Class 7 | Ganita Prakash Part-II | Chapter 2
➕ Operations with Integers
Add · Subtract · Multiply · Divide Integers
🔢 2.1 — Quick Recap of Integers
What are Integers?
..., –3, –2, –1, 0, 1, 2, 3, ... (all whole numbers, positive AND negative)
..., –3, –2, –1, 0, 1, 2, 3, ... (all whole numbers, positive AND negative)
- Positive integers: 1, 2, 3... (above zero)
- Zero: neither positive nor negative
- Negative integers: –1, –2, –3... (below zero)
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🏔️ Real Life Integers!
A mine shaft: 0m = ground level
+180m = 180 metres above ground
–175m = 175 metres underground
Integers help us show direction — above or below zero!
A mine shaft: 0m = ground level
+180m = 180 metres above ground
–175m = 175 metres underground
Integers help us show direction — above or below zero!
Adding Integers — Rules:
- Same signs → Add, keep the sign: (+4)+(+3) = +7
- Same signs → Add, keep the sign: (–4)+(–3) = –7
- Different signs → Subtract, keep sign of bigger: (–8)+(+5) = –3
- Subtracting = Adding the opposite: 5–(–3) = 5+(+3) = 8
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✖️ 2.2 — Multiplication of Integers
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Multiplication Rules — Sign Rules:
| Signs | Result | Example |
|---|---|---|
| (+) × (+) | Positive | 3 × 4 = 12 |
| (–) × (–) | Positive | (–3) × (–4) = 12 |
| (+) × (–) | Negative | 3 × (–4) = –12 |
| (–) × (+) | Negative | (–3) × 4 = –12 |
Easy memory trick! 🧠
Think of it like a friendship/enemy rule:
Friend of Friend = Friend (+)
Enemy of Enemy = Friend (+)
Friend of Enemy = Enemy (–)
Enemy of Friend = Enemy (–)
Think of it like a friendship/enemy rule:
Friend of Friend = Friend (+)
Enemy of Enemy = Friend (+)
Friend of Enemy = Enemy (–)
Enemy of Friend = Enemy (–)
Patterns in Multiplication:
3 × 3 = 9 3 × 2 = 6 3 × 1 = 3 3 × 0 = 0
3 × (–1) = –3 3 × (–2) = –6 3 × (–3) = –9
(each step DECREASES by 3)
(–3) × 3 = –9 (–3) × 2 = –6 (–3) × 1 = –3 (–3) × 0 = 0
(–3) × (–1) = 3 (–3) × (–2) = 6
(each step INCREASES by 3 → negative × negative = POSITIVE!)
3 × 3 = 9 3 × 2 = 6 3 × 1 = 3 3 × 0 = 0
3 × (–1) = –3 3 × (–2) = –6 3 × (–3) = –9
(each step DECREASES by 3)
(–3) × 3 = –9 (–3) × 2 = –6 (–3) × 1 = –3 (–3) × 0 = 0
(–3) × (–1) = 3 (–3) × (–2) = 6
(each step INCREASES by 3 → negative × negative = POSITIVE!)
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➗ Division of Integers
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Division Rules — Same as multiplication signs!
| Signs | Result | Example |
|---|---|---|
| (+) ÷ (+) | Positive | 12 ÷ 4 = 3 |
| (–) ÷ (–) | Positive | (–12) ÷ (–4) = 3 |
| (+) ÷ (–) | Negative | 12 ÷ (–4) = –3 |
| (–) ÷ (+) | Negative | (–12) ÷ 4 = –3 |
Division and Multiplication are LINKED:
If (–3) × 4 = –12 → then (–12) ÷ 4 = –3
If (–3) × (–4) = 12 → then 12 ÷ (–4) = –3
They are inverse operations!
If (–3) × 4 = –12 → then (–12) ÷ 4 = –3
If (–3) × (–4) = 12 → then 12 ÷ (–4) = –3
They are inverse operations!
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⚡ Properties of Integer Operations
| Property | For × | Example |
|---|---|---|
| Commutative | a × b = b × a | (–3) × 4 = 4 × (–3) = –12 |
| Associative | (a×b)×c = a×(b×c) | (2×(–3))×4 = 2×((–3)×4) |
| Distributive | a×(b+c)=a×b+a×c | (–2)×(3+4) = (–2)×3 + (–2)×4 |
| Identity | a × 1 = a | (–7) × 1 = –7 |
| Zero property | a × 0 = 0 | (–99) × 0 = 0 |
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📝 Quick Cheatsheet
(+)×(+) = + ✅
(–)×(–) = + ✅
(+)×(–) = – ❗
(–)×(–) = + ✅
(+)×(–) = – ❗
Same signs → +
Different signs → –
(for × and ÷)
Different signs → –
(for × and ÷)
Subtraction =
Add the opposite
5–(–3) = 5+3=8
Add the opposite
5–(–3) = 5+3=8
0 × anything = 0
1 × anything = itself
–1 × n = –n
1 × anything = itself
–1 × n = –n
🌟 Chapter 2 Complete! 🌟
Made with ❤️ by @edugrown
Made with ❤️ by @edugrown