Algebra
Algebraic Properties – Commutative, Associative, Distributive
Every time you rearrange, regroup, or expand an algebraic expression, you're relying on one of three fundamental properties: the commutative property (order doesn't matter for addition or multiplication), the associative property (grouping doesn't matter), and the distributive property (multiplication distributes over addition). These aren't new rules to memorise – they're the hidden justification behind moves you've already been making without necessarily naming them.
Knowing these properties by name matters because it lets you explain why an algebraic step is valid, not just that it happens to work. This becomes especially important once you reach more advanced manipulation, like collecting like terms or changing the subject of a formula, on the pages that follow.
The Three Core Properties
Commutative: a + b = b + a, and ab = ba. Associative: (a + b) + c = a + (b + c), and (ab)c = a(bc). Distributive: a(b + c) = ab + ac.
Swapping the order of an addition is the commutative property.
5(4 + 7) = 5 × 4 + 5 × 7 = 20 + 35 = 55.
Real-Life Application
- Mental maths: the distributive property is exactly how people multiply large numbers quickly in their heads.
- Spreadsheets: formula order often relies on these properties to give consistent, predictable results.
- Computer programming: compilers use these same properties to optimise how expressions are calculated.
Key Takeaways
- The commutative property says order doesn't matter for addition or multiplication.
- The associative property says grouping doesn't matter.
- The distributive property expands a(b + c) into ab + ac.