Properties of Integers

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