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Questions and Answers
What property of real numbers states that the result of adding, subtracting, multiplying, or dividing two real numbers is always a real number?
What property of real numbers states that the result of adding, subtracting, multiplying, or dividing two real numbers is always a real number?
Which property of real numbers ensures that the order of operations does not change the result when adding or multiplying three real numbers?
Which property of real numbers ensures that the order of operations does not change the result when adding or multiplying three real numbers?
What is the additive identity of real numbers?
What is the additive identity of real numbers?
What property of real numbers allows us to rewrite a × (b + c) as a × b + a × c?
What property of real numbers allows us to rewrite a × (b + c) as a × b + a × c?
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What is the multiplicative inverse of a nonzero real number a?
What is the multiplicative inverse of a nonzero real number a?
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Study Notes
Properties of Real Numbers
Closure Property
- The result of adding, subtracting, multiplying, or dividing two real numbers is always a real number.
Commutative Property
- The order in which two real numbers are added or multiplied does not change the result:
- a + b = b + a
- a × b = b × a
Associative Property
- The order in which three real numbers are added or multiplied does not change the result:
- (a + b) + c = a + (b + c)
- (a × b) × c = a × (b × c)
Distributive Property
- Multiplication distributes over addition:
- a × (b + c) = a × b + a × c
Identity Elements
- There exist two special real numbers:
- Additive Identity: 0 (a + 0 = a)
- Multiplicative Identity: 1 (a × 1 = a)
Inverse Elements
- For each real number a, there exist two special real numbers:
- Additive Inverse: -a (a + (-a) = 0)
- Multiplicative Inverse: 1/a (a × (1/a) = 1), except for a = 0
Properties of Real Numbers
- The result of adding, subtracting, multiplying, or dividing two real numbers is always a real number, due to the Closure Property.
Commutative Property
- The order of addends does not change the result of addition: a + b = b + a
- The order of factors does not change the result of multiplication: a × b = b × a
Associative Property
- The order in which three real numbers are added does not change the result: (a + b) + c = a + (b + c)
- The order in which three real numbers are multiplied does not change the result: (a × b) × c = a × (b × c)
Distributive Property
- Multiplication distributes over addition: a × (b + c) = a × b + a × c
Identity Elements
- Additive Identity: 0, since a + 0 = a
- Multiplicative Identity: 1, since a × 1 = a
Inverse Elements
- Additive Inverse: -a, since a + (-a) = 0
- Multiplicative Inverse: 1/a, since a × (1/a) = 1, except for a = 0
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Description
Learn about the properties of real numbers, including closure, commutative, and associative properties. Understand how these properties apply to addition, subtraction, multiplication, and division.
