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# Properties of Integers

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@MeticulousGroup

### What is the result of adding, subtracting, or multiplying two integers?

• Always a fraction
• Always a negative number
• Always an integer (correct)
• Always a real number
• ### What is the property of integers where the order of integers does not change the result when adding or multiplying?

• Distributive property
• Commutative property (correct)
• Associative property
• Closure property
• ### What is the property of integers where the order in which integers are added or multiplied does not change the result?

• Associative property (correct)
• Distributive property
• Commutative property
• ### What is the property of integers that states multiplication distributes over addition?

<p>Distributive property</p> Signup and view all the answers

### What is the additive identity of integers?

<p>0</p> Signup and view all the answers

### Which integers have a multiplicative inverse?

<p>Only 1 and -1</p> Signup and view all the answers

## 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

• Example: 2 × (3 + 4) = 2 × 3 + 2 × 4 (both equal 14)

• 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

• 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

### 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

• Example: 2 × (3 + 4) = 2 × 3 + 2 × 4 (both equal 14)

### Identities

• 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.

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