**Table of Contents:**

Commutative Property pertains specifically to addition and multiplication arithmetic operations. It asserts that altering the order or placement of two numbers being added or multiplied will not impact the final outcome. For instance, 4 + 5 equals 9, and 5 + 4 also equals 9. The sum is not influenced by the order of the two numbers. This principle applies similarly to multiplication. However, subtraction and division do not follow the commutative property as rearranging the numbers yields entirely different results.

**What is Commutative Property?**

The term ‘commutative’ comes from ‘commute’, signifying movement. Thus, the commutative property pertains to reordering numbers. In mathematical terms, if altering the order of operands does not alter the outcome of an arithmetic operation, that operation is commutative. The associative, distributive, and identity properties are distinct from the commutative property. This article will briefly examine the commutative property’s application in addition and multiplication.

### Commutative Property Formula

The commutative property formula for two given numbers A and B is as follows:

A + B = B + A |

A x B = B x A |

A – B ≠ B – A |

A ÷ B ≠ B ÷ A |

According to this formula, rearranging the order of two numbers being added or multiplied does not alter the outcome. However, when subtracting or dividing real numbers, the order of the numbers is crucial, and therefore cannot be modified. Therefore, the commutative property applies only to addition and multiplication operations, and not to subtraction and division.

**Commutative Property of Addition**

The commutative property of addition asserts that the order of the addends does not affect the sum’s value. Suppose ‘A’ and ‘B’ are two numbers. In that case, the commutative property of addition can be represented as illustrated in the figure below.

To better grasp the application of the commutative property of addition, let us consider an example. Suppose we have two numbers, 10 and 13. Adding them together yields 23, and switching their positions and adding them again still results in 23. Therefore, we can conclude that 10 + 13 = 13 + 10, as per the commutative property of addition.

## Commutative Property of Multiplication

The commutative property of multiplication states that rearranging the order in which we multiply two numbers does not affect the final product. The following diagram illustrates the commutative property of multiplication for two numbers.

Suppose we take the numbers 7 and 3. Multiplying them together yields 21, and switching their positions and multiplying them again still results in 21. Hence, we can conclude that 7 × 3 = 3 × 7. Therefore, the commutative property holds for the multiplication of numbers.

**Note:**

The commutative property only applies to addition and multiplication operations, not to subtraction and division.

9 – 5 ≠ 5 – 9.

9 ÷ 5 ≠ 5 ÷ 9.

**Commutative Property vs Associative Property**

In mathematics, there are four common properties of numbers: closure, commutative, associative, and distributive property. This section will focus on understanding the difference between the associative and commutative properties.

Both the associative and commutative properties state that the order of numbers does not affect the result of addition and multiplication. However, the two properties differ in their application. The commutative property deals with rearranging the order of the operands in a given arithmetic operation, while the associative property deals with changing the grouping of the operands.

Consider the table below, which shows the differences between the commutative and associative properties:

Property | Example | Formula |

Commutative | Addition: 5 + 3 = 3 + 5 | A + B = B + A |

Multiplication: 4 × 2 = 2 × 4 | A × B = B × A | |

Associative | Addition: (2 + 3) + 4 = 2 + (3 + 4) | (A + B) + C = A + (B + C) |

Multiplication: (5 × 6) × 2 = 5 × (6 × 2) | (A × B) × C = A × (B × C) |

As shown in the table, the commutative property applies to both addition and multiplication, while the associative property also applies to addition and multiplication, but deals with changing the grouping of operands.

**Important Notes:**

Below are some important points to keep in mind regarding the commutative property:

- The commutative property asserts that altering the order of the operands has no effect on the outcome.
- For addition, the commutative property is expressed as A + B = B + A.
- For multiplication, the commutative property is expressed as A × B = B × A.

**Commutative Property Examples**

**Example 1: **

Sam’s teacher asked him if the multiplication of two even numbers is an example of the commutative property. Can you help Sam find out whether it is commutative or not?

**Solution:**

We know that the commutative property of multiplication states that changing the order of the factors does not change the value of the product. Let’s take two even numbers, say 4 and 6.

4 x 6 = 24

But, if we switch the order of the factors, we get:

6 x 4 = 24

Since both the products are equal, we can say that the multiplication of two even numbers is an example of the commutative property.

**Example 2: **

Find the missing value: 96 ÷ 8 = ___ ÷ 12.

**Solution:**

The commutative property cannot be applied to division. However, we can use the properties of equality to solve this problem.

We know that if a = b, then a/c = b/c, provided c is not zero.

Let’s substitute the given values in the equation:

96 ÷ 8 = ___ ÷ 12

We can rewrite this equation as:

96/8 = x/12

To find the missing value, we need to isolate x on one side of the equation. We can do this by multiplying both sides of the equation by 12:

(96/8) × 12 = x

x = 144

∴ The missing number is 144.

**Example 3: **

Determine if the following statement is true or false.

“The commutative property applies to subtraction.”

**Solution:**

The commutative property only applies to addition and multiplication, and not to subtraction or division. Therefore, the given statement is false.

To further illustrate this point, let’s consider an example:

10 – 5 = 5

However, if we reverse the order of the operands, we get:

5 – 10 = -5

Since the two results are not equal, we can conclude that the commutative property does not apply to subtraction.

∴ The given statement is false.

**Can you apply the commutative property of addition/multiplication to 3 numbers?**Certainly. The commutative property is defined for two numbers, yet it also holds true for three numbers since it can be applied to any two numbers within the set of three in different permutations, resulting in the same outcome.

**Which operations do not follow commutative property?**The operations of subtraction and division do not adhere to the commutative property.

**What is the associative property of addition (or multiplication)?**The commutative property of addition (or multiplication) asserts that the sum (or product) of three or more whole numbers remains constant regardless of how the addends (or multiplicands) are grouped. Put simply,

(a + b) + c = a + (b + c) (a × b) × c = a × (b × c)

where a, b, and c are whole numbers.

**For which all operations does the associative property hold true?**The Associative property holds true for addition and multiplication.

**What is the distributive property of multiplication?**The distributive property of multiplication over addition states that if a number is multiplied by the sum of two or more addends, it yields the same outcome as if each addend is multiplied individually by the number, and the products are summed up. In other words,

a × (b + c) = (a × b) + (a × c)

where a, b, and c are whole numbers.