Properties of Addition — Definition, Formula & Examples
The properties of addition are rules that are always true when you add numbers. They tell you that you can change the order or grouping of numbers in a sum, and that adding zero changes nothing, without changing the total.
For all real numbers , , and , addition satisfies four properties. Commutative: . Associative: . Identity: , so is the additive identity. Inverse: for each there is a number with . Elementary courses usually teach only the first three, because the inverse property needs negative numbers.
Key Formula
Where:
- = Any real numbers
- = The additive identity, the number that leaves any sum unchanged
- = The additive inverse (opposite) of a
Worked Example
Problem: Use the properties of addition to find using mental math.
Step 1: Use the commutative property to swap 26 and 3. Now the numbers that make a friendly ten sit side by side.
Step 2: Use the associative property to group 17 and 3 first. Their sum is a round number.
Step 3: Add the round number to what remains.
Answer:
Why It Matters
These properties are the reason you can rearrange terms when you simplify expressions such as into in algebra. They also explain mental-math strategies like making tens, which you use when you add long lists of prices or scores.
Common Mistakes
Mistake: Assuming subtraction follows the same rules and writing .
Correction: Subtraction is not commutative or associative. For example, but . If you want to reorder, first rewrite subtraction as adding the opposite: .