Properties of Addition: Definition, Rules, and Examples

What are the properties of addition?

The main addition rules help us calculate sums more quickly and mentally rearrange complex equations.

These core rules include the commutative property, the associative property, the identity property, and the closure property. Each property defines a specific mathematical truth about how addition works, which remains consistent across whole numbers, fractions, and decimals.

Commutative property

The commutative property states that changing the order of the addends does not change the sum. You can add two numbers in any order and get the exact same result.

Symbolically, this is written as $a + b = b + a$.

Numerically, we see this property in action because $4 + 7 = 11$, and $7 + 4 = 11$.

Subtraction does not follow this rule. If you change the order of numbers during subtraction, the answer changes completely. For instance, $7 - 4 = 3$, but $4 - 7 = -3$.

Associative property

The associative property states that changing the grouping of the addends does not change the sum. When adding three or more numbers, you can group them differently using parentheses, and the final total remains the same.

Symbolically, this is written as $(a + b) + c = a + (b + c)$.

Numerically, grouping different numbers first gives the same total:

$(3 + 5) + 2 = 8 + 2 = 10$

$3 + (5 + 2) = 3 + 7 = 10$

Subtraction is not associative. If you group subtracted numbers differently, you arrive at different answers. For example, $(10 - 5) - 2 = 5 - 2 = 3$, but $10 - (5 - 2) = 10 - 3 = 7$.

Identity property

The identity property states that adding zero to any number does not change the value of that number. Because it preserves a number's identity, zero is known as the additive identity.

Symbolically, this is written as $a + 0 = a$ and $0 + a = a$.

Numerically, adding zero ensures the value remains completely unchanged: $15 + 0 = 15$.

Closure property

The closure property states that when you add two numbers from a specific group, the resulting sum will always belong to that exact same group.

For example, when you add two whole numbers, the result is always a whole number. You will never add two whole numbers and somehow produce a fraction or a decimal.

Numerically, $8 + 4 = 12$. Since $8$ and $4$ are whole numbers, and their sum $12$ is also a whole number, addition is considered closed for whole numbers.

How properties simplify calculations

We can use the commutative and associative properties together to simplify a multi-addend calculation. By rearranging and grouping numbers that are easy to add, we can often solve long problems mentally.

Suppose we want to add $24 + 17 + 76$.

First, we use the commutative property to rearrange the numbers so that compatible values sit directly next to each other. This creates the expression $24 + 76 + 17$.

Next, we use the associative property to group $24$ and $76$, because they combine to make a perfect multiple of $100$. This creates $(24 + 76) + 17$.

Finally, we add the grouped numbers and the remaining amount to reach $100 + 17 = 117$.

Worked examples

Example 1: Identifying the property used

Question: Which property is shown in the equation $45 + 0 = 45$?

Method:

1. Observe the numbers used in the equation.
2. Notice that zero is being added to a number.
3. The sum is exactly the original number, meaning its identity has not changed.

Answer: This shows the identity property of addition.

Check: The identity property states that $a + 0 = a$, which matches the structure of $45 + 0 = 45$.

Example 2: Finding a missing number

Question: Find the missing number using the associative property: $(12 + 19) + 8 = 12 + (\text{\_\_\_} + 8)$

Method:

1. Identify the addends on the left side of the equation: $12$, $19$, and $8$.
2. The associative property states that the exact same numbers can be grouped differently without changing the total sum.
3. The right side already has $12$ and $8$. The missing number must be the remaining addend.

Answer: The missing number is $19$.

Check: Calculate both sides manually. $(12 + 19) + 8 = 31 + 8 = 39$. On the right side, $12 + (19 + 8) = 12 + 27 = 39$. Both sides are equal.

Example 3: Simplifying a multi-addend sum

Question: Use the properties of addition to simplify $41 + 38 + 9$.

Method:

1. Look for addends that easily combine to make a multiple of $10$. Here, $41$ and $9$ combine to perfectly make $50$.
2. Use the commutative property to rearrange the numbers to place those addends together: $41 + 9 + 38$.
3. Use the associative property to group the compatible numbers: $(41 + 9) + 38$.
4. Add the grouped numbers first: $50 + 38$.

Answer: The simplified sum is $88$.

Check: Adding from left to right gives $41 + 38 = 79$, and $79 + 9 = 88$. The result is the exact same.

Frequently asked questions

What is the additive inverse?

The additive inverse is the mathematical opposite of a number. When you add a number and its additive inverse, the result is zero. For example, the additive inverse of $5$ is $-5$, because $5 + (-5) = 0$.

What is the difference between additive identity and additive inverse?

The additive identity is always zero, because adding zero to a number leaves the original number unchanged. The additive inverse is the opposite of a specific number, which is added to that number to intentionally yield zero.

What is the zero property of addition?

The zero property is simply another name for the identity property. It states that any number added to zero equals the original number itself.