✖️ We Distribute, Yet Things Multiply
Chapter 6: Quick Revision Notes | Class 8 Mathematics | Ganita Prakash
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📋 Table of Contents
🔶
🔶 Distributive Property — The Big Idea!
Distributivity connects multiplication and addition. Algebra lets us write patterns and proofs in a compact, powerful way using this property!
📖 Core Property
a × (b + c) = ab + ac
Also: (a + b) × c = ac + bc
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What is an Identity? 🤔
📖 Identity
A mathematical statement that is always true for all values of the letter-numbers.
Example: a(b + 8) = ab + 8a is an identity — works for ANY a and b!
Example: a(b + 8) = ab + 8a is an identity — works for ANY a and b!
💡 Key Idea — Increment in Products!
If product is ab and we increase b by 1:
→ New product = a(b+1) = ab + a (increases by a!)
If we increase a by 1 AND b by 1:
→ (a+1)(b+1) = ab + (a + b + 1) (increases by a+b+1!)
→ New product = a(b+1) = ab + a (increases by a!)
If we increase a by 1 AND b by 1:
→ (a+1)(b+1) = ab + (a + b + 1) (increases by a+b+1!)
⚠️ Tricky Case — One Up, One Down!
If a increases by 1 AND b decreases by 1:
→ (a+1)(b−1) = ab + (b − a − 1)
Product may increase OR decrease depending on values!
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→ (a+1)(b−1) = ab + (b − a − 1)
Product may increase OR decrease depending on values!
🧮
🧮 Identity 1 — The Master Formula
When a is increased by m and b is increased by n:
⭐ Identity 1
(a + m)(b + n) = ab + mb + an + mn
Increase = an + bm + mn
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🧠 Remember It As: "Each × Each"
Multiply every term of the first bracket with every term of the second bracket!
(a + m)(b + n) → a×b + a×n + m×b + m×n = 4 terms
(a + m)(b + n) → a×b + a×n + m×b + m×n = 4 terms
📝 Like Terms — Same letter-numbers!
Terms with the exact same letter-numbers are called Like Terms and can be added.
Example: ba + ab = ab + ab = 2ab (since ba = ab)
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Example: ba + ab = ab + ab = 2ab (since ba = ab)
🟨
🟨 Identity 1A — Square of a Sum
What is (a + b)²? It means (a + b) × (a + b). Let's expand it!
①(a+b)² = (a+b)(a+b)
②= a×a + a×b + b×a + b×b
③= a² + ab + ab + b² (since ba = ab)
④= a² + 2ab + b²
⭐ Identity 1A
(a + b)² = a² + 2ab + b²
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🚀 Quick Application!
65² = (60+5)² = 60² + 2×60×5 + 5² = 3600 + 600 + 25 = 4225 ✅
104²= (100+4)² = 10000 + 800 + 16 = 10816 ✅
37² = (40−3)² = use Identity 1B! → 1600 − 240 + 9 = 1369
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104²= (100+4)² = 10000 + 800 + 16 = 10816 ✅
37² = (40−3)² = use Identity 1B! → 1600 − 240 + 9 = 1369
🟩
🟩 Identity 1B — Square of a Difference
What is (a − b)²? We can think of it as (a + (−b))² and use Identity 1A!
①(a−b)² = (a+(−b))²
②= a² + 2×a×(−b) + (−b)²
③= a² − 2ab + b²
⭐ Identity 1B
(a − b)² = a² − 2ab + b²
🆚 Compare 1A vs 1B — Spot the Difference!
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
Only the sign of the middle term changes! Middle = ±2ab
(a − b)² = a² − 2ab + b²
Only the sign of the middle term changes! Middle = ±2ab
🚀 Quick Application!
99² = (100−1)² = 10000 − 200 + 1 = 9801 ✅
58² = (60−2)² = 3600 − 240 + 4 = 3364 ✅
55² = (60−5)² = 3600 − 600 + 25 = 3025 ✅
58² = (60−2)² = 3600 − 240 + 4 = 3364 ✅
55² = (60−5)² = 3600 − 600 + 25 = 3025 ✅
🌟 Cool Pattern!
2(a² + b²) = (a+b)² + (a−b)²
Twice the sum of squares = sum of (sum)² and (difference)²
Example: 2(5²+6²) = (5+6)² + (6−5)² = 121 + 1 = 122 ✅
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Twice the sum of squares = sum of (sum)² and (difference)²
Example: 2(5²+6²) = (5+6)² + (6−5)² = 121 + 1 = 122 ✅
🟥
🟥 Identity 1C — Difference of Squares
Pattern spotted from numbers:
📋 Observe the Pattern
9×9 − 1×1 = 10×8 → 81−1 = 80 ✅
8×8 − 6×6 = 14×2 → 64−36 = 28 ✅
Pattern: a² − b² = (a+b)(a−b)
8×8 − 6×6 = 14×2 → 64−36 = 28 ✅
Pattern: a² − b² = (a+b)(a−b)
①(a+b)(a−b) = a×a − a×b + b×a − b×b
②= a² − ab + ab − b²
③= a² − b² (since ab cancels!)
⭐ Identity 1C
(a + b)(a − b) = a² − b²
🚀 Fast Computation using 1C!
98 × 102 = (100−2)(100+2) = 100² − 2² = 10000 − 4 = 9996 ✅
45 × 55 = (50−5)(50+5) = 50² − 5² = 2500 − 25 = 2475 ✅
31² = (31+1)(31−1) + 1² = 32×30 + 1 = 960+1 = 961 ✅
45 × 55 = (50−5)(50+5) = 50² − 5² = 2500 − 25 = 2475 ✅
31² = (31+1)(31−1) + 1² = 32×30 + 1 = 960+1 = 961 ✅
🧠 Sridharacharya's Trick (750 CE)!
To find a² easily: use a² = (a+b)(a−b) + b²
Choose b to make (a+b) and (a−b) easy to multiply!
197² → b=3 → 200×194 + 9 = 38800 + 9 = 38809 🤯
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Choose b to make (a+b) and (a−b) easy to multiply!
197² → b=3 → 200×194 + 9 = 38800 + 9 = 38809 🤯
⚡
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⚡ Fast Multiplication Tricks
The distributive property gives us amazing mental math shortcuts! 🧠
🔢 Trick 1: Multiply by 11
Any number × 11 = number × (10+1) = number×10 + number
3 8 7 4 × 11
= 3 8 7 4 × 10 + 3 8 7 4
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3 (8+3) (7+8) (4+7) 4
= 4 2 6 1 4 = 42614
Rule: Add adjacent digits from right to left. Carry if sum > 9!
🔢 Trick 2: Multiply by 101, 1001, 10001...
Number × 101 = number × (100+1) = number×100 + number
d c b a × 101
= d c (b+d) (a+c) b a
Example: 3874 × 101 = 3 | 8 | (7+3) | (4+8) | 7 | 4
= 3 | 8 | 10 | 12 | 7 | 4 → carry → 391474
📜 Historical Note — Brahmagupta (628 CE)!
This method of fast multiplication using distributivity was called ista-gunana by Brahmagupta.
It was described in his work Brahmasphutasiddhanta (Verse 12.56).
Also used by Sridharacharya (750 CE) and Bhaskaracharya (1150 CE)! 🏛️
✖️
× 11
Add adjacent digits. Use (10+1) split.
💯
× 101
Use (100+1) split. Shift + add original.
⚡
× 99 / 999
Use (100−1) / (1000−1). Identity 1C!
🎯
Near 100s
98×102 = (100−2)(100+2) = 100²−4
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📊 Quick Summary — All Identities
🌟 The 3 Golden Identities of Chapter 6
| Identity | Formula | Middle Term | Use When |
|---|---|---|---|
| ⭐ 1A | (a+b)² = a² + 2ab + b² | +2ab | Squaring a sum |
| ⭐ 1B | (a−b)² = a² − 2ab + b² | −2ab | Squaring a difference |
| ⭐ 1C | (a+b)(a−b) = a² − b² | 0 (cancels!) | Product of sum×diff |
| ⭐ 1 | (a+m)(b+n) = ab+mb+an+mn | — | General expansion |
🔮 All Key Formulas:
a(b+c) = ab+ac
(a+b)² = a²+2ab+b²
(a−b)² = a²−2ab+b²
(a+b)(a−b) = a²−b²
(a+b)(c+d) = ac+ad+bc+bd
🔍 Identity 1A vs 1B vs 1C at a Glance:
| Check | 1A: (a+b)² | 1B: (a−b)² | 1C: (a+b)(a−b) |
|---|---|---|---|
| a² term | ✅ Yes | ✅ Yes | ✅ Yes |
| b² term | ✅ +b² | ✅ +b² | ❌ −b² |
| Middle term | +2ab | −2ab | None (=0) |
| Result terms | 3 terms | 3 terms | 2 terms |
| Shortcut for | 65², 104² | 99², 58² | 98×102 |
🌟 Golden Rules to Never Forget!
1️⃣ Distributive: a(b+c) = ab + ac — always!
2️⃣ (a+b)² ≠ a²+b² — middle term 2ab must NOT be forgotten!
3️⃣ (a+b)² and (a−b)² differ only in the sign of 2ab
4️⃣ (a+b)(a−b) = a²−b² — middle terms cancel out!
5️⃣ There are always multiple ways to solve — all lead to the same answer!
2️⃣ (a+b)² ≠ a²+b² — middle term 2ab must NOT be forgotten!
3️⃣ (a+b)² and (a−b)² differ only in the sign of 2ab
4️⃣ (a+b)(a−b) = a²−b² — middle terms cancel out!
5️⃣ There are always multiple ways to solve — all lead to the same answer!
📜 Historical Giants of Distributivity!
🏛️ Euclid (geometric form) • Āryabhaṭa (algebraic form)
📖 Brahmagupta (628 CE) — first explicit statement in Brahmasphutasiddhanta
✍️ Called it khanda-gunanam (multiplication by parts)
🔢 Sridharacharya (750 CE) • Bhaskaracharya (1150 CE)
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📖 Brahmagupta (628 CE) — first explicit statement in Brahmasphutasiddhanta
✍️ Called it khanda-gunanam (multiplication by parts)
🔢 Sridharacharya (750 CE) • Bhaskaracharya (1150 CE)
✏️ Notes by @edugrown • Ganita Prakash Grade 8 • Chapter 6