We Distribute, Yet Things Multiply Class 8 Notes

We Distribute, Yet Things Multiply Class 8 Notes

We Distribute, Yet Things Multiply Class 8 Notes

Distributive Property of Multiplication

Definition : When you multiply a number by a sum, you can distribute the multiplication to each addend.

Formula : a(b + c) = ab + ac.

Example : Let (a = 3), (b = 4), (c = 5);

3 (4 + 5) = 3 × 9 = 27 Using the distributive property :

3 (4+ 5) = 3 × 4 + 3 × 5 = 12 + 15 = 27

Commutativity of Multiplication

Definition : The order of multiplication does not affect the product.

Formula : ab = ba

Example : Let (a = 6) and (b = 7); 6 × 7 = 42 and 7 × 6 = 42

Increments in Products

Increasing One Number by 1 :

If (b) is increased by 1 : a (b + 1) = ab + a.

Increasing Both Numbers by 1 :

If both (a) and (b) are increased by 1 :

(a + 1) (b + 1) = ab + a + b + 1.

Increasing One Number and Decreasing the Other:

If (a) is increased by 1 and (b) is decreased by 1

(a + 1) (b - 1) = ab - a + b - 1.

Will the Product Always Increase ?

No, it can remain unchanged or decrease depending on the values of (a) and (b).

Examples : (a = 1, b = 2) : Remains unchanged. (a = 2, b = 1): Decreases to 0.

Identities

General Identity : (a + u) (b - v)

= ab + ub - av - uv

Changes in Products with Increments :

(a + m) (b + n) = ab + an + mb + mn

Like Terms

Definition : Terms that have the same-variable raised to the same power.

Example : In (3x + 4x + 5y), (3x) and (4x)

Expanding Expressions

Example 1 : (a + b) (a + b) = a² + 2ab + b²

Example 2 : (a-b) (a + b) = a² - b²

Special Cases of the Distributive Property

Square of the Sum: (a + b)² = a² + 2ab + b²

Square of the Difference:

(a - b)² = a² - 2ab + b² Difference of Squares:

(a + b)(a - b) = a² - b²

Ista-Gunana

A traditional method of multiplication in ancient Indian mathematics that emphasises the distributive property for faster calculations.

Some Properties of Multiplication

In mathematics, understanding the properties of multiplication is essential. Two important properties are the Distributive Property and Commutativity of Multiplication.

Distributive Property of Multiplication

The distributive property states that when you multiply a number by a sum, you can distribute the multiplication to each addend. This can be expressed as :

a(b + c) = ab + ac

Example: If we take (a - 3), (b - 4) and (c = 5) 3(4 + 5) = 3 × 9 = 27

Using the distributive property :

3(4 + 5) = 3 × 4 + 3 × 5 = 12 + 15 = 27

Both methods give the same result, demonstrating the distributive property.

Commutativity of Multiplication : The commutative property states that the order in which two numbers are multiplied does not change the product. This can be expressed as :

ab = ba

Example : If (a = 6) and (b = 7):

6 × 7 = 42

and 7 × 6 = 42

Both expressions yield the same product, confirming the commutative property.

Increments in Products : When we increase one or both numbers in a product by 1, we can analyse how the product changes.

1. Increasing One Number by 1:

If we have two numbers (a) and (b), and we increase (b) by 1, the new product becomes :

a(b + 1) = ab + a

This shows that the product increases by (a).

2. Increasing Both Numbers by 1:

If we increase both (a) and (b) by 1, the new product is :

(a + 1) (b + 1) = ab + a + b + 1

Here, the product increases by (a + b + 1).

Increasing One Number and Decreasing the Other: If one number is increased by 1 and the other is decreased by 1, we have :

(a + 1) (b - 1) = ab - a + b - 1

The change in the product can vary. It does not always increase; it depends on the values of (a) and (b).

Will the Product Always Increase?

No, the product does not always increase. Here are three examples where the product decreases :

1. Let (a = 1) and (b = 2):

Original product: (1 × 2 = 2)

New product: ((1 + 1)(2 -1) = 2 × 1 = 2) (remains unchanged)

2. Let (a = 2) and (b = 1):

Original product: (2 × 1 = 2)

New product : ((2 + 1)(1 - 1) = 3 × 0 = 0) (decreases)

3. Let (a = 2) and (6 = 3):

Original product: (2 × 3 = 6)

New product: ((2 + 1)(3 -1) = 3 × 2 = 6) (remains unchanged)

Identities : An identity is a mathematical statement that holds true for all values of the variables involved. For example, the identity :

(a + u) (b - v) = ab + ub - av - uv

This identity helps us understand how products change when the numbers being multiplied are increased or decreased by specific amounts.

Change in Products with Increments :

If one of the numbers is increased by (m) and the other by (n), the product changes as follows :

(a + m) (b + n) = ab + an + mb + mn

Like Terms : Like terms are terms in an expression that have the same variable raised to the same power. For example, in the expression (3x + 4x + 5y), the terms (3x) and (4x) are like terms, while (5y) is not.

Expanding Expressions

1. Expand ((a + b)(a + b)):

(a + b)(a + b) = a² + ab + ab + b²

= a² + 2ab + b²

2. Expand ((a - b)(a + b)):

(a - b)(a + b) = a² - b²

3. Expand ((a + b)(a² + 2ab + b²)):

(a + b)(a² + 2ab + b²) = a³ + 2a²b + ab² + ab² + b³

= a³ + 3a²b + 3ab² + b³

4. Expand ((a - b)(a² + ab + b²)):

(a - b)(a² + ab + b²) = a³ + a²b + ab² - a²b - ab² - b³

= a³ - b³.

A Pinch of History

Ista-Gunana is a traditional method of multi-plication used in ancient Indian mathematics. It emphasises the distributive property to simplify calculations.

1. It uses the distributive property to break down complex multiplications into simpler parts.
2. This method allows for faster calculations, especially with large numbers.

Special Cases of the Distributive Property

Square of the Sum of Two Numbers : The

Square of the Sum of two numbers is expressed by the identity:

(a + b)a² = a² + 2ab + b²

Explanation : When you square a sum, you multiply the sum by itself: (a + b) (a + b).

Using the distributive property, we expand this as follows:

First, multiply a by both terms in the second (a + b):

a. a + a.b = a² + ab.

Next, multiply b by both terms in the second (a + b):

b. a + b.b = ab + b².

Now, combine all these terms :

a² + ab + ab + b² = a² + 2 ab + b².

Example : Let’s say (a = 3) and (b = 4):

(3 + 4)² = 7² = 49

Using the identity:

3² + 2(3)(4) + 4² = 9 + 24 + 16 = 49

This confirms the identity holds true.

Square of the Difference of Two Numbers

The Square of the Difference of two numbers is given by the identity :

(a - b)² = a² - 2ab + b²

Explanation : Similar to the square of the sum, we expand the square of the difference :

(a - b) (a - b).

Using the distributive property :

First, multiply a by both terms in the second (a - b):

a.a - a.b = a² - ab.

Next, multiply -b by both terms in the second (a - b):

-b.a + b.b = -ab + b².

Combine all these terms :

a² - ab - ab + b² = a² - 2ab + b².

Example : Let’s say (a = 5) and (b =b2):

(5 - 2)² = 3² = 9

Using the identity :

5² - 2(5)(2) + 2² = 25 - 20 + 4 = 9

This confirms the identity holds true.

Investigating Patterns Pattern 1: The first pattern we observe is :

2 (a² + b²) = (a + b)² + (a - b)²

Explanation : This identity shows that if you take the sum of the squares of two numbers and double it, you will get the sum of the square of their sum and the square of their difference.

This can be verified by substituting the identities we discussed:

From the identities, we know :

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

Adding these gives :

(a² + 2ab + b²) + (a² - 2ab + b²) = 2a² + b²

Thus, we can see that:

2(a² + b²) = 2a² + 2b²

Pattern 2 : The second pattern is :

(a + b) (a - b) = a² - b²

Explanation:

1. This is known as the difference of squares identity.
2. It states that the product of the sum and difference of the same two terms equals the difference of their squares.
3. To verify this, we can use the distributive property: