Associative Property of Real Numbers

The associative property refers to how numbers are grouped in an operation. If changing the grouping of the numbers does not change the result, the operation is said to be associative.

Definition

For an operation $\ast$, the set is associative with respect to that operation if:

This means that the way you group the numbers doesn't affect the outcome.

Examples

• Associative: Addition and multiplication of real numbers
• Non-Associative: Subtraction and division of real numbers

Let's try with actual numbers

• $(2 + 3) + 4 = 2 + (3 + 4) = 9$
• $(2 \times 3) \times 4 = 2 \times (3 \times 4) = 24$
• $(5 - 2) - 1 = 2 \neq 5 - (2 - 1) = 4$

More Common Than Commutativeness

While Associativity feels like a next step from Commutativity, it is more common and basic. There are more sets that are Associative but not Commutative than there are sets that are Commutative but not Associative. The prime example is Matrix Multiplication which is Associative, but generally not commutative.

Try building the mathematical expressions