Algebra 2 Axioms/Properties Flashcards

Axioms and Properties of Addition

• Closure Axiom for Addition: The sum of any two real numbers (a + b) is also a real number.
• Associative Axiom for Addition: The grouping of numbers does not affect the sum, i.e., (a + b) + c = a + (b + c).
• Commutative Axiom for Addition: The order of addition does not change the result, i.e., a + b = b + a.
• Identity Axiom for Addition: Adding zero to any number does not change its value, i.e., a + 0 = a.
• Axiom of Additive Inverses: For every number a, there exists a unique number -a such that their sum is zero, i.e., a + (-a) = 0.

Axioms and Properties of Multiplication

• Closure Axiom for Multiplication: The product of any two real numbers (ab) is also a real number.
• Associative Axiom for Multiplication: The grouping of numbers does not affect the product, i.e., (ab)c = a(bc).
• Commutative Axiom for Multiplication: The order of multiplication does not change the result, i.e., ab = ba.
• Identity Axiom for Multiplication: Multiplying any number by one does not change its value, i.e., a * 1 = a.
• Axiom of Multiplicative Inverses: For every non-zero number a, there exists a unique number 1/a such that their product is one, i.e., a * (1/a) = 1.

Distributive and Other Properties

• Distributive Property: Multiplication distributes over addition, i.e., a(b + c) = ab + ac.
• Reflexive Property: Any number is equal to itself, i.e., a = a.
• Symmetric Property: If one number equals another, the reverse is also true, i.e., if a = b, then b = a.
• Transitive Property: If two numbers are equal to a third number, they are equal to each other, i.e., if a = b and b = c, then a = c.

Cancellation and Properties of Zero

• Cancellation Property of Addition: If two sums are equal, removing the same addend from both sides keeps them equal, i.e., if a + c = b + c, then a = b.
• Property of the Opposite of a Sum: The opposite of a sum is equal to the sum of the opposites, i.e., -(a + b) = -a + (-b).
• Cancellation Property of Multiplication: If two products are equal and one factor is shared, then the other factors are equal, i.e., if ac = bc, then a = b.
• Multiplicative Property of Zero: Multiplying any number by zero results in zero, i.e., a * 0 = 0.

Properties of Negation

• Multiplicative Property of -1: Multiplying any number by -1 gives its additive inverse, i.e., (-1)a = -a.
• Properties of Opposites in Products: Multiple relationships hold true for opposites in multiplication: (-a)b = -ab, a(-b) = -ab, and (-a)(-b) = ab.

Definitions of Operations

• Definition of Subtraction: Subtraction can be defined as adding the additive inverse, i.e., a - b = a + (-b).
• Definition of Division: Division is defined as multiplying by the reciprocal, i.e., a / b = a * (1 / b).

Properties of Equality

• Additive Property of Equality: If two values are equal, adding the same number to both sides retains equality, i.e., if a = b, then a + c = b + c.
• Multiplicative Property of Equality: If two values are equal, multiplying both sides by the same non-zero number retains equality, i.e., if a = b, then ac = bc.