Commutative Property of Addition: Definition, Formula, and Examples

What is the commutative property of addition?

The commutative property of addition is a mathematical rule showing that numbers can be swapped in an addition problem without affecting the final answer. The numbers we add are the addends, and the total is the sum. You can learn more about addends and sum to understand addition vocabulary.

When you rearrange addends, their combined value is identical. For instance, if you add $4$ and $5$, the sum is $9$. If you reverse the order and add $5$ and $4$, the sum is still $9$.

The order of addends does not change the sum.

This property only applies when adding numbers. The word "commutative" comes from the word "commute," which means to move around or travel. The addends can "travel" to different positions in the equation while keeping the total perfectly balanced.

Formula and visual model

The commutative property is expressed with a simple algebraic formula showing that two variables equal the same sum regardless of which one is written first.

Formula: $a+b=b+a$

In this formula, $a$ and $b$ represent any real numbers. No matter what values you substitute for these letters, the left side of the equation will always equal the right side.

The number line visualizes this perfectly. A jump of $5$ followed by a jump of $3$ lands exactly on $8$. Similarly, a jump of $3$ followed by a jump of $5$ also lands on $8$.

Why the property works

The commutative property works because addition is simply the process of combining quantities into a single total. The actual amount of items you are counting never changes.

If you have a group of blue blocks and a group of yellow blocks, pushing the two groups together creates the same pile, no matter which group you move first. The physical quantity is preserved.

Counting the objects from left to right gives the same result as counting them from right to left. The arrangement is different, but the final count is identical.

How to use it

The commutative property is one of the foundational properties of addition that makes calculating faster and easier. You can use it to rearrange equations so they are simpler to solve mentally.

When faced with a string of numbers, you can look for friendly pairs that add up to $10$ or $100$. For example, solving $14+29+6$ in order can be tricky. Using the commutative property, you can swap the addends to group $14$ and $6$ together:

$14+6+29=20+29=49$

You can also use this property to check your work. If you add a list of numbers from top to bottom and want to verify your answer, you can add the same list from bottom to top. If the sums match, your calculation is likely correct.

What is not commutative?

While addition allows numbers to be swapped freely, subtraction does not. Subtraction is strictly non-commutative because the order of the numbers changes the result entirely.

If you swap the order of the numbers in a subtraction problem, you do not get the same difference. Subtracting a small number from a large number gives a positive result. Subtracting a large number from a small number gives a negative result.

For example, $5-2=3$. However, $2-5=-3$. Because $3$ is not equal to $-3$, the equation $5-2=2-5$ is false.

Worked examples

Example 1: Finding a missing addend

Question: Find the missing value that makes the equation true: $34+81=81+\text{\_\_\_}$

Method:

1. Look at the numbers on the left side of the equals sign: $34$ and $81$.
2. Look at the right side of the equals sign. We have $81$ and a missing number.
3. Apply the commutative property ($a+b=b+a$).

Answer: The missing number is $34$.

Check: The equation $34+81=81+34$ is perfectly balanced.

Example 2: Rearranging three addends

Question: Use the commutative property to write the equation $7+12+3=22$ in a different sequence that is easier to add.

Method:

1. Identify pairs of numbers that add up to a multiple of ten. Here, $7+3=10$.
2. Rearrange the addends so $7$ and $3$ are next to each other.
3. Move the $12$ to the end of the sequence.

Answer: $7+3+12=22$.

Check: Adding the friendly numbers first gives $10+12=22$, which matches the original sum.

Example 3: Comparing sums in word problems

Question: Liam bought $4$ red pens and $9$ blue pens. Sarah bought $9$ red pens and $4$ blue pens. Did they buy an equal total number of pens?

Method:

1. Write an addition equation for Liam's pens: $4+9$.
2. Write an addition equation for Sarah's pens: $9+4$.
3. Compare the two equations using the commutative property.

Answer: Yes, they bought an equal total number of pens.

Check: Calculate both sums manually. Liam has $13$ pens, and Sarah has $13$ pens.

Frequently asked questions

Can you apply the commutative property to three or more numbers?

Yes. While the basic formula uses only two numbers ($a+b=b+a$), the property holds true for any amount of addends. You can completely scramble a list of five, ten, or a hundred numbers in an addition problem, and the final sum will remain unchanged.

Which other mathematical operations are commutative?

Multiplication is the only other basic arithmetic operation that is commutative. For example, $4\times 5=20$ and $5\times 4=20$. Subtraction and division are not commutative because order matters for both operations.

How is the commutative property different from the associative property?

The commutative property is about moving numbers into a different order. The associative property is about grouping numbers differently using parentheses without changing their order. Both properties are often used together to solve complex math problems efficiently.