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# 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.**

## FAQs on Commutative Property Formula

### What Is the Commutative Property Formula for Addition?

The commutative property formula for addition is defined asthe sum of two or more numbers that remain the same, irrespective of the order of the operands. For addition, the commutative property formula is expressed as (A + B) = (B + A)

### What Is the Commutative Property Formula for Multiplication?

The commutative property formula for multiplication is defined as** t**he product of two or more numbers that remain the same, irrespective of the order of the operands. For multiplication, the commutative property formula is expressed as (A × B) = (B × A).

### What Is the Commutative Property Formula for Rational Numbers?

The commutative property formula for rational numbers can be expressed only for addition and multiplication. The general form is given as (P + Q) = (Q + P) or (P × Q) = (Q × P). Here the values of P, Q are in form of a/b, where b ≠ 0.

### Which Two Operations Satisfy the Condition of Commutative Property Formula?

The two operations which satisfy the condition of the commutative property formula are addition and multiplication.

### What is the Commutative Property? Give Examples?

The commutative property states that if the order of position of numbers is interchanged while performing addition or multiplication, the sum or the product obtained does not change. It is to be noted that commutative property holds good only for addition and multiplication and not for subtraction and division. For example, 6 + 7 is equal to 13 and 7 + 6 is also equal to 13. Similarly, 6 × 7 = 42, and 7 × 6 = 42.

### What is the Commutative Property of Addition?

According to the commutative property of addition, when two numbers are added in any order the sum remains the same. This property applies even when more than two numbers are added, and the order of the numbers is changed, the sum still remains the same. For example, 3 + 4 + 5 is equal to 12, and 4 + 3 + 5 is also equal to 12. In both cases, the sum is the same.

### What is the Commutative Property of Multiplication?

According to the commutative property of multiplication, the order of multiplication of numbers does not change the product. For example, 4 × 5 is equal to 20 and 5 × 4 is also equal to 20. Though the order of numbers is changed the product is 20. This property holds true even when more than 2 numbers are multiplied.

### Can Commutative Property be Applied for Subtraction and Division? List Out the Reasons.

Commutative property cannot be applied for subtraction and division, because the changes in the order of the numbers while doing subtraction and division do not produce the same result. For example, 5 – 2 is equal to 3, whereas 3 – 5 is not equal to 3. In the same way, 10 divided by 2, gives 5, whereas, 2 divided by 10, does not give 5. Therefore, commutative property is not true for subtraction and division.

### What is the Difference Between Commutative Property and Associative Property?

The commutative property states that the change in the order of numbers in an addition or multiplication operation does not change the sum or the product. The commutative property of addition is written as A + B = B + A. The commutative property of multiplication is written as A × B = B × A. The associative property states that the grouping or combination of two or more numbers that are being added or multiplied does not change the sum or the product. The associative property of addition is written as: (A + B) + C = A + (B + C). The associative property of multiplication is written as (A × B) × C = A × (B × C).

### What is the Difference Between Commutative Property and Distributive Property?

The commutative property states that the change in the order of numbers for the addition or multiplication operation does not change the result. The commutative property of addition for two numbers ‘A’ and ‘B’ is A + B = B + A. The distributive property means multiplying a number with every number inside the parentheses. The numbers inside the parentheses are separated by an addition or a subtraction symbol. The distributive property of addition for two numbers ‘A’, ‘B’ is: A(B + C) = AB + AC.

### What are Commutative Laws?

Commutative law is another word for the commutative property that applies to addition and multiplication. The commutative law of addition states that the order of adding two or more numbers does not change the sum (A + B = B + A). The commutative property of multiplication states that the order of multiplying two or more numbers does not change the product (A × B = B × A).

### What is commutative property? Give examples.

In Mathematics, a commutative property states that if the position of integers are moved around or interchanged while performing addition or multiplication operations, then the answer remains the same.

Examples are:

4+5 = 5+4 and 4 x 5 = 5 x 4

9 + 2 = 2 + 9 and 9 x 2 = 2 x 9

### What is commutative property of addition?

When two numbers are added together, then if we swap the positions of numbers, the sum of the two remains the same.

For example, 3+4 = 4+3 = 7

### What is the commutative property of multiplication?

When two numbers are multiplied together and if we interchange their positions, then the product of the two remains the same.

For example, 5 x 3 = 3 x 5 = 15

### What are the major four properties in Maths?

The four major properties in Maths are:

Identity property, commutative property, associative property and Distributive property

### What is the difference between commutative and associative property?

Commutative property holds regardless of order of numbers while addition or multiplication. Whereas associative property holds regardless of grouping of numbers.

**Example 1:** **Which of the following obeys commutative law?**

**3 × 12****4 + 20****36 ÷ 6****36 – 6****-3 × 4**

**Solution: **Options 1, 2 and 5 follow the commutative law

**Explanation:**

- 3 × 12 = 36 and

12 x 3 = 36

=> 3 x 12 = 12 x 3 (commutative)

- 4 + 20 = 24 and

20 + 4 = 24

=> 4 + 20 = 20 + 4 (commutative)

- 36 ÷ 6 = 6 and

6 ÷ 36 = 0.167

=> 36 ÷ 6 ≠ 6 ÷ 36 (non commutative)

- 36 − 6 = 30 and

6 – 36 = – 30

=> 36 – 6 ≠ 6 – 36 (non commutative)

- −3 × 4 = -12 and

4 x -3 = -12

=> −3 × 4 = 4 x -3 (commutative)

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

Sol: Given that, a = 10 and b = 9

LHS = a+b = 10 + 9 = 19 ……(1)

RHS = b + a = 9 + 10 = 19 ……(2)

By equation 1 and 2, as per commutative property of addition, we get;

LHS = RHS

Hence, proved.

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

Sol: Given, A = 4 and B = 3.

A.B = 4.3 = 12 ….. (1)

B.A = 3.4 = 12 …..(2)

By eq.(1) and (2), as per the commutative property of multiplication, we get;

LHS = RHS

A.B = B.A

Hence, proved.

# Commutative Property

The commutative property deals with the arithmetic operations of addition and multiplication. It means that changing the order or position of numbers while adding or multiplying them does not change the end result. For example, 4 + 5 gives 9, and 5 + 4 also gives 9. The order of numbers being added does not affect the sum. The same concept applies to multiplication too. The commutative property does not hold for subtraction and division, as the end results are completely different in changing the order of numbers. Let us look at all these concepts in detail.

## What is Commutative Property?

The word ‘commutative’ originates from the word ‘commute’, which means to move around. Hence, the commutative property deals with moving the numbers around. So mathematically, if changing the order of the operands does not change the result of the arithmetic operation then that particular arithmetic operation is commutative. Apart from this, there are other properties of numbers: the associative property, the distributive property, and the identity property. They are different from the commutative property of numbers. Let us discuss the commutative property of addition and multiplication briefly.

## Commutative Property of Addition

The commutative property of addition says that changing the order of the addends does not change the value of the sum. There are cases where we need to add more than two numbers. The commutative property is true even when there are more than two numbers being added. For example, 10 + 20 + 30 + 40 = 100, and 40 + 30 + 20 + 10 is also equal to 100. The sum is 100 in both cases even when the order of numbers is changed. If ‘A” and ‘B’ are two numbers, then the commutative property of numbers can be represented as shown in the figure below.

## Commutative Property of Multiplication

The commutative property of multiplication says that the order in which we multiply the numbers does not change the final product. Similar to the commutative property of addition, the commutative property holds good if there are more than 2 numbers to be multiplied. For example, 6 × 7 × 8 = 336. The same result is obtained when we multiply 8 × 7 × 6 = 336. The product in both cases is 336. So, it is evident that the order or the position of numbers being multiplied does not change the product. The image given below represents the commutative property of the multiplication of two numbers.

**Important Notes**

Some key points to remember about the commutative property are given below.

- The Commutative property states that “
**changing the order of the operands does not change the result.”** - The commutative property for addition is A + B = B + A
- The commutative property for multiplication is A × B = B × A

## Examples of Commutative Property

**Example 1: Jacky’s mother asked him whether x + y = y + x is an example of the commutative property. Can you help Jacky find out whether it is commutative or not?**

**Solution**

We know that the commutative property of addition states that changing the order of the addends does not change the value of the sum. In this case, x + y = y + x, is an example of the commutative property which says A + B = B + A.

**Example 2: Benny was given a question in his assignment: 3 + 5 + 7 = 15, in which he was asked to check whether the commutative property of addition was applied or not. Can you help Bran with his assignment?**

**Solution:**

By simple rules of addition, the sum of the given numbers can be calculated as: Sum = 3 + 5 + 7 = 15

Now, we if we change the order of the numbers and add them: Sum = 5 + 3 + 7 = 15

Let us arrange this in another order: Sum = 7 + 5 + 3 = 15

∴ The sum is the same in all the cases. Thus, the commutative property is applied here.

## Associative, Commutative, and Distributive Properties

There are three basic properties of numbers, and your textbook will probably have just a little section on these properties, somewhere near the beginning of the course, and then you’ll probably never see them again (until the beginning of the next course). My impression is that covering these properties is a holdover from the “New Math” fiasco of the 1960s. While the topic will start to become relevant in matrix algebra and calculus (and become amazingly important in advanced math, a couple years after calculus), they really don’t matter a whole lot now.

Why not? Because every math system you’ve ever worked with has obeyed these properties! You have never dealt with a system where *a*×*b* did not in fact equal *b*×*a*, for instance, or where (*a*×*b*)×*c* did not equal *a*×(*b*×*c*). Which is why the properties probably seem somewhat pointless to you. Don’t worry about their “relevance” for now; just make sure you can keep the properties straight so you can pass the next test. The lesson below explains how I keep track of the properties.

## Distributive Property

The Distributive Property is easy to remember, if you recall that “multiplication distributes over addition”. Formally, they write this property as “*a*(*b* + *c*) = *ab* + *ac*“. In numbers, this means, for example, that 2(3 + 4) = 2×3 + 2×4. Any time they refer in a problem to using the Distributive Property, they want you to take something through the parentheses (or factor something out); any time a computation depends on multiplying through a parentheses (or factoring something out), they want you to say that the computation used the Distributive Property.

- Why is the following true? 2(
*x*+*y*) = 2*x*+ 2*y*

Since they distributed through the parentheses, this is true **by the Distributive Property.**

- Use the Distributive Property to rearrange: 4
*x*– 8

The Distributive Property either takes something through a parentheses or else factors something out. Since there aren’t any parentheses to go into, you must need to factor out of. Then the answer is:

By the Distributive Property, 4*x* – 8 = 4(*x* – 2).

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“But wait!” I hear you cry; “the Distributive Property says multiplication distributes over addition, not over subtraction! What gives?” You make a good point. This is one of those times when it’s best to be flexible. You can either view the contents of the parentheses as the subtraction of a positive number (“*x* – 2″) or else as the addition of a negative number (“*x* + (–2)”). In the latter case, it’s easy to see that the Distributive Property applies, because you’re still adding; you’re just adding a negative.

The other two properties come in two versions each: one for addition and the other for multiplication. (Yes, the Distributive Property refers to both addition and multiplication, too, but it refers to both of the operations within just the one rule.)

## Associative Property

The word “associative” comes from “associate” or “group”; the Associative Property is the rule that refers to grouping. For addition, the rule is “*a* + (*b* + *c*) = (*a* + *b*) + *c*“; in numbers, this means 2 + (3 + 4) = (2 + 3) + 4. For multiplication, the rule is “*a*(*bc*) = (*ab*)*c*“; in numbers, this means 2(3×4) = (2×3)4. Any time they refer to the Associative Property, they want you to regroup things; any time a computation depends on things being regrouped, they want you to say that the computation uses the Associative Property.

- Rearrange, using the Associative Property: 2(3
*x*)

They want me to regroup things, not simplify things. In other words, they do not want me to say “6*x*“. They want to see me do the following regrouping:

(2×3)*x*

- Simplify 2(3
*x*), and justify your steps.

In this case, they do want me to simplify, but I have to say why it’s okay to do… just exactly what I’ve always done. Here’s how this works:

2(3*x*) : original (given) statement

(2×3) *x* : by the Associative Property

6*x* : simplification of 2×3

- Why is it true that 2(3
*x*) = (2×3)*x*?

Since all they did was regroup things, this is true **by the Associative Property.**

## Commutative Property

The word “commutative” comes from “commute” or “move around”, so the Commutative Property is the one that refers to moving stuff around. For addition, the rule is “*a* + *b* = *b* + *a*“; in numbers, this means 2 + 3 = 3 + 2. For multiplication, the rule is “*ab* = *ba*“; in numbers, this means 2×3 = 3×2. Any time they refer to the Commutative Property, they want you to move stuff around; any time a computation depends on moving stuff around, they want you to say that the computation uses the Commutative Property.

- Use the Commutative Property to restate “3×4×
*x*” in at least two ways.

They want me to move stuff around, not simplify. In other words, my answer should not be “12*x*“; the answer instead can be any two of the following:

4 × 3 × *x*

4 × *x* × 3

3 × *x* × 4

*x* × 3 × 4

*x* × 4 × 3

- Why is it true that 3(4
*x*) = (4*x*)(3)?

Since all they did was move stuff around (they didn’t regroup), this statement is true **by the Commutative Property.**

## Worked examples

- Simplify 3
*a*– 5*b*+ 7*a*. Justify your steps.

I’m going to do the exact same algebra I’ve always done, but now I have to give the name of the property that says its okay for me to take each step. The answer looks like this:

3*a* – 5*b* + 7*a* : original (given) statement

3*a* + 7*a* – 5*b* : Commutative Property

(3*a* + 7*a*) – 5*b* : Associative Property

*a*(3+7) – 5*b* : Distributive Property

*a*(10) – 5*b* : simplification (3 + 7 = 10)

10*a* – 5*b* : Commutative Property

The only fiddly part was moving the “– 5*b*” from the middle of the expression (in the first line of my working above) to the end of the expression (in the second line). If you need help keeping your negatives straight, convert the “– 5*b*” to “+ (–5*b*)”. Just don’t lose that minus sign!

- Simplify 23 + 5
*x*+ 7*y*–*x*–*y*– 27. Justify your steps.

I’ll do the exact same steps I’ve always done. The only difference now is that I’ll be writing down the reasons for each step.

23 + 5*x* + 7*y* – *x* – *y* – 27 : original (given) statement

23 – 27 + 5*x* – *x* + 7*y* – *y* : Commutative Property

(23 – 27) + (5*x* – *x*) + (7*y* – *y*) : Associative Property

(–4) + (5*x* – *x*) + (7*y* – *y*) : simplification (23 – 27 = –4)

(–4) + *x*(5 – 1) + *y*(7 – 1) : Distributive Property

–4 + *x*(4) + *y*(6) : simplification (5 – 1 = 4, 7 – 1 = 6)

–4 + 4*x* + 6*y* : Commutative Property

- Simplify 3(
*x*+ 2) – 4*x*. Justify your steps.

3(*x* + 2) – 4*x* : original (given) statement

3*x* + 3×2 – 4*x* : Distributive Property

3*x* + 6 – 4*x* : simplification (3×2 = 6)

3*x* – 4*x* + 6 : Commutative Property

(3*x* – 4*x*) + 6 : Associative Property

*x*(3 – 4) + 6 : Distributive Property

*x*(–1) + 6 : simplification (3 – 4 = –1)

–*x* + 6 : Commutative Property

- Why is it true that 3(4 +
*x*) = 3(*x*+ 4)?

All they did was move stuff around.

**Commutative Property**

- Why is 3(4
*x*) = (3×4)*x*?

All they did was regroup.

**Associative Property**

- Why is 12 – 3
*x*= 3(4 –*x*)?

They factored.

**Distributive Property**

## Commutative Property Over Addition

Commutative property over addition says the sum of two numbers is said to be commutative for addition if their sum remains the same even if the order of the addition is changed.

Therefore, we can say the natural numbers, whole numbers, integers, and rational numbers are commutative over addition.

## Commutative Property Over Addition Formula

The commutative property says if A and B be the operands, then changing its position does not change the result of the addition.Therefore, A+B=B+A.

## Commutative Property Over Multiplication

Commutative property over multiplication says if the product two whole numbers or integers is said to be commutative for multiplication if their product remains the same even if the order of the multiplication is changed.

Therefore, we can say the natural numbers, whole numbers, integers, and rational numbers are commutative over multiplication also.

## Commutative Property Over Multiplication Formula

The commutative property says if A and BB be the operands, then changing its position does not change the result of the multiplication.Therefore, A×B=B×A.

## Commutative Property Over Subtraction

Let us see whether the commutative property is applicable to the subtraction of numbers or not.

Therefore, the difference between the two rational numbers is not the same if their places are interchanged.

Thus, the rational numbers are not commutative under subtraction.

As the integers and rational numbers are not commutative under subtraction, the natural numbers and the whole numbers are also not commutative under subtraction

## Commutative Property Over Division

Let us see whether the commutative property is applicable to the division

Therefore, the result of the division of the two rational numbers is not the same if their places are interchanged.

Thus, the rational numbers are not commutative under division.

As the integers and rational numbers are not commutative under the division then, the natural numbers and the whole numbers are not commutative under division.

## Solved Examples – Commutative Property Formula

**Q.1. Prove that the addition of 2020 and 3030 comes under the commutative property.Ans:**

**The commutative property says if A and B be the operands, then changing its position does not change the result of the addition.**

Therefore, the difference between the two numbers is not the same if their places are interchanged.

Thus, the given numbers are not commutative under subtraction.

**Commutative Property**

The commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer. The property holds for Addition and Multiplication, but not for subtraction and division.

The above examples clearly show that we can apply the commutative property on addition and multiplication. However, we cannot apply commutative property on subtraction and division. If you move the position of numbers in subtraction or division, it changes the entire problem.

Therefore, if a and b are two non-zero numbers, then:

The commutative property of addition is:

**a + b = b + a**

The commutative property of multiplication is:

**a × b = b × a**

In short, in commutative property, the numbers can be added or multiplied to each other in any order without changing the answer.

**Let us see some examples to understand commutative property.**

**Example 1: Commutative property with addition**

Myra has 5 marbles, and Rick has 3 marbles. How many marbles they have in total?

To find the answer, we need to add 5 and 3.

Hence, we can see whether we add 5 + 3 or 3 + 5, the answer is always 8.

**Example 2:** Commutative property with subtraction.

Alvin has 12 apples. He gives 8 apples to his sister. How many apples are left with Alvin?

Here, we subtract 8 from 12 and get the answer as 4 apples. However, we cannot subtract 12 from 8 and get 8 as the answer.

**Example 3:** Commutative property with multiplication.

Sara buys 3 packs of buns. Each pack has 4 buns. How many buns did she buy?

Here, if we multiply 3 by 4 or 4 by 3, in both cases we get the answer as 12 buns.

So, the commutative property holds for multiplication.

So, commutative property holds true for multiplication.

**Example 4:** Commutative property with division.

If you have to divide 25 strawberries to 5 kids, each kid will receive 5 strawberries. However, if you have to divide 5 strawberries amongst 25 children, every kid will get a tiny fraction of the strawberry. Therefore, we cannot apply the commutative property with the division.

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