Integers - Class 7 Mathematics (CBSE)
Based on the 2025-26 NCERT syllabus for Class 7 Mathematics. This chapter covers the properties of addition and subtraction, multiplication and division of integers, and their applications through word problems.
1. Why this chapter matters
Integers are the building blocks of mathematics. Every calculation involving temperature, elevation, profit-loss, and bank balances uses integers. Understanding integer properties helps students avoid sign errors that persist into higher classes. In CBSE exams, this chapter contributes 6-10 marks through direct calculations, properties-based questions, and word problems.
2. Properties of addition and subtraction
Closure property
The sum or difference of any two integers is always an integer.
Examples:
- 5 + (-3) = 2 (integer)
- -7 - (-4) = -3 (integer)
Commutative property
Addition is commutative: a + b = b + a for all integers.
- 8 + (-5) = 3 and (-5) + 8 = 3
Subtraction is NOT commutative:
- 8 - 3 = 5 but 3 - 8 = -5. The order matters.
Associative property
Addition is associative: (a + b) + c = a + (b + c).
- (4 + (-6)) + 2 = (-2) + 2 = 0
- 4 + ((-6) + 2) = 4 + (-4) = 0
Subtraction is NOT associative.
Additive identity
Zero is the additive identity: a + 0 = a = 0 + a.
Additive inverse
For every integer a, there exists (-a) such that a + (-a) = 0.
3. Properties of multiplication and division
Closure property
Product of any two integers is always an integer:
- (-4) x 6 = -24 (integer) But division of integers is NOT always an integer: (-5) / 2 is not an integer.
Commutative property
Multiplication is commutative: a x b = b x a.
- (-3) x 7 = -21 and 7 x (-3) = -21
Division is NOT commutative.
Associative property
Multiplication is associative: (a x b) x c = a x (b x c).
Multiplicative identity
1 is the multiplicative identity: a x 1 = a.
Distributive property
a x (b + c) = a x b + a x c.
Example: (-2) x (5 + 3) = (-2) x 8 = -16 and (-2) x 5 + (-2) x 3 = -10 + (-6) = -16.
Division property
Any integer divided by zero is undefined. Zero divided by any non-zero integer is zero.
4. Multiplication and division rules
Sign rules for multiplication
| Sign combination | Result | Example |
|---|---|---|
| Positive x Positive | Positive | 4 x 3 = 12 |
| Negative x Negative | Positive | (-4) x (-3) = 12 |
| Positive x Negative | Negative | 4 x (-3) = -12 |
| Negative x Positive | Negative | (-4) x 3 = -12 |
Sign rules for division
Same rules apply: same signs give positive, different signs give negative.
5. Worked examples
Example 1: Evaluate (-15) + 8 - (-3)
Step 1: (-15) + 8 = -7 Step 2: -7 - (-3) = -7 + 3 = -4 Answer: -4
Example 2: Evaluate (-12) x 5 / (-3)
Step 1: (-12) x 5 = -60 Step 2: -60 / (-3) = 20 Answer: 20
Example 3: Temperature word problem
The temperature in Shimla was -2 C at 6 AM. It rose by 5 C by noon and then fell by 7 C by midnight. What is the midnight temperature?
Step 1: Temperature at noon = -2 + 5 = 3 C Step 2: Temperature at midnight = 3 - 7 = -4 C Answer: -4 C
Example 4: Verify (-3) x [4 + (-2)] = (-3) x 4 + (-3) x (-2)
Left side: (-3) x [4 + (-2)] = (-3) x 2 = -6 Right side: (-3) x 4 + (-3) x (-2) = -12 + 6 = -6 Hence verified. This shows the distributive property.
6. Common mistakes and how to fix them
| Mistake | Fix |
|---|---|
| Thinking -5 + 3 = -8 | Moving right from -5 by 3 gives -2, not -8 |
| Writing -3 - 5 = 2 | -3 - 5 = -8. Convert to -3 + (-5) |
| Saying (-2) x (-3) = -6 | Same signs multiply to positive: (-2) x (-3) = 6 |
| Forgetting BODMAS with integers | Multiply/divide before add/subtract always |
| Dropping brackets around negatives | Always write (-5), never -5 alone in operations |
7. CBSE exam focus
| Question type | Marks | Frequency |
|---|---|---|
| Direct integer calculation | 1-2 marks | 2-3 questions |
| Property identification and verification | 2 marks | 1 question |
| Word problem (temperature, finance) | 3 marks | 1 question |
| Activity-based integer operations | 3-5 marks | Occasional |
8. Self-test
- Evaluate: (-25) + 14 - (-6).
- Find the product: (-8) x (-7) x (-2).
- Simplify using properties: (-15) x 8 + (-15) x 2.
- The temperature in a city is 10 C. It drops by 3 C every hour for 5 hours. What is the final temperature?
- A shopkeeper gains Rs. 15 per toy sold and loses Rs. 8 per pencil sold. He sells 4 toys and 6 pencils. What is his net profit or loss?
- Is (-2) - 3 equal to 3 - (-2)? Why or why not?
9. Answer key
- (-25) + 14 = -11. Then -11 - (-6) = -11 + 6 = -5.
- (-8) x (-7) = 56. Then 56 x (-2) = -112.
- Using distributive property: (-15) x (8 + 2) = (-15) x 10 = -150.
- Drop after 5 hours = 3 x 5 = 15 C. Final = 10 - 15 = -5 C.
- Profit from toys = 4 x 15 = Rs. 60. Loss from pencils = 6 x 8 = Rs. 48. Net profit = 60 - 48 = Rs. 12.
- (-2) - 3 = -5, but 3 - (-2) = 5. They are not equal because subtraction is NOT commutative.
10. Quick revision
- Properties of addition: closure, commutative, associative, identity (0), inverse.
- Properties of multiplication: closure, commutative, associative, identity (1), distributive.
- Division by zero is undefined.
- Same signs in multiplication/division give positive; different signs give negative.
- Always show steps in word problems for partial marks in CBSE exams.
