Number Properties

The properties of operations are rules that are always true for all numbers. They explain why you can rearrange addition problems, why long multiplication works, and why mental math shortcuts are valid. Understanding these properties now builds the foundation for algebra, where they become essential.

Commutative Property

Changing the order does not change the result.

$a + b = b + a$

$a \times b = b \times a$

Examples

$7 + 3 = 3 + 7 = 10$

$4 \times 6 = 6 \times 4 = 24$

Important: Subtraction and division are NOT commutative.

$8 - 3 = 5 \qquad \text{but} \qquad 3 - 8 = -5$

Associative Property

Changing the grouping does not change the result.

$(a + b) + c = a + (b + c)$

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

Examples

$(2 + 3) + 4 = 5 + 4 = 9$ $2 + (3 + 4) = 2 + 7 = 9$

$(5 \times 2) \times 3 = 10 \times 3 = 30$ $5 \times (2 \times 3) = 5 \times 6 = 30$

This property is why you can regroup numbers strategically for mental math (combined with the commutative property to reorder first):

$(25 \times 7) \times 4 = 25 \times (7 \times 4) = 25 \times 28 = 700$

Important: Subtraction and division are NOT associative.

$(10 - 3) - 2 = 5 \qquad \text{but} \qquad 10 - (3 - 2) = 9$

Distributive Property

Multiplication distributes over addition (and subtraction).

$a \times (b + c) = a \times b + a \times c$

$a \times (b - c) = a \times b - a \times c$

Example 1: Use distribution to simplify $6 \times 14$

$6 \times 14 = 6 \times (10 + 4) = 6 \times 10 + 6 \times 4 = 60 + 24 = 84$

Example 2: Use distribution to simplify $8 \times 97$

$8 \times 97 = 8 \times (100 - 3) = 800 - 24 = 776$

The distributive property is the reason long multiplication works — you multiply by each digit (each place value) separately, then add the results.

Identity Properties

An identity leaves a number unchanged.

Additive Identity (0):

$a + 0 = a$

Adding zero to any number gives back that number.

Multiplicative Identity (1):

$a \times 1 = a$

Multiplying any number by 1 gives back that number.

Zero Property of Multiplication

$a \times 0 = 0$

Any number multiplied by zero equals zero. This is different from the identity property — multiplying by 0 does not preserve the number; it annihilates it.

Inverse Properties

An inverse undoes an operation.

Additive Inverse: For any number $a$, there exists $-a$ such that:

$a + (-a) = 0$

Multiplicative Inverse: For any nonzero number $a$, there exists $\frac{1}{a}$ such that:

$a \times \frac{1}{a} = 1$

Summary Table

Applying Properties to Mental Math

These properties are not just abstract rules — they are mental math tools:

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Commutative: order does not matter for addition and multiplication
• Associative: grouping does not matter for addition and multiplication
• Distributive: multiplication distributes over addition and subtraction
• Neither subtraction nor division is commutative or associative
• Identity elements: 0 for addition, 1 for multiplication
• These properties are the foundation for mental math tricks and algebraic manipulation

Return to Arithmetic for more foundational math topics.