Mathematics deals with four operations: addition, subtraction, multiplication, and division. Since the area of focus is the commutative property of multiplication, multiplication will be kept in focus. It is a mathematical operation in which whole numbers are repeated in addition to yield a new number. It is often called repeated addition. The inverse operation of multiplication is division.

Both multiplication and division play an important role in the fundamentals of various living organisms. Living beings multiply to increase their population, and microorganisms, like yeast and bacteria, divide to increase their growth.

Since we have teased what multiplication is, it’s time to focus on one of its special properties, i.e. commutativity.

**What is a commutative property?**

The term “commutative” comes from the word “commute,” which means “to move around.” As a result, the commutative property is concerned with shifting the numbers. So, if altering the sequence of the inputs does not influence the outcomes of the mathematical operations, that arithmetic operation is commutative. Other qualities of integers include the associative property, the distributive property, and the identity property. They are not the same as the commutative property of numbers.

Let us understand commutative property by an example. In the given figure, two inlets can be seen. The first inlet is x and the second inlet is y. Both the inlets fuse in a jar, and the outlet becomes x and y. If the inlets are interchanged, as seen in the next figure, the outlet becomes y and x. However, the outlet in both the figures is the same, i.e. x and y = y and x. This property is known as the commutative property.

Commutativity is followed by addition and multiplication only. It is a binary operation in which changing the order of the operand does not change the result. As seen in the above example, even if you change the inlets, the outlet remains the same, i.e. x and y = y and x.

Note: The commutative property does not hold for subtraction and division operations. Let us understand this by the examples given below.

6 – 2 = 4, but 2 – 6 = -4. Thus, 6 – 2 ≠ 2 – 6.

6 ÷ 2 = 3, but 2 ÷ 6 = 1/3. Thus, 6 ÷ 2 ≠ 2 ÷ 6

**Commutative property of multiplication definition **

The commutative property of multiplication deals with the changing order or interchange of their places, and still, the answer does not change. For instance, 6 x 5 = 30 and also 5 x 6 = 30. Changing the position of the numbers does not affect the outcome.

**What is commutative property of multiplication? **

The commutative property of multiplication states that the sequence wherein two integers are multiplied does not affect the complete outcome. The graphic below depicts the commutative property of 2 different multiplications.

Let’s take the example of 10 and 2. The product of 10 x 2 is 20. Now, interchange the position of the integers. The previous becomes 2 x 10, giving 20 as a result. So, this proves that changing the integer sequence does not change the result. As a result, the commutative condition stands accurate for integer multiplication.

**Formula for the commutative property of multiplication**

If two integers are given, let’s say A and B, then they can be written as

A x B = B x A

Where A and B follow the commutative property, the changed order of the integers does not affect the result, and the product of the integers remains the same.

The commutative property of multiplication is widely used in the function composition of linear functions. In this, the real numbers to the real numbers are arranged in such a way that they yield the same so that the value of variables can be found.

For example, (f o g) (x) = f g(x) = 2 (3x + 7) + 1 = 6x + 15, and (g o f) (x) = g f(x) = 3 (2x + 1) + 7 = 6x + 10. Once you know the value of such functions, you can take out the value of variables present in the equations.

**Commutative property of multiplication example in everyday life**

The commutative property of multiplication plays a crucial role in everyday life. It is associated with the property that holds for a pair of elements under a certain binary operation. Below are a few examples where you can see the commutative property of multiplication in everyday life.

- If you want to put extra toppings on the pizza, and they are priced the same, you can add them using the commutative property of multiplication. For instance, if you want to add pepperoni and jalapeno in 4 x 5 parts, and they have the same price, you can put them either way, depending upon the choice.
- Learning lessons to perform quizzes in a better way. You can practice quizzes depending on the schedule of your choice. If you want to complete two subjects and attempt only two or three quizzes from each subject, you can choose them either way in a specific ratio.
- Taking out currency from the ATM. You can withdraw currency from the ATM by using the commutative property of multiplication. If you want to take out 1000 dollars and the ATM has only limited currency and withdrawal limits, then, for instance, you can take out 200 dollars 5 times or take out 500 dollars 2 times.
- Physical examination of students’ health in a camp. This is the area where the commutative property of multiplication is widely used. The physicians multiply the number of students with the average reading of a particular age group to take out the average of various physical aspects of different age groups. This way, they get to know the health situation of that age group.

**Commutative property of multiplication examples **

**Example 1: Riya made a vertical row of 5 balls and a horizontal row of 8 balls. How can she rearrange the rows and columns of the balls to get the same number of balls?**

**Solution: **Since Riya has made 5 vertical rows and 8 horizontal rows, it means there are a total of 5 x 8 = 40 balls with Riya. To let her make a new row with the same number of balls, she can make 8 vertical rows and 5 horizontal rows. This is because, according to the commutative property of multiplication, the product of 5 x 8 = 8 x 5. Hence, the new arrangement will be of 8 vertical rows and 5 horizontal rows.

**Example 2: State whether true or false – Division of 12 by 4 satisfies the commutative property.**

**Solution:** The commutative property does not hold for division operations. So, the given statement is false. How? You can verify it by doing the calculation as done below.

12 ÷ 4 = 3

4 ÷ 12 = 1/3 = 0.33

⇒ 12 ÷ 4 ≠ 4 ÷ 12

Therefore, the given statement is false.

**Example 3: Find the missing value: 131 x 56 x 72 = 72 x __ x 56**

**Solution:** The commutative property of multiplication states that if there are three numbers x, y, and z, then x × y × z = z × y × x = y × z × x or another possible arrangement can be made. If you observe the given equation, you will find that the commutative property can be applied. If x = 131, and y = 56 and z = 72, then we know that 131 x 56 x 72 = 72 x 131 x 56.

Therefore, the missing number is 131.

**Frequently asked questions on the Commutative Property of Multiplication **

**Q1. What is the commutative property of multiplication? **

The commutative property of multiplication is the mathematical operation in which if you change the sequence of operands, the output will remain the same. The final product does not change even if you change the sequence of the numbers.

**Q2. Why should I study the commutative property of multiplication?**

It is necessary to study the commutative property of multiplication because it is used in various situations in everyday life, such as taking out money from money, making rows and columns to fill data in a spreadsheet, and many more such areas.

**Q3. Is multiplication always commutative?**

Yes, multiplication is always commutative. Since multiplication follows that the inductive definition and the Cartesian-products definition are equivalent, the multiplication (defined inductively) is commutative.

**Q4. Is the matrix multiplication ever commutative?**

The commutative property of multiplication over matrices does not hold. Since matrices deal with linear transformations’ composition, the composition of functions is not commutative. Moreover, you can switch the order of the factors that will yield the same results. However, if it contains vector notions, then the commutative property of multiplication might not hold because vector multiplication changes the direction of the resultant vector.