Commutative Property Formula

The commutative property formula deals with moving the numbers around. So mathematically, we can define any binary operation as a commutative one, if changing the order of the operands does not change the result of the operation. Let's learn about the commutative property formula with a few solved examples in the end.

What Is the Commutative Property Formula?

The commutative property says that the order of operands does not change the final result. The commutative property formulas for addition and multiplication are as given below.

Commutative Property Formula

Commutative property formula for addition: The commutative property of addition says that changing the order of the addends will not change the value of summation.

A + B = B + A

Commutative property formula for multiplication: The commutative property of multiplication says that the order in which we multiply the numbers does not change the final product.

A × B = B × A

Verification of Commutative Property Formula

Let us try to justify how and why the commutative property formula is only valid for addition and multiplication operations. We will apply the commutative property formula individually on the four basic operations.

For Addition: The general commutative property formula for addition is expressed as a + b = b + a. Let us try to verify the same. For example, (1 + 4) = (4 + 1) = 5. We say that the addition is commutative for the given set of numbers.

For Subtraction: The general commutative property formula for subtraction is expressed as (A - B) ≠ (B - A). Let us try to verify the same. For example, (1 - 4) ≠ (4 - 1) i.e., -3 ≠ 3. We say that subtraction is not commutative for the given set of numbers.

For Multiplication: For any set of two numbers (A, B) commutative property for multiplication is given as A × B = B × A. For example, (2 × 4) = (4 × 2) = 8. Here we find that multiplication is commutative for the given set of numbers.

For Division: For any two numbers (A, B) commutative property for division is given as A ÷ B ≠ B ÷ A. For example, (6 ÷ 3) ≠ (3 ÷ 6) = 2 ≠ 1/2. You will find that expressions on both sides are not equal. So division is not commutative for the given numbers.

Examples on Commutative Property Formula

Let us take a look at a few examples to better understand the formula of the commutative property.

Example 1: If (6 × 4) = 24, then prove (4 × 6) also results in 24 using commutative property formula

Solution:

Since multiplication satisfies the commutative property formula
Hence (6 × 4) = (4 × 6) = 24.

Example 2: Jacky's mother asked him whether x + y = y + x is an example of the commutative property formula. Can you help Jacky find out whether it is commutative or not?

Solution:     

We know that the commutative property formula for addition states that changing the order of the addends will not change the value of summation.
So, we see that changing the order will not alter the summation value.
Hence, the summation remains the same. So this is an example of the commutative property.

Answer: x + y = y + x is an example of the commutative property.

Example 3: Rinny walked 5 miles to the North, and then 3 miles to the East. Will she end up at the same place if she would have walked 3 miles to the East first, and then 5 miles to the North?

Solution:     

We try to convert this on a coordinate plane. We assume the miles he walked to the North to be the y-coordinate, while miles in the East to be the x-coordinate. By simple rules of vector addition, we can say that the final position of Rinny can be calculated as:
Final position 1 = 5 miles to the North + 3 miles to the East = (3,5)
Therefore the final coordinate = (3,5)
Now, we can also say: Final position 2 = 3 miles to the East + 5 miles to the North = (3,5)
Therefore the final coordinate = (3,5)
Thus, in both cases, our answer for the final position coordinate is the same.

Answer: Rinny will end up in the same place.