Commutative Property Formula with Examples

Key Points

• The commutative property formula is $a + b = b + a$ for addition and $a \times b = b \times a$ for multiplication.

• It says order does not affect the result, and it holds for every real number without exception.

• Subtraction and division are not commutative, since $5 - 3$ is 2 while $3 - 5$ is $-2$.

• Matrix multiplication is not commutative either, so $AB$ and $BA$ are generally different products.

• Commutativity is about order while associativity is about grouping, which is why the two properties are stated separately.

What Is the Commutative Property Formula?

The commutative property formula states that swapping the order of two numbers leaves the result unchanged:

$$a + b = b + a, \qquad a \times b = b \times a$$

Strictly it is a property rather than a formula, which is why it is written as an identity holding for all values rather than as something to compute.

Why Order Can Be Changed for Some Operations and Not Others

Addition counts a combined total, and a pile of three objects joined to a pile of five contains eight however the piles are named. Nothing in the counting depends on which came first.

Multiplication has the same symmetry through area. A rectangle 3 units by 5 units has the same area as one 5 by 3, because rotating it does not change the space it covers.

Subtraction has no such symmetry, because it asks how far one number is above another, and that is a directed question. Swapping the inputs reverses the direction and therefore the sign, which is why $5-3$ and $3-5$ differ.

Variable Key

Examples of the Commutative Property Formula

Example 1

Show that addition is commutative for 7 and 12.

$$7 + 12 = 19, \qquad 12 + 7 = 19$$

Final answer: Both give 19, so $7+12 = 12+7$.

Example 2

Apply the commutative property to $9 - 4$.

The property swaps the order of the two values, so applying it here gives $4 - 9$.

$$9 - 4 = 5, \qquad 4 - 9 = -5$$

These are not equal, so the swap was not valid. Subtraction is not a commutative operation, and the property cannot be applied to it.

Final answer: $9 - 4 = 5$ only. Swapping gives $-5$, which shows the property does not extend to subtraction.

Example 3

Use the commutative property to make $4 \times 17 \times 25$ easier.

Reorder so the convenient pair multiply first:

$$4 \times 25 \times 17 = 100 \times 17 = 1700$$

Final answer: 1,700

This is the property's main practical use: reordering a product so an easy pair comes together.

Example 4

Check whether matrix multiplication is commutative, using $A = \begin{pmatrix}1&1\0&1\end{pmatrix}$ and $B = \begin{pmatrix}1&0\1&1\end{pmatrix}$.

$$AB = \begin{pmatrix}2&1\1&1\end{pmatrix}, \qquad BA = \begin{pmatrix}1&1\1&2\end{pmatrix}$$

Final answer: $AB \neq BA$, so matrix multiplication is not commutative.

The multiplication symbol is the same, but the property belongs to the operation rather than to the symbol.

Example 5

Simplify $3x + 7 + 5x$ using the commutative property.

Reorder the terms so like terms sit together, then add:

$$3x + 5x + 7 = 8x + 7$$

Final answer: $8x + 7$

Collecting like terms depends on commutativity. Without it, the terms could not legitimately be moved past one another.

Commutative, Associative and Distributive Compared

Commutative and associative are the pair most often confused. One moves values past each other, the other moves the brackets, and an operation can have one without the other.

Where Do Students Trip Up on the Commutative Property?

Mistake 1: Applying it to subtraction or division

Where it slips in: The property is often summarised as "order does not matter", without naming which operations that covers.

Don't do this: $9 - 4 = 4 - 9$

The correct way: Restrict it to addition and multiplication. For subtraction the two answers differ in sign, and for division they are reciprocals, as $\frac{8}{2} = 4$ against $\frac{2}{8} = 0.25$.

Students who name the operation out loud before invoking any property stop misapplying it, because the property is a fact about one operation rather than a general permission. The habit matters more later, when non-commutative multiplication appears with matrices.

Mistake 2: Confusing it with the associative property

Where it slips in: Both let an expression be rewritten and both feel like permission to rearrange.

Don't do this: Call $(2+3)+4 = 2+(3+4)$ the commutative property.

The correct way: That is associativity, since only the brackets moved. Commutativity would be $2+3 = 3+2$, where the values swapped places.

Mistake 3: Assuming all multiplication commutes

Where it slips in: Numbers commute under multiplication, so the symbol itself seems to carry the property.

Don't do this: Assume $AB = BA$ for matrices.

The correct way: Check what is being multiplied. Matrix multiplication is not commutative, and neither is function composition, so the property belongs to the operation and the objects rather than to the symbol.

Key Takeaways

• The commutative property formula is $a+b = b+a$ and $a \times b = b \times a$.

• It holds for addition and multiplication of numbers, and fails for subtraction and division.

• Its practical value is reordering a sum or product so convenient pairs come together.

• Matrix multiplication and function composition are not commutative, so the property attaches to the operation rather than the symbol.

• To work through algebraic properties with a teacher, explore a 6th grade math tutor or math classes online.

A Practical Next Step

Use the property to compute $2 \times 37 \times 50$ in one line. Then decide which of $12 \div 3$, $8+5$ and $6-1$ allow their order to be swapped, and say why.

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Read More

• Properties of Equality

• Properties of Matrices

• Multiplication of Matrices

• What Is an Integer

• What Is a Natural Number