Questions and Answers
What does the statement 'a + b is a unique whole number' represent?
What does the equation '(a+b)+c=a+(b+c)' represent?
What does the equation 'a+b=b+a' represent?
What does the equation 'a+0=a' represent?
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What does the equation 'a+(-a)=0' represent?
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What does the statement 'ab is a unique number' represent?
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What does the equation '(ab)c=a(bc)' represent?
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What does the equation 'ab=ba' represent?
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What does the equation 'a*1=a' represent?
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What does the equation 'a*(1/a)=1' represent?
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What does the equation 'a(b+c)=ab+ac' represent?
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What does the equation 'a=a' represent?
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What does the statement 'If a=b, then b=a' represent?
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What does the statement 'If a=b and b=c, then a=c' represent?
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What does the statement 'If a+c=b+c, then a=b' represent?
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What does the equation '-(a+b)=(-a)+(-b)' represent?
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What does the equation '-(-a)=a' represent?
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What does the statement 'If ac=bc, then a=b' represent?
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What does the equation '1/ab=(1/a)*(1/b)' represent?
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What does the equation 'a*0=0' represent?
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What does the equation '(-1)a=-a' represent?
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What does the equation '(-a)b=-ab, a(-b)=-ab, (-a)(-b)=ab' represent?
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What does the equation 'a-b=a+(-b)' represent?
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What does the equation 'a/b=a*(1/b)' represent?
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What does the statement 'If a=b, a+c=b+c' represent?
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Study Notes
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.
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Description
Test your understanding of the fundamental axioms and properties of addition in algebra. This quiz covers crucial concepts including closure, associativity, commutativity, identity, and inverses. Perfect for reinforcing your knowledge in Algebra 2.
