs. 5 (that is -5) and someone cancels your debt (subtracts the -5), you are Rs. 5 better off. Mathematically, a - (-b) = a + b because the two negatives (the subtraction sign and the negative sign) cancel each other out. Two negatives make a positive.
Q2. How do I remember the rules for subtracting integers?
Remember just one rule: Change the sign of the number being subtracted, then add. Specifically: change subtraction to addition, and flip the sign of the second number. Then use the addition rules you already know. Some students remember it as: Keep-Change-Change (keep the first number, change subtraction to addition, change the sign of the second number).
Q3. Is subtraction of integers commutative?
No. Subtraction is not commutative. 5 - 3 = 2, but 3 - 5 = -2. The results are different. In general, a - b and b - a are opposites of each other (they have the same absolute value but different signs). This is an important difference from addition, which is commutative.
Q4. Can I convert every subtraction problem to addition?
Yes! That is the beauty of the rule a - b = a + (-b). Every subtraction can be rewritten as addition. This is especially helpful when you have a long expression with mixed addition and subtraction — convert everything to addition, then group positives and negatives separately for easy calculation.
Q5. What is the difference between minus sign and negative sign?
Technically, the minus sign (-) between two numbers means subtraction (an operation): 8 - 3. The negative sign (-) before a single number means the number is negative (a property): -5. However, since subtraction and adding a negative give the same result (8 - 3 = 8 + (-3) = 5), the two uses of the minus sign are closely connected. In Class 6, you can think of them as interchangeable.
Q6. How do you find the distance between two integers?
The distance between two integers a and b on the number line is |a - b| (the absolute value of their difference). For example, the distance between -3 and 5 is |(-3) - 5| = |-8| = 8. Or equivalently, |5 - (-3)| = |8| = 8. Distance is always positive, which is why we use absolute value.
Q7. Why is subtraction harder than addition with integers?
Subtraction has two complicating factors: (1) it is not commutative, so order matters, and (2) subtracting a negative can be confusing at first. The trick is to convert subtraction to addition (using the additive inverse). Once you do this conversion, you only need to use addition rules, which are simpler and more familiar.
Q8. What does 0 - a equal?
0 - a = -a. For example, 0 - 5 = -5 and 0 - (-3) = 3. This makes sense: subtracting a from zero gives you the opposite (additive inverse) of a. On the number line, 0 - a takes you from zero to the opposite side of a.
Q9. Can the result of subtracting two negative numbers be positive?
Yes! If the number being subtracted has a larger absolute value, the result is positive. Example: (-3) - (-10) = (-3) + 10 = 7 (positive). Think of it as: you owe Rs. 3, and Rs. 10 of debt is forgiven — after clearing your Rs. 3 debt, you have Rs. 7 extra.
Q10. How do I check if my subtraction answer is correct?
Use the relationship: if a - b = c, then c + b should equal a. For example, if you calculated 8 - (-3) = 11, check: 11 + (-3) = 8 ✓. If you calculated (-5) - 7 = -12, check: (-12) + 7 = -5 ✓. This reverse check catches mistakes quickly.
