Commutative Property of Real Numbers

The commutative property refers to the ability to change the order of the numbers in an operation without changing the result.

Definition

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

This is read as, "For all a and b in the Real Number set, a operating on b is equal to b operating on a". This means the order of the operands does not matter.

Examples

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

Let's try with actual numbers

• $2 + 5 = 5 + 2 = 7$
• $3 \times 4 = 4 \times 3 = 12$
• $5 - 2 = 3 \neq 2 - 5 = -3$

Not Always True

Not all operations are commutative. For example, matrix multiplication and function composition are not commutative, even though they might be associative. We can describe that property as that there exists an $a$ and $b$ in the given set such that $a \ast b \ne b \ast a$.

Try building the mathematical expressions