Algebra: Commutative and Associative Properties

Commutative Property

• Applies to addition and multiplication
• States that the order of numbers being added or multiplied does not change the result
• Examples:
  • 2 + 3 = 3 + 2
  • 2 × 3 = 3 × 2
• Can be expressed as:
  • a + b = b + a
  • a × b = b × a

Associative Property

• Applies to addition and multiplication
• States that when three or more numbers are added or multiplied, the order in which the operations are performed does not change the result
• Examples:
  • (2 + 3) + 4 = 2 + (3 + 4)
  • (2 × 3) × 4 = 2 × (3 × 4)
• Can be expressed as:
  • (a + b) + c = a + (b + c)
  • (a × b) × c = a × (b × c)

Distributive Property

• Applies to multiplication over addition
• States that multiplication distributes over addition
• Examples:
  • 2 × (3 + 4) = 2 × 3 + 2 × 4
  • 5 × (2 - 3) = 5 × 2 - 5 × 3
• Can be expressed as:
  • a × (b + c) = a × b + a × c
  • a × (b - c) = a × b - a × c

Commutative Property

• The commutative property applies to both addition and multiplication operations
• Changing the order of numbers being added or multiplied does not affect the result
• Examples of the commutative property in action:
  • 2 + 3 = 3 + 2
  • 2 × 3 = 3 × 2
• The commutative property can be expressed mathematically as:
  • a + b = b + a
  • a × b = b × a

Associative Property

• The associative property also applies to both addition and multiplication operations
• The order in which three or more numbers are added or multiplied does not change the result
• Examples of the associative property in action:
  • (2 + 3) + 4 = 2 + (3 + 4)
  • (2 × 3) × 4 = 2 × (3 × 4)
• The associative property can be expressed mathematically as:
  • (a + b) + c = a + (b + c)
  • (a × b) × c = a × (b × c)

Distributive Property

• The distributive property applies to multiplication over addition
• Multiplication distributes over addition, meaning that the operation can be applied to each addend separately
• Examples of the distributive property in action:
  • 2 × (3 + 4) = 2 × 3 + 2 × 4
  • 5 × (2 - 3) = 5 × 2 - 5 × 3
• The distributive property can be expressed mathematically as:
  • a × (b + c) = a × b + a × c
  • a × (b - c) = a × b - a × c

Learn about the commutative and associative properties in algebra, including their definitions, examples, and expressions. Understand how these properties apply to addition and multiplication.