Commutative Property Formula with Solved Examples

In mathematics, the Commutative Property , also known as the commutative law, states that when performing arithmetic operations, such as addition and multiplication, the order of terms does not affect the result. In other words, you can freely change the positions or swap the numbers when adding or multiplying any two numbers. This property holds true for integers and is a fundamental concept in mathematics.

For instance, it means that 1 + 2 is equivalent to 2 + 1, and 2 x 3 is equal to 3 x 2.

Commutative Property:

A + B = B + A   (Addition)

A x B = B x A    (Multiplication)

What is Commutative Property?

The commutative property, also known as the commutative law, is a fundamental concept in mathematics that applies to certain arithmetic operations, specifically addition and multiplication. This property states that the order in which you perform these operations does not affect the result. In other words:

Commutative Property of Addition: For any two numbers, a and b, a + b = b + a. This means you can swap the positions of the numbers when adding them, and the result remains the same. For example, 3 + 5 is equal to 5 + 3, and both equal 8.

Commutative Property of Multiplication: For any two numbers, a and b, a * b = b * a. This means you can swap the positions of the numbers when multiplying them, and the result remains the same. For example, 2 * 4 is equal to 4 * 2, and both equal 8.

These properties are fundamental in mathematics and provide a foundation for various mathematical operations and concepts. They help simplify calculations and algebraic expressions by allowing you to rearrange terms without changing the outcome.

History

While the formal recognition of the commutative property dates back to the late 18th century, its principles were understood even in ancient times.

The term "Commutative" derives from the French word "commute" or "commuter," which means to interchange or move around. When combined with the suffix "-ative," it conveys the idea of a tendency toward something. Therefore, the literal meaning of "commutative" is "tending to switch or move around." This concept implies that when we exchange the positions of integers, the outcome remains unchanged.

Commutative Property of Addition

The Commutative Property of Addition states that when you add two numbers, the order in which you add them doesn't change the result. In other words, you can swap the numbers being added, and the sum remains the same.

Here are some examples of the Commutative Property of Addition in action:

Example 1:

2 + 3 = 5

Now, let's swap the numbers:

3 + 2 = 5

The result is still 5.

Example 2:

7 + 9 = 16

If we reverse the order:

9 + 7 = 16

The sum remains 16.

Example 3:

(-4) + 12 = 8

When we switch the numbers:

12 + (-4) = 8

The sum is still 8.

In all of these examples, you can see that changing the order of the numbers being added doesn't affect the result. This property holds true for all real numbers and is a fundamental concept in arithmetic.

Also Check – Rational Formula

Commutative Property of Multiplication

The Commutative Property of Multiplication states that when you multiply two numbers, the order in which you multiply them doesn't change the result. In other words, you can swap the numbers being multiplied, and the product remains the same.

Here's an example of the Commutative Property of Multiplication in action:

Example:

4 x 7 = 28

Now, let's swap the numbers:

7 x 4 = 28

The product remains 28.

In this example, you can see that changing the order of the numbers being multiplied does not affect the result. This property holds true for all real numbers and is a fundamental concept in arithmetic.

Also Read – Linear Equation Formula

Important Facts of Commutative Property

• The Commutative property applies exclusively to two fundamental arithmetic operations: Addition and Multiplication.
• This property states that altering the order of operands in these operations does not have any impact on the final result.
• For addition, it is expressed as the Commutative Property of Addition: A + B = B + A.
• For multiplication, it is expressed as the Commutative Property of Multiplication: A * B = B * A.

Non-Commutative Property

Indeed, some mathematical operations are non-commutative, meaning that changing the order of their operands leads to different results. Two such non-commutative operations are subtraction and division.

For subtraction:

When you subtract one number from another, the order of the numbers matters. For example:

4 - 3 equals 1.

However, 3 - 4 equals -1.

These results are distinct integers, demonstrating that subtraction is non-commutative.

For division:

Division also does not adhere to the commutative property. For instance:

6 ÷ 2 equals 3.

But, 2 ÷ 6 equals 1/3.

Consequently, 6 ÷ 2 is not equal to 2 ÷ 6, underscoring the non-commutative nature of division.

These examples illustrate that not all mathematical operations share the commutative property, and it is essential to recognize this distinction when working with different arithmetic operations.

Also Check – Rational Numbers Formulas

Solved Examples on Commutative Property

Example 1: Identify which of the following operations obey the commutative law:

3 × 12

4 + 20

36 ÷ 6

36 – 6

(-3) × 4

Solution: Options 1, 2, and 5 adhere to the commutative law.

Explanation:

For multiplication:

3 × 12 = 36, and

12 × 3 = 36.

So, 3 × 12 = 12 × 3 (commutative).

For addition:

4 + 20 = 24, and

20 + 4 = 24.

Hence, 4 + 20 = 20 + 4 (commutative).

For division:

36 ÷ 6 = 6, but

6 ÷ 36 = 0.167.

Therefore, 36 ÷ 6 ≠ 6 ÷ 36 (non-commutative).

For subtraction:

36 − 6 = 30, while

6 − 36 = −30.

Thus, 36 − 6 ≠ 6 − 36 (non-commutative).

For multiplication with negative numbers:

(-3) × 4 = -12, and

4 × (-3) = -12.

Consequently, (-3) × 4 = 4 × (-3) (commutative).

Q.2: Prove that a + b = b + a if a = 10 and b = 9.

Solution: Given a = 10 and b = 9.

LHS = a + b = 10 + 9 = 19 ……(1)

RHS = b + a = 9 + 10 = 19 ……(2)

By equations 1 and 2, in accordance with the commutative property of addition, we have:

LHS = RHS

Hence, it is proven.

Q.3: Prove that A * B = B * A, if A = 4 and B = 3.

Solution: Given A = 4 and B = 3.

A * B = 4 * 3 = 12 ….. (1)

B * A = 3 * 4 = 12 …..(2)

By equations (1) and (2), following the commutative property of multiplication, we conclude:

LHS = RHS

A * B = B * A

Thus, it is proven.