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Questions and Answers
What is the result of the operation a × 0 in real numbers?
Which of the following properties is NOT true for division of real numbers?
What is the relationship between addition and subtraction in real numbers?
According to the properties of operations, what is the result of any operation on real numbers?
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What is the property that states a + (b - c) = a + b - c in real numbers?
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What is the order of operations in real numbers?
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What is the purpose of the additive identity in real numbers?
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Which property of real numbers states that the order of the numbers does not change the result of the operation?
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What is the result of multiplying a real number by its multiplicative inverse?
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Which property of real numbers allows for the rearrangement of numbers in an operation without changing the result?
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What is the purpose of the additive inverse of a real number?
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What is the result of adding the additive inverse of a real number to itself?
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Which property of real numbers allows for the multiplication of a real number by a sum of real numbers?
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What is the relationship between the multiplicative identity and the multiplicative inverse of a real number?
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Study Notes
Operations With Real Numbers
Addition and Subtraction
- Commutative Property:
a + b = b + aanda - b = -(b - a) - Associative Property:
(a + b) + c = a + (b + c)and(a - b) - c = a - (b + c) - Distributive Property:
a + (b - c) = a + b - c - The sum of two real numbers is a real number
- The difference of two real numbers is a real number
Multiplication
- Commutative Property:
a × b = b × a - Associative Property:
(a × b) × c = a × (b × c) - Distributive Property:
a × (b + c) = a × b + a × c - The product of two real numbers is a real number
- Multiplication by 0:
a × 0 = 0 × a = 0
Division
- Not commutative:
a ÷ b ≠b ÷ a(except whena = b) - Not associative:
(a ÷ b) ÷ c ≠a ÷ (b ÷ c) - Division by 0 is undefined:
a ÷ 0is not a real number - The quotient of two real numbers is a real number (except when dividing by 0)
Properties of Operations
- Closure: The result of any operation on real numbers is always a real number
- The order of operations (PEMDAS) applies: parentheses, exponents, multiplication and division, and addition and subtraction
Inverse Operations
- Addition and subtraction are inverse operations:
a + b = cimpliesa = c - bandb = c - a - Multiplication and division are inverse operations:
a × b = cimpliesa = c ÷ bandb = c ÷ a
Operations With Real Numbers
Properties of Addition and Subtraction
- Commutative Property of Addition: order of numbers does not change the result
- Associative Property of Addition: grouping numbers does not change the result
- Distributive Property of Addition: combining like terms is allowed
- Result of adding or subtracting two real numbers is always a real number
Properties of Multiplication
- Commutative Property of Multiplication: order of numbers does not change the result
- Associative Property of Multiplication: grouping numbers does not change the result
- Distributive Property of Multiplication: multiplying a number by a sum is the same as multiplying by each addend
- Product of two real numbers is a real number
- Multiplying a number by 0 results in 0
Properties of Division
- Division is not commutative (except when numbers are equal)
- Division is not associative
- Dividing by 0 is undefined
- Result of dividing two real numbers is a real number (except when dividing by 0)
General Properties
- Closure: result of any operation on real numbers is always a real number
- Order of operations (PEMDAS) applies: parentheses, exponents, multiplication and division, and addition and subtraction
Inverse Operations
- Addition and subtraction are inverse operations: adding a number can be undone by subtracting it
- Multiplication and division are inverse operations: multiplying by a number can be undone by dividing by it
Properties of Real Numbers
Closure Property
- The set of real numbers is closed under addition, subtraction, multiplication, and division, meaning that the result of combining two real numbers using these operations is always a real number.
Commutative Property
- Real numbers follow the commutative property, meaning that the order of the numbers does not change the result of addition and multiplication:
- Addition: a + b = b + a
- Multiplication: a × b = b × a
Associative Property
- Real numbers follow the associative property, meaning that the order in which operations are performed does not change the result:
- Addition: (a + b) + c = a + (b + c)
- Multiplication: (a × b) × c = a × (b × c)
Distributive Property
- Real numbers follow the distributive property, meaning that multiplication can be distributed over addition:
- a × (b + c) = a × b + a × c
Identities and Inverses
- Additive Identity: 0 is the additive identity, meaning a + 0 = a
- Multiplicative Identity: 1 is the multiplicative identity, meaning a × 1 = a
- Additive Inverses: every real number has an additive inverse, meaning a + (-a) = 0
- Multiplicative Inverses: every non-zero real number has a multiplicative inverse, meaning a × (1/a) = 1
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Description
This quiz covers the properties of addition, subtraction, and multiplication of real numbers, including commutative, associative, and distributive properties.
