Commutative Law — Definition, Formula & Examples

The Commutative Law says you can add or multiply numbers in any order and the answer stays the same. For example, 3 + 5 gives the same result as 5 + 3.

For all real numbers $a$ and $b$ , the commutative law of addition states $a + b = b + a$ , and the commutative law of multiplication states $a \times b = b \times a$ . This property does not hold for subtraction or division.

Key Formula

a + b = b + a \qquad a \times b = b \times a

Where:

$a$ = Any number
$b$ = Any number

How It Works

When you add or multiply two numbers, try swapping their positions — the result will not change. With addition,

7 + 2 = 2 + 7 = 9

. With multiplication,

4 \times 6 = 6 \times 4 = 24

. However, if you swap numbers in subtraction or division, you get a different answer:

10 - 3 = 7

but

3 - 10 = -7

. So the Commutative Law applies only to addition and multiplication.

Worked Example

Problem: Show that 8 × 5 and 5 × 8 give the same product.

Step 1: Multiply in the original order.

8 \times 5 = 40

Step 2: Swap the two numbers and multiply again.

5 \times 8 = 40

Step 3: Compare the results. Both equal 40, confirming the Commutative Law of Multiplication.

8 \times 5 = 5 \times 8 = 40

Answer: Both expressions equal 40, so the order of multiplication does not matter.

Why It Matters

The Commutative Law lets you rearrange numbers to make mental math easier — for instance, computing

2 \times 37 \times 5

is simpler if you first swap to get

2 \times 5 \times 37 = 370

. It is also the foundation for rewriting and simplifying algebraic expressions throughout algebra courses.

Common Mistakes

Mistake: Assuming the Commutative Law works for subtraction or division (e.g., thinking 10 − 4 = 4 − 10).

Correction: The Commutative Law only applies to addition and multiplication. Subtraction and division change their result when you swap the numbers.