Questions and Answers
What is the Commutative Property for Addition?
What is the Commutative Property for Multiplication?
What is the Associative Property for Addition?
What is the Identity Property for Addition?
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What is the Distributive Property?
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What is the Property of the Opposite of a Sum?
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What does the Cancellation Property of Addition state?
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What is the Definition of Subtraction?
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What does the Reflexive Property of Equality state?
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What is the Multiplicative Inverse Property?
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What is the Multiplication Property of 0?
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Study Notes
Properties of Algebra
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Commutative Property
- For addition: ( a + b = b + a )
- For multiplication: ( a(b) = b(a) )
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Associative Property
- For addition: ( a + (b + c) = (a + b) + c )
- For multiplication: ( (ab)c = a(bc) )
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Distributive Property
- Multiplies a number by a sum: ( a(b + c) = ab + ac )
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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) )
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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 ))
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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.
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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 )
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Equality with Operations
- Addition Property: If ( a = b ), then ( a + c = b + c )
- Multiplication Property: If ( a = b ), then ( ac = bc )
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Cancellation Properties
- Addition: If ( a + c = b + c ), then ( a = b )
- Multiplication: If ( ab = ac ), then ( b = c )
- Additive Inverse: ( -(-a) = a )
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Special Multiplication Properties
- Multiplying by -1 reverses the sign: ( -1b = -b )
- Multiplying by 0 results in 0: ( 0(a) = 0 )
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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} )
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Subtraction and Division Definitions
- Subtraction: ( a - b = a + (-b) )
- Division: ( \frac{a}{b} = a \times \frac{1}{b} )
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Substitution Principle
- If ( x = 8 - 5 ), then ( x = 3 )
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Product with Opposites
- ( x(-y) = -xy ) demonstrates the product of a number and a negative is negative.
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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.