Algebra
Expressions
Properties of numbers
20 practice questions
2 video lessons
Theory + worked examples
Theory
The properties of numbers justify rewriting expressions:
- Commutative: \(a+b=b+a\), \(ab=ba\).
- Associative: \((a+b)+c=a+(b+c)\).
- Distributive: \(a(b+c)=ab+ac\).
- Identity: \(a+0=a\), \(a\cdot1=a\).
Commutative and associative work for \(+\) and \(\times\), not for \(-\) or \(\div\).
The four number properties.
What each property lets you do.
The properties:
\[a+b=b+a,\quad (a+b)+c=a+(b+c),\quad a(b+c)=ab+ac\]
The distributive property links multiplication and addition.
How to use the properties
- Reorder terms with the commutative property.
- Regroup with the associative property.
- Multiply across a sum with the distributive property.
- Drop \(+0\) and \(\times1\) with identities.
Example 1 — Commutative
Rewrite \(3+x\) using the commutative property.
Solution
Order can switch for addition.
| \(3+x\) | \(=\) | \(x+3\) |
Example 2 — Associative
Regroup \((2+x)+5\).
Solution
Grouping can change for addition.
| \((2+x)+5\) | \(=\) | \(2+(x+5)\) |
| \(=\) | \(x+7\) |
Example 3 — Distributive
Apply the distributive property to \(4(x+2)\).
Solution
Multiply \(4\) across the sum.
| \(4(x+2)\) | \(=\) | \(4x+8\) |
Example 4 — Identity
Simplify \(x+0\) and \(1\cdot y\).
Solution
Adding \(0\) or multiplying by \(1\) changes nothing.
| \(x+0\) | \(=\) | \(x\) |
| \(1\cdot y\) | \(=\) | \(y\) |
Common pitfalls
Subtraction and division are not commutative: \(5-3\neq3-5\).
Distribute to every term inside the parentheses.
Identity for \(+\) is \(0\); for \(\times\) it is \(1\).
Frequently asked questions
What is the commutative property?
Order doesn't matter for addition or multiplication: \(a+b=b+a\).
What is the associative property?
Grouping doesn't matter: \((a+b)+c=a+(b+c)\).
What is the distributive property?
\(a(b+c)=ab+ac\).
Is subtraction commutative?
No — \(5-3\) does not equal \(3-5\).
More in Expressions