Definition
Associative property
The associative property says regrouping addends or factors does not change their sum or product.
Continue from this idea into connected AXIO math.
Defining relationship
Same order, new grouping.
For addition, (a + b) + c = a + (b + c). For multiplication, (a × b) × c = a × (b × c). The order stays fixed; only the grouping changes.
- 1
group the first pair
- 2
group the second pair
- Order stays 2, 3, 4
- Only the parentheses move
- The total stays 9
What it means
Parentheses tell you which pair to combine first. With addition or multiplication, you can move the parentheses and keep the same order of numbers. The answer stays the same, so you can choose the easier grouping.
Remember it
Same order, new grouping.
Example
For 17 + 3 + 8, (17 + 3) + 8 = 20 + 8 = 28, and 17 + (3 + 8) = 17 + 11 = 28.
Watch for
Subtraction and division are not associative in general: (12 − 5) − 2 = 5, but 12 − (5 − 2) = 9.
Common mix-up
Moving numbers is commutative; moving only the grouping is associative.
Exact meaning
For addition, (a + b) + c = a + (b + c). For multiplication, (a × b) × c = a × (b × c). The order stays fixed; only the grouping changes.