# What is the Commutative Property?

## What Is the Commutative Property?

In mathematics, a **binary operation ** is a rule that takes two numbers and produces another. **Addition **and **multiplication** are both binary operators, since we can add or multiply two numbers together, and the answer is another number.

An operation is said to be **commutative** if the order of the two inputs values does not change the output value. Symbolically, a binary operation represented by {eq}\star {/eq} is commutative if:

$$a\star b = b\star a $$

Commutativity is one of several properties that mathematical operations like addition, multiplication, subtraction, and division may or may not have. Another property that, at first glance, can be easily confused for commutativity is the **associative **property. An operation is associative if a series of several operations can be performed in any order, expressed symbolically this means:

$$a\star (b\star c) = (a \star b) \star c $$

In other words, associativity means the placement of brackets does not change the result. The brackets do change the order in which the operations are performed, but the left-to-right order of the inputs does *not *change, so associativity is different from commutativity.

The **distributive **property involves both addition and multiplication. Multiplying a number with a sum can be decomposed into the sum of two products, as shown below. Multiplication is said to distribute over addition.

$$a\cdot (b + c) = a\cdot b + a\cdot c $$

### Commutative Property of Addition

The addition of numbers is commutative, meaning that for any numbers {eq}a {/eq} and {eq}b {/eq} we have:

$$a+b = b+a $$

The commutative property of addition is easiest to understand with whole-valued positive numbers, but it applies to *all* numbers, including negative numbers. This means, for example, that:

$$5 + (-2) = (-2) + 5 = 3 $$

The addition of a negative value can equally be expressed using the separate operation of **subtraction**. However, by considering the same example we can see that subtraction is *not* commutative. Changing the order of the subtraction produces a different, opposite value:

$$5 -2 =3 \qquad\quad 2-5 = -3 $$

### Commutative Property of Multiplication

Multiplication of numbers is also commutative: for any numbers {eq}a {/eq} and {eq}b {/eq} we have:

$$a\cdot b = b\cdot a $$

As for addition, the commutative property can be visualized with examples involving whole-valued, positive numbers. For example, the diagram below shows how:

$$3\cdot 2 = 2\cdot 3 =6 $$

Again, the property does apply to all numbers, including negative numbers. Multiplying a positive with a negative creates a negative value, while multiplying two negatives creates a positive, but in either case the order of the numbers does not change the answer.

The operation of **division** is equivalent to multiplication by the reciprocal of the second number, so for example:

$$10\div 5 =10 \cdot \frac{1}{5} = 2 $$

Multiplication is of course commutative, including when fractional values are involved, but the operation of division is not. For the previous example, we can see that changing the order in the division yields a different answer:

$$5 \div 10 = 5 \cdot \frac{1}{10} = 0.5 $$

## The Long Commute

If you commute to school or work, you know that sometimes the traffic can drive you crazy. Maybe, like me, you have four or so different routes you can take to work that all are about the same distance and take roughly the same amount of time (depending on if you catch the lights right or not).

Obviously, you have decided on each of these routes to your destination because each of them will get you where you want to go. There probably is no chance that you will drive in a pattern that will not get you to school (even if you would like to take the day off). That is the whole point of the commute - to get you there, regardless of the pattern you take to get you there.

## Commutative Property Examples

Here are some more complicated examples which can be simplified using the commutative property.

### Example 1

Solve {eq}24+5 -24 + 2 +10 = ? {/eq}

Because addition is commutative, these numbers can be added in *any* order, and a different order may involve simpler steps. In this case:

$$\begin{eqnarray} 24+5 -24 + 2 +10 &=& 24 + (-24) + 5 + 10 + 2 \\ &=& 0 + 15 + 2 \\ &=& 17 \end{eqnarray} $$

The subtraction exactly cancels the first term, and multiples of 5 are relatively easy to add, so can be done before finally adding the 2.

### Example 2

Solve {eq}5\div 2 \cdot 8 \div 5 \cdot 3 = ? {/eq}

Multiplication is commutative, so these numbers can likewise be multiplied in any order. The divisions can also be performed in a different order than originally given, so long as the indicated divisor remains the divisor.

$$\begin{eqnarray} 5\div 2 \cdot 8 \div 5 \cdot 3 &=& 5 \div 5 \cdot 8 \div 2 \cdot 3 \\ &=& 1 \cdot 4 \cdot 3 \\ &=& 12 \end{eqnarray} $$

5 divided by itself is 1, and the even number 8 can be easily divided by 2, so these operations are best done first.

## Applications of the Commutative Property

The commutative property is a fundamental fact about both addition and multiplication, basic operations which can describe many real-world situations. The commutative property can therefore be used to simplify complicated equations that can appear in all sorts of technical fields, from engineering to finance.

For example, the commutative property shows that a property or investment that increases in value by 10% one year, and 5% the next, reaches the same value as if it grew first by 5% and then by 10%. This is because percentage growth is calculated through multiplication:

$$(1+0.10)\cdot(1+0.05) = (1+0.05) \cdot (1+0.10) = 1.155 \quad \implies \quad +15.5\% $$

The commutative property may or may not apply to other operations which are more complex than simply addition and multiplication. Movements in a horizontal plane can be thought of as a commutative operation: moving 1 km north and then 1 km west brings you to the same location as first moving west and then north. Rotations in the plane are also commutative, but rotations in three-dimensions are not. Knowing when operations are *not * commutative becomes an important factor in higher mathematics.

## Lesson Summary

The **commutative** property is one possible attribute of a ** binary operation**, which takes two numbers (or other inputs) and calculates another. **Addition **and **multiplication** are important examples of basic operations which are commutative. This means that the order of either of these operations can be reversed, without changing the final result.

Addition and multiplication are also **associative**, meaning a sequence of the operations can also be performed in any order. Together they also satisfy the **distributive ** property: multiplication distributes over addition. The operations of **subtraction **and **division** are not commutative. Subtraction can be expressed as addition of a negative value, and division as multiplication by a reciprocal value, and this allows for correct rearrangement of expressions using the commutativity of addition and multiplication.

## A Mathematical Commute

Right now you might be sitting there thinking, what on earth does this have to do with math? Keep listening, and you will find out.

In mathematics, there are three basic principles for how equations work. They form the backbone of all higher math. These properties are:

• The commutative property

• The associative property

• The distributive property

## The Commutative Property

This lesson focuses on the **commutative property**. It states that you can swap terms in an equation and still get the same answer. Just like your commute where you can take different routes to get to the same place, in addition and multiplication, you can swap the order of your terms and still get the same answer.

Let's look at a simple example: 2 + 5 = 7.

When you swap the terms, in this case the 2 and 5, you will still get the same answer: 5 + 2 = 7.

It works for multiplication as well. 4 * 6 = 24 is the same as 6 * 4 = 24.

As my high school algebra teacher said: the commutative property means that 'order doesn't matter' (for addition and multiplication).

It also doesn't matter how long your problem is. 5 + 3 + 9 + 12 is the same as 12 + 3 + 5 + 9. The answer is 29 both times.

Or 2 * 7 * 5 * 1 is the same as 7 * 2 * 1 * 5. Again, the answer will be 70, no matter what order the numbers are in.

The commutative property can also work with subtraction, if you are very careful. If you remember, subtraction is the opposite of addition. Because of this, you can turn any subtraction problem into an addition problem. This means that 6 - 3 is the same as 6 + (-3).

When you turn your subtraction problem into an addition problem, you can apply the commutative property. 7 - 4 can be changed to 7 + (-4) and then swapped to equal (-4) + 7. No matter which way you arrange it, the answer is 3.

## So What?

With simple problems such as these, you might be scratching your head, wondering why mathematicians go to all this trouble. The reason they do so is for the complicated problems. When mathematicians or scientists or engineers are working with a complex equation, it can help them with the solution if they are sure that they can arrange the terms without affecting the problem.

## Lesson Summary

The commutative property states that in addition and multiplication problems, the order of the terms does not matter to the final outcome of the problem. You can arrange the terms in any order and still obtain the correct answer. The property is also true for subtraction if you convert your subtraction problem to an addition problem and are very careful to keep the negative with the correct number.

## Learning Outcomes

After you have finished with this lesson, you should be able to:

- Utilize the commutative property when solving addition and multiplication problems
- Apply the commutative property to subtraction problems

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## The Long Commute

If you commute to school or work, you know that sometimes the traffic can drive you crazy. Maybe, like me, you have four or so different routes you can take to work that all are about the same distance and take roughly the same amount of time (depending on if you catch the lights right or not).

Obviously, you have decided on each of these routes to your destination because each of them will get you where you want to go. There probably is no chance that you will drive in a pattern that will not get you to school (even if you would like to take the day off). That is the whole point of the commute - to get you there, regardless of the pattern you take to get you there.

## A Mathematical Commute

Right now you might be sitting there thinking, what on earth does this have to do with math? Keep listening, and you will find out.

In mathematics, there are three basic principles for how equations work. They form the backbone of all higher math. These properties are:

• The commutative property

• The associative property

• The distributive property

## The Commutative Property

This lesson focuses on the **commutative property**. It states that you can swap terms in an equation and still get the same answer. Just like your commute where you can take different routes to get to the same place, in addition and multiplication, you can swap the order of your terms and still get the same answer.

Let's look at a simple example: 2 + 5 = 7.

When you swap the terms, in this case the 2 and 5, you will still get the same answer: 5 + 2 = 7.

It works for multiplication as well. 4 * 6 = 24 is the same as 6 * 4 = 24.

As my high school algebra teacher said: the commutative property means that 'order doesn't matter' (for addition and multiplication).

It also doesn't matter how long your problem is. 5 + 3 + 9 + 12 is the same as 12 + 3 + 5 + 9. The answer is 29 both times.

Or 2 * 7 * 5 * 1 is the same as 7 * 2 * 1 * 5. Again, the answer will be 70, no matter what order the numbers are in.

The commutative property can also work with subtraction, if you are very careful. If you remember, subtraction is the opposite of addition. Because of this, you can turn any subtraction problem into an addition problem. This means that 6 - 3 is the same as 6 + (-3).

When you turn your subtraction problem into an addition problem, you can apply the commutative property. 7 - 4 can be changed to 7 + (-4) and then swapped to equal (-4) + 7. No matter which way you arrange it, the answer is 3.

## So What?

With simple problems such as these, you might be scratching your head, wondering why mathematicians go to all this trouble. The reason they do so is for the complicated problems. When mathematicians or scientists or engineers are working with a complex equation, it can help them with the solution if they are sure that they can arrange the terms without affecting the problem.

## Lesson Summary

The commutative property states that in addition and multiplication problems, the order of the terms does not matter to the final outcome of the problem. You can arrange the terms in any order and still obtain the correct answer. The property is also true for subtraction if you convert your subtraction problem to an addition problem and are very careful to keep the negative with the correct number.

## Learning Outcomes

After you have finished with this lesson, you should be able to:

- Utilize the commutative property when solving addition and multiplication problems
- Apply the commutative property to subtraction problems

To unlock this lesson you must be a Study.com Member.

Create your account

#### What is the commutative property in math?

An operation, like addition, has the commutative property if the order of the numbers in the operation does not change the result. Addition and multiplication are both commutative, for example.

#### How do you use the commutative property?

If a mathematical operation like addition is commutative, the order of the numbers in the operation does not matter. This means that the order of numbers can be changed freely. A different order of numbers or algebraic terms may be easier to calculate or simplify than whatever order was originally given.

#### What is an example of the commutative property?

Addition and multiplication of numbers are examples of commutative operations. For example, 1+2=2+1=3, and 3x5=5x3=15. The order of the numbers added or multiplied does not change the answer.

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