Swapping (Commutative Property)
When you add or (multiply) two or more numbers, you can change their order, and the sum or (product) will stay the same.
Example:
1. \(8 + (-3) = 5\)
\((-3) + 8 = 5\)
2. \(8 \times (-3) = -24\)
\((-3) \times 8 = -24\)
So, changing the order of terms does not change the value.
Grouping (Associative Property)
When you add or (multiply) three or more numbers, you can group them in any way, and the sum or (product) will still be the same.
Example:
1. \((-7) + 10 + (-4)\)
\(=\) \([(-7) + 10] + (-4) = 3 + (-4) = -1\)
\(=\) \((-7) + [10 + (-4)] = (-7) + 6 = -1\)
2. \((2 \times 3) \times 4 = 6 \times 4 = 24\)
\(2 \times (3 \times 4) = 2 \times 12 = 24\)
So, changing the grouping of numbers does not change the value.
In addition and multiplication you can swap or group numbers any way you like - the final sum and product remains the same.
Important!
Subtraction is neither commutative nor associative because changing the order or grouping changes the difference.
Division is neither commutative nor associative because changing the order or grouping changes the quotient.