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Questions and Answers
What is the result of adding, subtracting, or multiplying two integers?
What is the result of adding, subtracting, or multiplying two integers?
What is the property of integers where the order of integers does not change the result when adding or multiplying?
What is the property of integers where the order of integers does not change the result when adding or multiplying?
What is the property of integers where the order in which integers are added or multiplied does not change the result?
What is the property of integers where the order in which integers are added or multiplied does not change the result?
What is the property of integers that states multiplication distributes over addition?
What is the property of integers that states multiplication distributes over addition?
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What is the additive identity of integers?
What is the additive identity of integers?
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Which integers have a multiplicative inverse?
Which integers have a multiplicative inverse?
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Study Notes
Properties of Integers
Closure Property
- The result of adding, subtracting, or multiplying two integers is always an integer.
- Example: 2 + 3 = 5 (integer), 4 × 5 = 20 (integer)
Commutative Property
- The order of integers does not change the result when adding or multiplying.
- Example: 2 + 3 = 3 + 2 (both equal 5), 4 × 5 = 5 × 4 (both equal 20)
Associative Property
- The order in which integers are added or multiplied does not change the result.
- Example: (2 + 3) + 4 = 2 + (3 + 4) (both equal 9), (4 × 5) × 2 = 4 × (5 × 2) (both equal 40)
Distributive Property
- Multiplication distributes over addition.
- Example: 2 × (3 + 4) = 2 × 3 + 2 × 4 (both equal 14)
Additive Identity
- The additive identity is 0, which means that when 0 is added to any integer, the result is the same integer.
- Example: 5 + 0 = 5
Multiplicative Identity
- The multiplicative identity is 1, which means that when 1 is multiplied by any integer, the result is the same integer.
- Example: 5 × 1 = 5
Additive Inverse
- Every integer has an additive inverse, which is the same integer with a negative sign.
- Example: The additive inverse of 5 is -5, because 5 + (-5) = 0
Multiplicative Inverse
- Not every integer has a multiplicative inverse, but those that do are the numbers 1 and -1.
- Example: The multiplicative inverse of 1 is 1, because 1 × 1 = 1, and the multiplicative inverse of -1 is -1, because (-1) × (-1) = 1
Properties of Integers
Closure Property
- The result of adding, subtracting, or multiplying two integers is always an integer.
- Example: 2 + 3 = 5 (integer), 4 × 5 = 20 (integer)
Commutative Property
- The order of integers does not change the result when adding or multiplying.
- Example: 2 + 3 = 3 + 2 (both equal 5), 4 × 5 = 5 × 4 (both equal 20)
Associative Property
- The order in which integers are added or multiplied does not change the result.
- Example: (2 + 3) + 4 = 2 + (3 + 4) (both equal 9), (4 × 5) × 2 = 4 × (5 × 2) (both equal 40)
Distributive Property
- Multiplication distributes over addition.
- Example: 2 × (3 + 4) = 2 × 3 + 2 × 4 (both equal 14)
Identities
- Additive identity: 0
- Multiplicative identity: 1
- Example: 5 + 0 = 5, 5 × 1 = 5
Inverses
- Additive inverse: every integer has an additive inverse, which is the same integer with a negative sign.
- Example: The additive inverse of 5 is -5, because 5 + (-5) = 0
- Multiplicative inverse: only 1 and -1 have multiplicative inverses.
- Example: The multiplicative inverse of 1 is 1, because 1 × 1 = 1, and the multiplicative inverse of -1 is -1, because (-1) × (-1) = 1
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Description
Learn about the fundamental properties of integers, including closure, commutative, and associative properties, with examples and explanations.
