Commutative and associative with subtraction

A very common mistake is to apply the properties of addition to subtractions as well. Beware: it does not work like that.

Beware — Commutative and associative with subtraction

The commutative and associative properties hold only for addition. To apply them when there are subtractions, you must first rewrite each subtraction as the addition of the opposite: $3 - 2 = 3 + (-1)\cdot 2, \qquad -7 - 4 = (-1)\cdot 7 + (-1)\cdot 4.$

Example. $3 - 4 + 7 - 1 = 3 + (-1)\cdot 4 + 7 + (-1)\cdot 1$.

• ✓ Correct commutative: $(-1)\cdot 4 + 3 + (-1)\cdot 1 + 7$
• ⚠ Wrong commutative: $4 - 3 + 1 - 7$   NO!
• ✓ Correct associative: $3 + [(-1)\cdot 4 + 7] + (-1)\cdot 1$
• ⚠ Wrong associative: $3 - (4+7) - 1$   NO!

The practical rule: every "$-$" sign belongs to the term that follows it and must travel with it when rearranging or grouping.

Links

Topics: Numbers and operations
Concepts: Associative property · Commutative property
Skills: Calculating