Algebra 2 Properties Flashcards

Properties of Algebra

• Commutative Property

  • For addition: ( a + b = b + a )
  • For multiplication: ( a(b) = b(a) )

• Associative Property

  • For addition: ( a + (b + c) = (a + b) + c )
  • For multiplication: ( (ab)c = a(bc) )

• Distributive Property

  • Multiplies a number by a sum: ( a(b + c) = ab + ac )

• Identity Property

  • For addition: Adding zero does not change the value: ( a + 0 = a )
  • For multiplication: Multiplying by one does not change the value: ( a = 1(a) )

• Inverse Property

  • Additive inverse: A number plus its negative equals zero: ( a + (-a) = 0 )
  • Multiplicative inverse: A number times its reciprocal equals one, ( a(1/a) = 1 ) (where ( a \neq 0 ))

• Closure Property

  • For addition: The sum of two real numbers is a real number.
  • For multiplication: The product of two real numbers is a real number.

• Equality Properties

  • Reflexive: ( a + b = a + b )
  • Symmetric: If ( a = b + c ), then ( b + c = a )
  • Transitive: If ( a = b + c ) and ( b + c = d ), then ( a = d )

• Equality with Operations

  • Addition Property: If ( a = b ), then ( a + c = b + c )
  • Multiplication Property: If ( a = b ), then ( ac = bc )

• Cancellation Properties

  • Addition: If ( a + c = b + c ), then ( a = b )
  • Multiplication: If ( ab = ac ), then ( b = c )
  • Additive Inverse: ( -(-a) = a )

• Special Multiplication Properties

  • Multiplying by -1 reverses the sign: ( -1b = -b )
  • Multiplying by 0 results in 0: ( 0(a) = 0 )

• Properties of Sums and Products

  • Opposite of a Sum: ( -(a + b) = -a + (-b) )
  • Reciprocals of a Product: ( \frac{1}{ab} = \frac{1}{a} \cdot \frac{1}{b} )

• Subtraction and Division Definitions

  • Subtraction: ( a - b = a + (-b) )
  • Division: ( \frac{a}{b} = a \times \frac{1}{b} )

• Substitution Principle

  • If ( x = 8 - 5 ), then ( x = 3 )

• Product with Opposites

  • ( x(-y) = -xy ) demonstrates the product of a number and a negative is negative.

Explore the essential properties of Algebra 2 through flashcards, including the Commutative, Associative, and Distributive properties. Each card provides a definition and an example to enhance your understanding of these fundamental concepts in mathematics.