Commutative Property of Multiplication: Definition, Method and Examples

What is the commutative property of multiplication?

In mathematics, the word "commute" means to move around or travel. The commutative property of multiplication means that numbers can move or swap their positions in a multiplication sentence without altering the final answer.

Changing the order of the numbers being multiplied will never change their total product.

For example, if a notebook costs $4$ dollars and you buy $3$ of them, you can find the total cost by multiplying $3 \times 4$, which equals $12$ dollars. If you instead multiply the cost per notebook by the number of notebooks, $4 \times 3$, the result is still $12$ dollars.

This commutative property multiplication rule applies to whole numbers, fractions, decimals, and algebraic variables.

Formula and array proof

The mathematical formula for the commutative property, also known as the commutative law, is written algebraically using two variables.

Often stated simply as a times b equals b times a, the relationship is written as:

$a \times b = b \times a$

We can prove this using multiplication arrays by changing the orientation of the rows and columns. When an array is rotated, the total number of objects inside it does not change.

As the model shows, creating $3$ groups of $5$ items produces the same total as creating $5$ groups of $3$ items.

Turn-around facts

Learning the turn-around property cuts the number of multiplication facts and times tables you need to memorize in half.

Because of this property, multiplication tables mirror themselves. Once you learn that $6 \times 7 = 42$, you automatically know that $7 \times 6 = 42$.

These matching pairs of equations are commonly called turn around facts because you simply turn the factors around to read the new equation. Recognizing turn around facts builds confidence and speeds up mental mathematics.

How to use the property

The commutative law is one of the foundational properties of multiplication that allows you to reorganize complicated problems.

When an equation contains large numbers or multiple factors, you can commute the numbers to create simpler combinations. This is especially helpful when multiplying three or more numbers together.

For example, calculating $2 \times 37 \times 5$ from left to right requires multiplying $74 \times 5$, which is difficult to do mentally. However, by changing the order of the factors to $2 \times 5 \times 37$, you can first multiply $2 \times 5$ to get $10$. Multiplying $10 \times 37$ leaves a simple mental product of $370$.

What is not commutative?

While the order property multiplication applies consistently to addition and multiplication, it does not apply to subtraction or division.

If you reverse the order of the numbers in a subtraction or division problem, you create a completely different mathematical expression with a different result.

For example, dividing $10$ items evenly into $2$ groups gives $5$ items per group ($10 \div 2 = 5$). However, attempting to divide $2$ items into $10$ groups produces the fraction $\dfrac{2}{10}$, which simplifies to the decimal $0.2$. Because $5$ is not equal to $0.2$, division is never commutative.

Worked examples

Example 1: Finding a missing factor

Question: Find the missing value in the equation: $15 \times 7 = \square \times 15$.

Method:

1. Identify the two given factors on the left side of the equation.
2. Apply the commutative property to reverse their order on the right side.

Answer: The missing value is $7$.

Check: Calculate both sides of the equation independently. $15 \times 7 = 105$ and $7 \times 15 = 105$. The products match exactly.

Example 2: Reordering three numbers

Question: Use the commutative property to rewrite the expression $4 \times 9 \times 25$ so that it is easier to calculate mentally, then find the final product.

Method:

1. Identify the factors that are easiest to multiply together first, such as $4$ and $25$, which make a multiple of $100$.
2. Commute the factors to place $4$ and $25$ next to each other.
3. Multiply $4 \times 25$, and then multiply the result by $9$.

Answer: The rewritten expression is $4 \times 25 \times 9$. The product is $100 \times 9 = 900$.

Check: Multiplying left to right in the original order gives $36 \times 25$. Calculating $36 \times 25$ using a standard algorithm also equals $900$.

Example 3: Checking if division is commutative

Question: A student claims that $20 \div 4$ will have the same result as $4 \div 20$ because of the commutative property. Is the student correct?

Method:

1. Calculate the first expression, $20 \div 4$.
2. Calculate the second expression, $4 \div 20$, by writing it as a fraction.
3. Compare the two results to determine if they are equal.

Answer: The student is incorrect because $20 \div 4 = 5$, but $4 \div 20 = \dfrac{4}{20}$, which simplifies to $0.2$.

Check: Reversing a division problem changes the quotient. The commutative property only applies to addition and multiplication.

Frequently asked questions

Does the commutative property apply to addition?

Yes, addition is completely commutative. Just as with multiplication, changing the order of the addends does not change the total sum. For example, $8 + 3 = 11$ and $3 + 8 = 11$.

What is the difference between the commutative property and the associative property?

The commutative property is about moving or changing the order of the numbers. The associative property of multiplication is about how numbers are grouped using parentheses. The associative property states that changing the groupings of factors does not change the product, such as $(2 \times 3) \times 4 = 2 \times (3 \times 4)$.

Can we commute more than two numbers?

Yes, the commutative property can be applied to equations with three or more numbers. You can rearrange any number of factors in a multiplication sequence, and the final product will remain unchanged.