# Commutative property

If we analyze the mathematical operators we can verify that they have different properties. These tell us about the way in which numbers can be distributed, associated or switched when carrying out an operation. This time we are going to focus on the **commutative property** which allows us, if we wish, to **alter the order of the addends in the addition and of the factors in the multiplication without the result being altered** . Something that we must highlight is the importance of this property, because thanks to it **we can avoid errors** when solving equations and mathematical operations.

**Quick example**

**Sum**- 2 + 6 + 1 = 9
- 6 + 1 + 2 = 9

**Multiplication**- 3 x 2 x 4 = 24
- 4 x 3 x 2 = 24

## What is the commutative property?

The **commutative property** is a mathematical property that both **addition** and **multiplication have** . This tells us that **it does not matter in the order in which the addends are added or the factors multiplied,** as the result obtained will not be altered.

- Commutative property of addition
- Commutative Property of Multiplication
- Commutative Property of Subtraction
- Commutative property of division
- Commutative property of matrices
- What is it for
- History
- To which number systems does the commutative property apply?
- Related properties
- Related topics

## Commutative property of addition

Definition:“The order of the addends does not alter the sum”

The **commutative property of the addition** , also called the **property of the order of the addition** , is one that **allows us that the addends or figures of a sum or addition can be exchanged without the result being affected** . That is, we can do the addition in the order we want because the result that we will obtain will always be identical.

#### Examples

**We are going to add 2, 3 and 5 in different order to see that the result is the same:**

- 2 + 5 + 3 = 10
- 3 + 5 + 2 = 10
- 5 + 3 + 2 = 10

Therefore we have to:

2 + 5 + 3 = 3 + 5 + 2 = 5 + 3 + 2 = 10

As we can see, the commutative property is fulfilled because no matter how much we alter the order, the result is always 8. And it is that the order in which we add the different numbers does not matter because in no case does it affect the result.

Let’s look at another example.

**Now we will add the numbers 2, 4 and 6 altering the order to check if it gives the same result:**

- 2 + 4 + 6 = 12
- 4 + 6 + 2 = 12
- 6 + 2 + 4 = 12

Namely:

2 + 4 + 6 = 4 + 6 + 2 = 6 + 2 + 4 = 12

As in the previous example, we see that whatever the order in which we decide to perform the operation, the result is exactly the same.

#### Training

[HDmaybe quiz = “870”]

## Commutative Property of Multiplication

Definition:“The order of the factors does not alter the product”

The **commutative property of multiplication** or **property of the order of multiplication** , is the mathematical property that **tells us that the different factors of a multiplication can be arranged in the order we want** , because whatever this is, the result we will obtain will be the same .

#### Examples

**In this example we will multiply the numbers 2, 3 and 5 in a different order.**

- 2 x 3 x 5 = 30
- 3 x 5 x 2 = 30
- 5 x 2 x 3 = 30

Therefore:

2 x 3 x 5 = 3 x 5 x 2 = 5 x 2 x 3 = 30

**We multiply 3, 4 and 6 by modifying their order. Is the result the same?**

- 3 x 4 x 6 = 72
- 6 x 3 x 4 = 72
- 4 x 6 x 3 = 72

We obtain that:

3 x 4 x 6 = 6 x 3 x 4 = 4 x 6 x 3 = 72

#### Training

[HDmaybe quiz = “871”]

## Commutative Property of Subtraction

Subtraction is made up of two parts, the minuend and the subtrahend, and if we try to alter their order we will see how the result we obtain is not the same and **the commutative property does not apply in the case of subtraction** . Given this, we can conclude that any change we make in the order of the elements of a subtraction will cause the result of the same to change as well.

#### Example

**We are going to subtract 8 and 2 in different order to see if the result we obtain is the same or not:**

- 8 – 2 = 6
- 2 – 8 = -6

So we have to:

8 – 2 ≠ 2 – 8

As we can see, the result of subtracting 2 from 8 is not the same as that of subtracting 8 from 2, so the fact that the commutative property does not apply to subtraction is satisfied.

## Commutative property of division

The same thing happens with **division** as with subtraction, since **the commutative property cannot be applied** since if we change the dividend for the divisor we will see how the result will not be the same. Let’s illustrate this with an example.

#### Example

**In this example we are going to divide 8 and 4 alternating the order and we will check if the result obtained is the same or not.**

- 8: 4 = 2
- 4: 8 = 0.5

As you can see, the result is different in each case, so we cannot apply the commutative property to the division.

## Commutative property of matrices

In the case of matrices, **this property will be fulfilled** only in the case that what we do is a **sum** . If we are going to perform **matrix multiplication** , we must know that the **commutative property cannot be applied ** except in very specific situations. Let’s see this with an example.

#### Example

**We are going to multiply matrix A and matrix B by modifying their order and we will analyze the result obtained.**

If we look at the result of both multiplications we will realize that the results are different so that A · B ≠ B · A so the commutative property is not fulfilled.

## What is it for

This mathematical property **makes our lives easier when performing operations** since it allows us to order the numbers in the way that best suits us when carrying out additions or multiplications and all this without modifying the final result. One of the main uses it receives is to **help us solve equations** with unknowns, as this property makes it easier for **us to solve** the most complicated parts. It is also especially useful when we try to **add matrices, polynomials or vectors** , and it can even be used in fields other than mathematics such as **propositional logic** in operations related to **set theory** , or that of the **physics** where it can be applied to the** uncertainty principle** .

## History

The history of the **commutative property** is based on **algebra** , which is the branch of mathematics that studies how to combine the elements of structures according to certain rules, whether they are **numbers** or **quantities** . The roots of the word algebra can be traced back to **ancient Babylonian mathematics** where it was possible to develop an arithmetic system with which they were able to perform calculations algorithmically. Commutativity was already known in **ancient times**in elementary operations, add and subtract. Through the consequent extensions that the concept of number underwent, the scope of the operations of adding and multiplying was broadened.

## To which number systems does the commutative property apply?

The commutative property of addition and multiplication is fulfilled in the different number systems so it can be applied to **natural, integer, real, rational, algebraic and complex** numbers .

## Related properties

Symmetry and associativity are two properties that are closely related to the commutative property.

The first one, **symmetry** , tells us that **a mathematical operation is symmetric when after making an alteration in it, the result obtained is the same** . This is precisely what happens with the addition and multiplication operators in which the commutative property is applied. In these cases we alternate the order of the factors or the addends and despite this, the result we achieve is the same.

As for the **associative property** , it tells us that when doing an **addition or a multiplication of three numbers or more, the order in which we carry out the operations does not matter** because we will always arrive at the same result. That is, if we add 2 + 1 + 3, we can first do 2 + 1 and then add 3, or first add 1 + 3 and then add 2 and the result will be the same.

(2 + 1) + 3 = 2 + (1 + 3)

As we have seen, the commutative property is similar to the asocitarive, but in this case it refers to the order of the terms and not the order in which the operation is performed. We must emphasize that, **in general , in the operations in which commutativity is fulfilled** , **associativity is also fulfilled** , although in reality the fact that one is given does not imply that the other is also given.