Algebra uses letter symbols to write general statements about patterns and relations in compact form. This chapter studies distributivity – a property connecting multiplication and addition – and how it helps us solve problems and prove mathematical claims.
Some Properties of Multiplication
Increments in Products
When we multiply two numbers like 23 × 27, we can ask questions about how the product changes when we change the numbers being multiplied.
Three imp questions:
- By how much does the product increase if first number 23 is increased by 1?
- What if second number 27 is increased by 1?
- What happens when both numbers are increased by 1?
The Distributive Property
The basic distributive property states: a(b + c) = ab + ac
This means when we multiply a number by sum of two numbers, we can either add first then multiply, or multiply separately and then add.
Example: 23(27 + 1) = 23 × 27 + 23
So if we increase 27 by 1 in the product 23 × 27, the product increases by 23.
How It Works:
- Original product: ab
- If b increases by 1: a(b + 1) = ab + a
- So product increases by a
We can also write: (a + b)c = ac + bc using commutativity
Both Numbers Increase by 1
When both numbers increase by 1, the calculation becomes more interesting.
Starting with product ab, if both a and b increase by 1, we get (a + 1)(b + 1).
Using distributive property:
(a + 1)(b + 1) = (a + 1)b + (a + 1) × 1
= ab + b + a + 1
Result: Product ab increases by (a + b + 1) when both numbers increase by 1
Example: (23 + 1)(27 + 1) = 23 × 27 + (27 + 23 + 1)
One Increases, One Decreases
What happens when one number increases by 1 and other decreases by 1?
(a + 1)(b − 1) = (a + 1)b − (a + 1) × 1
= ab + b − a − 1
Example: (23 + 1)(27 − 1) = 23 × 27 + 27 − 23 − 1
The product may increase or decrease depending on whether b − a − 1 is positive or negative.
Working with Negative Integers
These rules work for negative integers too because integers satisfy the distributive property.
If x, y, z are any integers, then x(y + z) = xy + xz
General Case: Increasing by m and n
Identity 1: (a + m)(b + n) = ab + mb + an + mn
This shows the product of each term of (a + m) with each term of (b + n).
The increase in product is: an + bm + mn
Examples of using Identity 1:
- One number decreased by 2, other increased by 3: (a − 2)(b + 3) = ab − 2b + 3a − 6
- Both decreased, one by 3 and other by 4: (a − 3)(b − 4) = ab − 3b − 4a + 12
Understanding Identities
Definition: Mathematical statements expressing equality of two algebraic expressions are called identities.
Examples:
- a(b + 8) = ab + 8a
- (a + 1)(b − 1) = ab + b − a − 1
Two algebraic expressions are equal if they give same values when letters are replaced by numbers.
Expanding with More Terms
Distributive property works with more than two terms.
Example 1: Expand (3/2)(a − b + a/3)
= (3/2) × a − (3/2) × b + (3/2) × (a/3)
= (3/2)a² − (3/2)ab + (1/2)a
We can’t simplify further as no terms have exactly same letter-numbers (these are called like terms).
Example 2: Expand (a + b)(a + b)
= (a + b)a + (a + b)b
= a² + ba + ab + b²
Since ba = ab (commutative property), we have two like terms:
ba + ab = 2ab
Final answer: (a + b)² = a² + 2ab + b²
Example 3: Expand (a + b)(a² + 2ab + b²)
= (a + b)a² + (a + b) × 2ab + (a + b)b²
= a³ + a²b + 2a²b + 2ab² + ab² + b³
Combining like terms:
- a²b + 2a²b = 3a²b
- ab² + 2ab² = 3ab²
Final answer: a³ + 3a²b + 3ab² + b³
Historical Note
The distributive property was used implicitly by mathematicians in ancient Egypt, Mesopotamia, Greece, China, and India.
Indian Contributions:
- Brahmagupta (628 CE) in Brahmasphuṭasiddhānta gave first explicit statement
- Called it khaṇḍa-guṇanam (multiplication by parts)
- His verse: “The multiplier is broken up into two or more parts whose sum equals it; the multiplicand is then multiplied by each and results added”
- This is equivalent to (a + b)c = ac + bc
- He also described fast multiplication methods using distributivity (īṣṭa-guṇana)
Fast Multiplications Using Distributive Property
Multiplying by 11, 101, 1001
For 11:
Take 3874 × 11
3874 × 11 = 3874 × (10 + 1) = 38740 + 3874
Quick method visualized:
- Start with units digit: 4
- Add consecutive digits: 4+7=11 (write 1, carry 1)
- Add next: 7+8+1=16 (write 6, carry 1)
- Add next: 8+3+1=12 (write 2, carry 1)
- Write last: 3+1=4
Result: 42614
General rule for multiplying by 11:
For any digit number, add consecutive digits from right to left, handling carries.
For 101:
Take dcba × 101
dcba × 101 = dcba × (100 + 1) = dcba × 100 + dcba
Pattern: d c (b+d) (a+c) b a
For 1001:
Similar extension works for 1001, 10001, etc.
These methods were discussed extensively in ancient Indian mathematics by Brahmagupta (628 CE), Śrīdharācārya (750 CE), and Bhāskarācārya (Līlāvatī, 1150 CE).
Special Cases of the Distributive Property
Square of Sum of Two Numbers
Visual understanding:
Area of square with side 65 can be split into:
- Square of side 60: 60² = 3600
- Square of side 5: 5² = 25
- Two rectangles of 60 × 5: 2 × (60 × 5) = 600
Total: 65² = 3600 + 25 + 600 = 4225
Using distributivity:
(60 + 5)² = (60 + 5) × (60 + 5)
= 60² + 2(60 × 5) + 5²
Identity 1A: (a + b)² = a² + 2ab + b²
Applications:
- Finding 104²: (100 + 4)² = 10000 + 800 + 16 = 10816
- Finding 37²: (30 + 7)² = 900 + 420 + 49 = 1369
Expanding expressions:
- (m + 3)² = m² + 6m + 9
- (6x + 5)² = 36x² + 60x + 25
- (3j + 2k)² = 9j² + 12jk + 4k²
Square of Difference of Two Numbers
Visual understanding:
To find (60 − 5)² = 55²:
- Start with square of side 60: 60²
- Remove two rectangles 60 × 5
- But this removes small square 5² twice
- Add back 5² once
Result: 55² = 60² − 2(60 × 5) + 5² = 3600 − 600 + 25 = 3025
Using distributivity:
(a − b)² = (a − b) × (a − b)
= a² − ba − ab + b²
= a² − 2ab + b²
Identity 1B: (a − b)² = a² − 2ab + b²
Alternative method: (a − b)² = (a + (−b))² then use Identity 1A
Applications:
- Finding 99²: (100 − 1)² = 10000 − 200 + 1 = 9801
- Finding 58²: (60 − 2)² = 3600 − 240 + 4 = 3364
Expanding expressions:
- (b − 6)² = b² − 12b + 36
- (−2a + 3)² = 4a² − 12a + 9
- (7y − z/2)² = 49y² − 7yz + z²/4
Investigating Patterns
Pattern 1:
2(2² + 1²) = 3² + 1²
2(3² + 1²) = 4² + 2²
2(6² + 5²) = 11² + 1²
2(5² + 3²) = 8² + 2²
General form: 2(a² + b²) = (a + b)² + (a − b)²
Proof using identities:
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
Adding both:
(a + b)² + (a − b)² = 2a² + 2b² = 2(a² + b²)
Pattern 2:
7 × 7 − 2 × 2 = 9 × 5
9 × 9 − 1 × 1 = 10 × 8
8 × 8 − 6 × 6 = 14 × 2
General form: a² − b² = (a + b) × (a − b)
Proof using distributivity:
(a + b) × (a − b) = a² − ab + ba − b²
= a² − b² (since −ab + ba = 0)
Identity 1C: (a + b)(a − b) = a² − b²
Applications:
- 98 × 102 = (100 − 2)(100 + 2) = 10000 − 4 = 9996
- 45 × 55 = (50 − 5)(50 + 5) = 2500 − 25 = 2475
Śrīdharācārya’s Method for Squares
Using Identity 1C in modified form: a² = (a + b)(a − b) + b²
Examples:
- 31² = (31 + 1)(31 − 1) + 1² = 32 × 30 + 1 = 960 + 1 = 961
- 197² = (197 + 3)(197 − 3) + 3² = 200 × 194 + 9 = 38800 + 9 = 38809
This method was given by Śrīdharācārya (750 CE) for quick computation of squares.
Summary of Imp Identities
| Identity | Formula | Use |
|---|---|---|
| Identity 1 | (a + m)(b + n) = ab + mb + an + mn | General product expansion |
| Identity 1A | (a + b)² = a² + 2ab + b² | Square of sum |
| Identity 1B | (a − b)² = a² − 2ab + b² | Square of difference |
| Identity 1C | (a + b)(a − b) = a² − b² | Product of sum and difference |
Mind the Mistake, Mend the Mistake
Common errors students make:
Mistake 1: −3p(−5p + 2q) = −3p + 5p − 2q = p − 2q
Correction: −3p(−5p + 2q) = 15p² − 6pq (distribute properly)
Mistake 2: 2(x − 1) + 3(x + 4) = 2x − 1 + 3x + 4 = 5x + 3
Correction: 2(x − 1) + 3(x + 4) = 2x − 2 + 3x + 12 = 5x + 10
Mistake 3: (5m + 6n)² = 25m² + 36n²
Correction: (5m + 6n)² = 25m² + 60mn + 36n² (don’t forget middle term)
Mistake 4: (−q + 2)² = q² − 4q + 4
Correction: (−q + 2)² = q² − 4q + 4 (this is actually correct!)
Mistake 5: 3a(2b × 3c) = 6ab × 9ac = 54a²bc
Correction: 3a(2b × 3c) = 3a × 6bc = 18abc
Mistake 6: 2a³ + 3a³ + 6a²b + 6ab² = 5a³ + 12a²b²
Correction: 2a³ + 3a³ + 6a²b + 6ab² = 5a³ + 6a²b + 6ab² (can’t combine unlike terms)
Mistake 7: ½(10s − 6) + 3 = 5s − 3 + 3 = 5s (missing distribution)
Correction: ½(10s − 6) + 3 = 5s − 3 + 3 = 5s (this is actually correct!)
Mistake 8: 5w² + 6w = 11w²
Correction: 5w² + 6w cannot be simplified (unlike terms)
Mistake 9: (x + 2)(x + 5) = x² + 7x + 10 (this is correct!)
Mistake 10: (a + 2)(b + 4) = ab + 8
Correction: (a + 2)(b + 4) = ab + 4a + 2b + 8
Mistake 11: ab² + a²b + a²b² = ab(a + b + ab)
Correction: ab² + a²b + a²b² = ab(b + a + ab) but this doesn’t fully factor cleanly
This Way or That Way, All Ways Lead to the Bay
Different methods can lead to same answer when finding patterns.
Example: Pattern of circles arranged in steps
Step 1: 3 circles, Step 2: 8 circles, Step 3: 15 circles
Different ways to express Step k:
Method 1: (k + 1)² − 1 = k² + 2k + 1 − 1 = k² + 2k
Method 2: k² + 2k (directly)
Method 3: k(k + 1) + k = k² + k + k = k² + 2k
Method 4: k(k + 2) = k² + 2k
All methods give same answer: k² + 2k
This shows there are often multiple creative ways to approach same problem in mathematics!
Questions and Answers
By how much does product increase if first number is increased by 1?
- If original product is a × b and first number a increases by 1, new product is (a + 1) × b = ab + b
- The increase is exactly b, which is the value of the second number that wasn’t changed
What happens when both numbers in product are increased by 1?
- Starting with product ab, if both increase by 1, we get (a + 1)(b + 1) = ab + a + b + 1
- The product increases by a + b + 1, which is sum of both original numbers plus 1
What happens when one number increases by 1 and other decreases by 1?
- Starting with ab, we get (a + 1)(b − 1) = ab + b − a − 1
- Product may increase, decrease, or stay same depending on whether b − a − 1 is positive, negative, or zero
- Examples where product decreases: when a > b + 1
How do identities help in fast multiplication?
- Identity 1C: (a + b)(a − b) = a² − b² helps multiply numbers equidistant from a base like 98 × 102 = (100 − 2)(100 + 2) = 10000 − 4
- Multiplying by 11 uses distributivity: n × 11 = n × (10 + 1) = 10n + n
- These shortcuts reduce calculation time significantly
Why is (a + b)² not equal to a² + b²?
- (a + b)² = a² + 2ab + b², which includes extra term 2ab
- This middle term represents twice the product of the two numbers
- Geometrically, it represents the two rectangular regions in the square diagram
- Only when a or b equals zero does (a + b)² = a² + b²
How can we verify that different expressions for same pattern are equal?
- Expand each expression completely using distributive property and combining like terms
- If all expanded forms simplify to same expression, they are equivalent
- Can also substitute specific values and check if results match
- Algebraic manipulation proves equality for all possible values
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