Properties of Multiplication: Definitions, Rules, and Examples

What are the properties of multiplication?

Multiplication properties are standardized laws that govern how numbers interact during multiplication. They prove that certain manipulations, such as changing the sequence of factors or splitting numbers apart, will reliably produce the same result.

There are five primary rules of multiplication:

• Commutative property: The order of factors does not change the product.
• Associative property: The grouping of factors does not change the product.
• Distributive property: Multiplication distributes over addition and subtraction.
• Identity property: Multiplying by $1$ leaves a number unchanged.
• Zero property: Multiplying by $0$ results in $0$.

Commutative property

The commutative property of multiplication states that the order in which two numbers are multiplied does not change their product.

Rule: $a \times b = b \times a$

For example, calculating $3 \times 4$ yields $12$, and reversing the order to $4 \times 3$ also yields $12$. A visual array perfectly demonstrates this rule. Rotating a grid of dots changes its orientation but does not change the total number of dots.

Associative property

The associative property of multiplication states that when multiplying three or more numbers, the way the numbers are grouped does not change the final product. Parentheses are used to indicate which numbers are grouped to be multiplied first.

Rule: $a \times (b \times c) = (a \times b) \times c$

For example, when calculating the product of $2$, $3$, and $4$, you can group the last two numbers: $2 \times (3 \times 4)$. Solving the grouped part first gives $3 \times 4 = 12$, and multiplying by the remaining number gives $2 \times 12 = 24$.

Using the alternative grouping $(2 \times 3) \times 4$, the first step becomes $2 \times 3 = 6$. Multiplying that result by $4$ gives $6 \times 4 = 24$. Both groupings produce the same final value. This property allows learners to group numbers strategically to make mental calculations simpler.

Together, the commutative and associative properties allow multiplication to be performed in any sequence.

Distributive property

The distributive property of multiplication states that multiplying a number by a sum or difference gives the same result as multiplying that number by each part individually and then adding or subtracting the products.

Over Addition: $a \times (b + c) = (a \times b) + (a \times c)$

Over Subtraction: $a \times (b - c) = (a \times b) - (a \times c)$

For example, $4 \times (5 + 2)$ can be solved by adding the numbers inside the parentheses first to get $4 \times 7 = 28$. Alternatively, distributing the multiplication gives $(4 \times 5) + (4 \times 2) = 20 + 8 = 28$. This property is highly effective for breaking down larger numbers into manageable pieces.

Identity and zero properties

The identity property of multiplication states that any number multiplied by $1$ remains unchanged. Because it preserves the identity of the original number, the number $1$ is known as the multiplicative identity.

Rule: $a \times 1 = a$

For instance, $15 \times 1 = 15$. This rule holds because one group of any amount is simply that original amount.

The zero property of multiplication states that any number multiplied by $0$ always results in a product of $0$.

Rule: $a \times 0 = 0$

For example, $42 \times 0 = 0$. Representing zero groups of any number, or any number of empty groups, will always yield a total of nothing.

Which operations do not share these properties?

While multiplication properties provide immense flexibility, division and subtraction do not share these exact rules. Assuming they behave the same way is a common mathematical mistake.

• Not commutative: Subtraction and division produce entirely different results when the order changes. For subtraction, $10 - 5 = 5$, but $5 - 10 = -5$. For division, $10 \div 2 = 5$, but $2 \div 10 = 0.2$.
• Not associative: Grouping fundamentally changes the result in division. Calculating $(12 \div 4) \div 2$ becomes $3 \div 2 = 1.5$. Changing the grouping to $12 \div (4 \div 2)$ makes it $12 \div 2 = 6$.

Additionally, the distributive property uniquely applies to multiplication distributing over addition or subtraction. Addition does not distribute over multiplication.

Worked examples

Example 1: Identifying the missing property

Question: Which mathematical property is demonstrated by $8 \times (5 \times 2) = (8 \times 5) \times 2$?

Method:

1. Observe the numbers on both sides of the equation.
2. Note that the sequence of the numbers remains identical: $8$, then $5$, then $2$.
3. Identify the structural change. The parentheses have moved, changing how the numbers are grouped for the first step.

Answer: This demonstrates the associative property.

Check: Evaluate both sides. The left side is $8 \times 10 = 80$. The right side is $40 \times 2 = 80$. Both equal $80$.

Example 2: Distributing to calculate products

Question: Use the distributive property to calculate $6 \times 14$.

Method:

1. Break $14$ into a simpler sum, such as $10 + 4$.
2. Rewrite the expression as $6 \times (10 + 4)$.
3. Distribute the $6$ to both parts of the sum: $(6 \times 10) + (6 \times 4)$.
4. Calculate the partial products: $60$ and $24$.
5. Add the results to find the total product.

Answer: $6 \times 14 = 84$.

Check: Multiply directly. $6 \times 10$ is $60$, and $6 \times 4$ is $24$. Adding them confirms $84$.

Example 3: Reasoning with zero

Question: Find the missing value in $7 \times (9 \times x) = 0$ if $x$ is an integer, and identify the property that forces this result.

Method:

1. Notice that the final product of the equation is $0$.
2. Recall the zero property, which states that a product is $0$ if and only if at least one factor is $0$.
3. Since neither $7$ nor $9$ is zero, the missing value $x$ must be $0$.

Answer: The missing value is $0$, governed by the zero property.

Check: Substitute $0$ back into the expression. The inner parentheses become $9 \times 0 = 0$. The outer expression becomes $7 \times 0 = 0$.

Frequently asked questions

What is the multiplicative inverse property?

The multiplicative inverse property states that multiplying a real number by its reciprocal results in exactly $1$. For example, multiplying $5$ by $\dfrac{1}{5}$ gives $1$. The number $0$ has no reciprocal, so this property does not apply to zero.

Does the closure property apply to multiplication?

Yes. The closure property states that multiplying two numbers from a specific set (such as integers or rational numbers) always results in a product that belongs to the same set. For example, the product of any two integers will always be an integer.

What are the parts of a multiplication equation?

The numbers being multiplied are called factors. Specifically, the first number is often called the multiplicand, and the second is the multiplier. The final result is called the product.