Properties of Integers: Rules and Examples

What are the properties of integers?

The core properties of integers form the foundation for all integer operations. They determine whether the order of numbers can be changed, how groups of numbers can be combined, and which operations always produce another integer.

These rules are closely related to the general properties of addition and properties of multiplication, but they apply specifically to the entire set of positive numbers, negative numbers, and zero.

The main properties are:

• Closure property: Determines if an operation always results in an integer.
• Commutative property: Determines if the order of the numbers changes the result.
• Associative property: Determines if the grouping of the numbers changes the result.
• Distributive property: Describes how multiplication interacts with addition or subtraction.
• Identity property: Identifies a value that leaves another integer unchanged.

Closure property

The closure property of integers states that when an operation is performed on any two integers, the result is always another integer.

If $a$ and $b$ are integers, then for the operation to be closed, the answer must also belong to the exact same numerical set. Addition, subtraction, and multiplication are all closed operations.

• Addition: $(-5) + 9 = 4$
• Subtraction: $2 - 7 = -5$
• Multiplication: $(-3) \times (-4) = 12$

Division is not closed because dividing two integers often results in a fraction or decimal, which is not an integer. For example, $(-6) \div 4 = -1.5$.

The closure property applies to addition, subtraction, and multiplication, but not division.

Commutative and associative properties

The commutative property states that the order of the numbers does not change the result. For any integers $a$ and $b$, this rule applies to addition and multiplication:

• Addition: $a + b = b + a$
• Multiplication: $a \times b = b \times a$

For example, $(-6) + 4 = -2$, and $4 + (-6) = -2$. The sum is identical regardless of which integer is written first.

The associative property states that the grouping of three or more numbers does not change the final result. Grouping is typically shown using parentheses. For any integers $a$, $b$, and $c$:

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

For example, $(-2) + (3 + 5)$ evaluates to $-2 + 8 = 6$. Changing the grouping to $((-2) + 3) + 5$ evaluates to $1 + 5 = 6$. The final sum remains exactly the same.

Distributive property

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

For any integers $a$, $b$, and $c$:

• Over addition: $a \times (b + c) = (a \times b) + (a \times c)$
• Over subtraction: $a \times (b - c) = (a \times b) - (a \times c)$

This property breaks complex calculations into simpler mental steps. To calculate $-4 \times 102$, you can rewrite $102$ as a sum inside parentheses, $100 + 2$:

$-4 \times (100 + 2) = (-4 \times 100) + (-4 \times 2) = -400 + (-8) = -408$.

Identity and additive inverse

An identity element is a specific value that leaves another integer perfectly unchanged when an operation is applied to it.

• Additive identity: The number $0$ is the additive identity because adding zero to any integer does not change its value. For any integer $a$, the absolute rule is $a + 0 = a$ and $0 + a = a$.
• Multiplicative identity: The number $1$ is the multiplicative identity because multiplying any integer by one does not change its value. The rule is $a \times 1 = a$ and $1 \times a = a$.

The additive inverse of an integer is its exact opposite value on the number line. When you add any integer to its additive inverse, the result is always the additive identity, which is $0$.

For example, the additive inverse of $4$ is $-4$, because $4 + (-4) = 0$.

Which properties do not apply?

Subtraction and division behave fundamentally differently from addition and multiplication. They are rigidly ordered, meaning they do not share all the same algebraic flexibility.

• No commutative property: Changing the order of subtraction or division changes the mathematical result. For example, $8 - 5 = 3$, but $5 - 8 = -3$. Similarly, $10 \div 2 = 5$, but $2 \div 10 = \dfrac{1}{5}$.
• No associative property: Changing the grouping of subtraction or division alters the final answer. For example, $(12 - 6) - 2 = 4$, but $12 - (6 - 2) = 8$.
• No identity element: Subtracting zero only works correctly in one direction. While $9 - 0 = 9$, the reverse equation $0 - 9 = -9$ changes the original value.

Because these operations lack these properties, you must always evaluate subtraction and division strictly from left to right unless mathematical grouping symbols specify otherwise.

Worked examples

Example 1: Identifying the integer property

Question: Which integer property is demonstrated by the equation $(-8) \times (-5) = (-5) \times (-8)$?

Method:

1. Examine the values on both sides of the equals sign. Both sides contain $-8$ and $-5$.
2. Identify the mathematical operation. Both sides use multiplication.
3. Notice what has changed. The order of the integers is reversed, but the grouping has not changed.
4. Match this observation to the correct property rule.

Answer: The equation demonstrates the commutative property of multiplication.

Check: Evaluate both sides. $40 = 40$. The order of multiplication does not change the result.

Example 2: Using the distributive property

Question: Use the distributive property to mentally calculate $-7 \times 99$.

Method:

1. Rewrite the large number as a simple subtraction involving a multiple of $100$. Write $99$ as $(100 - 1)$.
2. Substitute this expression into the original problem: $-7 \times (100 - 1)$.
3. Multiply the outside integer by each inside value separately.
4. Multiply $-7 \times 100$ to get $-700$.
5. Multiply $-7 \times 1$ to get $-7$.
6. Subtract the second product from the first: $-700 - (-7)$.

Answer: $-700 + 7 = -693$.

Check: Standard multiplication confirms that $-7 \times 99 = -693$.

Example 3: Proving division is not commutative

Question: Provide a counterexample to show that the commutative property does not apply to the division of integers.

Method:

1. Choose two different integers that divide evenly in one direction, such as $12$ and $4$.
2. Write the equation with the larger integer first: $12 \div 4$.
3. Calculate the quotient. $12 \div 4 = 3$.
4. Reverse the order of the integers to test the commutative rule: $4 \div 12$.
5. Calculate the new quotient. $4 \div 12 = \dfrac{1}{3}$.

Answer: Because $3$ is not equal to $\dfrac{1}{3}$, the equation $12 \div 4 = 4 \div 12$ is false. The order matters.

Check: Any pair of distinct non-zero integers will produce different quotients when reversed.

Frequently asked questions

What are the four main properties of integers?

The four most frequently used properties of integers are the closure property, the commutative property, the associative property, and the distributive property.

Why are the properties of integers important?

These properties act as the fundamental rules of arithmetic. They allow you to safely rearrange terms, simplify mental math, and solve complex algebraic expressions without altering the final value of the equation.

Is division closed for integers?

No, division is not a closed operation for integers. Dividing two integers frequently results in a fraction or decimal. For example, $5 \div 2 = 2.5$, which is not an integer.