Chapter 6: We Distribute, Yet Things Multiply
The Geometry & Algebra of Distributivity: Increments, Standard Identities, Mental Math & Visual Generalisation
📦 1. Distributivity over Addition
The FoundationMultiplication distributes over addition geometrically as an area of contiguous rectangular blocks:
a(b + c) = ab + ac
• One-sided shift: a(b + 1) = ab + a (increases by a).
• Integer Validity: Holds universally for positive integers, zero, negative integers, and fractions!
📈 2. Increments in Products
Product DynamicsHow does the product of two numbers change when their factors are adjusted?
- Both Increased by 1:
(a + 1)(b + 1) = ab + (a + b + 1)
→ Increase is a + b + 1. - One +1, Other −1:
(a + 1)(b − 1) = ab + (b − a − 1)
→ Increases only if b > a + 1. - General Shifts (+m, +n):
(a + m)(b + n) = ab + an + bm + mn.
📜 3. Indian Mathematical Heritage
Khanda-gunanamBrahmagupta (628 CE) codified the distributive law in Brahmasphutasiddhanta (Verse 12.55) as Khanda-gunanam (multiplication by parts):
Later developed as Ista-gunana by Sridharacharya (750 CE) and Bhaskaracharya (1150 CE) for rapid mental calculations.
📐 4. Identity 1A: (a + b)2
Square of a Sum
Mental Math: 1042 = (100 + 4)2
= 10000 + 2(400) + 16 = 10,816.
✂️ 5. Identity 1B: (a − b)2
Square of a DifferenceGeometric Removal: Take whole square a2, cut strips a × b twice. Because the corner b2 is removed twice, add back b2!
Mental Math: 992 = (100 − 1)2
= 10000 − 200 + 1 = 9,801.
Note: (a − b)2 is always equal to (b − a)2.
🔄 6. Identity 1C: Difference of Squares
Conjugate Multiplication
Sridharacharya's Squaring Method (750 CE):
a2 = (a + b)(a − b) + b2
Rapid Computation:
1972 = (197 + 3)(197 − 3) + 32
= 200 × 194 + 9 = 38800 + 9 = 38,809!
⚡ Fast Mental Multiplications Using Distributivity (Ista-gunana)
How expanding into base powers simplifies arithmetic into single-line mental math:
| Multiplier | Decomposition Form | One-Line Step Algorithm | Worked Example |
|---|---|---|---|
| × 11 | N × (10 + 1) = 10N + N | Write ends; add adjacent digit pairs with carry-overs. | 3874 × 11 = 42,614 |
| × 101 | N × (100 + 1) = 100N + N | Shift by 2 places and sum: dcba00 + dcba. | 3874 × 101 = 3,91,274 |
| × 99 | N × (100 − 1) = 100N − N | Append two zeros and subtract original number N. | 9734 × 99 = 9,63,666 |
| × 1001 | N × (1000 + 1) = 1000N + N | Shift by 3 places and sum: dcba000 + dcba. | 265831 × 1001 = 26,60,96,831 |
🎨 7. Multiple Interpretations of Patterns
A single geometric growth pattern can be formulated in multiple creative ways, all reducing to the exact same algebraic identity:
- Method 1 (Large Square − 1): (k + 1)2 − 1 = k2 + 2k
- Method 2 (Inner Square + Strip): k2 + 2 × k = k2 + 2k
- Method 3 (Columns + Extra): k(k + 1) + k = k2 + 2k
- Method 4 (Split Rectangle): k × (k + 2) = k2 + 2k
Tadang vs Yusuf Interior Area: (m + n)2 − 4mn = (n − m)2!
📅 8. Calendar & Number Invariants
The Calendar 2 × 2 Cross Invariant:
In any monthly calendar, take a 2 × 2 date block. The difference between diagonal products is always exactly 7!
Label entries: a, (a + 1), (a + 7), (a + 8).
Diagonal Difference = (a + 7)(a + 1) − a(a + 8)
= (a2 + 8a + 7) − (a2 + 8a) = 7 (Constant!).
Consecutive Square Trick: Square the middle of three consecutive numbers and subtract product of ends: n2 − (n − 1)(n + 1) = 1!
⚠️ "Mind the Mistake, Mend the Mistake" — Exam Pitfalls
The Missing Middle Term
Mistake: (5m + 6n)2 = 25m2 + 36n2.
Correct: 25m2 + 60mn + 36n2.
Never forget the cross term 2ab!
Negative Sign Distribution
Mistake: −3p(−5p + 2q) = −3p + 5p − 2q.
Correct: 15p2 − 6pq.
Multiply the outer factor to every term inside!
Adding Unlike Terms
Mistake: 5w2 + 6w = 11w3 (or 11w2).
Correct: Cannot be combined! Only terms with identical variable powers are like terms.