ACT® · Math: Integrating Essential Skills
Properties of Real Numbers
The properties of real numbers are the rules that say which moves are legal when you rearrange or expand an expression. They let you reorder sums, regroup products, and split or combine terms through multiplication, but never through subtraction, division, or square roots.
Addition and multiplication are forgiving. The commutative property says you can swap the order of terms you're adding or factors you're multiplying: 3 + 5 gives the same answer as 5 + 3, and 4 times 9 gives the same answer as 9 times 4. The associative property says you can regroup which terms you combine first: (2 + 3) + 7 equals 2 + (3 + 7). Subtraction and division don't get this freedom. 10 - 4 does not equal 4 - 10, and changing the order changes the answer, so you can never silently swap or regroup across a minus sign.
The distributive property is the bridge between multiplication and addition: a(b + c) = ab + ac. It's the reason you can expand 7(38 + 62) into 7 times 38 plus 7 times 62, and the reason you can reverse the move, pulling a common factor back out of a sum. This single rule licenses nearly every step you take when simplifying an algebraic expression, combining like terms, or factoring.
These properties matter because they're the hidden justification behind every shortcut you take with rates, percents, and multi-step word problems elsewhere in this unit. When a problem hands you numbers that look ugly to multiply directly, these properties are your signal to regroup or factor first. Spotting when a problem is built to reward factoring instead of brute-force computation saves real time.
Key terms
- Commutative property
- You can swap the order of terms in addition or factors in multiplication without changing the result.
- Associative property
- You can regroup which terms you add first or which factors you multiply first without changing the result.
- Distributive property
- Multiplying a sum by a number equals multiplying each term by that number and adding: a(b + c) = ab + ac.
- Factoring
- Reversing the distributive property by pulling a common factor out of every term in a sum.
Watch it explained
- How it worksHow do the commutative, associative and distributive properties license every step of an algebra move?
- Worked exampleFind the value of 7 times 38 plus 7 times 62 without multiplying either product.
- Common mistakeWhy isn't the distributive property allowed to reach across a plus sign inside a square root?
Check yourself
End of topic.