Integers: Definition and Properties

Integers

Definition

• An integer is a whole number, either positive, negative, or zero, without a fractional part.
• Examples: ..., -3, -2, -1, 0, 1, 2, 3, ...

Properties

• Closure: The sum, difference, product, and quotient (except division by zero) of two integers is always an integer.
• Commutative Property: The order of integers in an operation does not change the result.
• Associative Property: The order in which integers are grouped in an operation does not change the result.
• Distributive Property: The multiplication of an integer over the sum of two integers is equal to the sum of their individual products.

Operations

• Addition: The sum of two integers is always an integer.
• Subtraction: The difference of two integers is always an integer.
• Multiplication: The product of two integers is always an integer.
• Division: The quotient of two integers is not always an integer, but it can be expressed as an integer or a decimal.

Types of Integers

• Even Integers: divisible by 2, e.g., ..., -4, -2, 0, 2, 4, ...
• Odd Integers: not divisible by 2, e.g., ..., -3, -1, 1, 3, 5, ...
• Prime Integers: only divisible by 1 and themselves, e.g., 2, 3, 5, 7, ...

Integers

Definition and Examples

• An integer is a whole number with no fractional part, which can be positive, negative, or zero.
• Examples of integers include ..., -3, -2, -1, 0, 1, 2, 3, ...

Properties of Integers

Closure

• The sum of two integers is always an integer.
• The difference of two integers is always an integer.
• The product of two integers is always an integer.
• The quotient of two integers is an integer except for division by zero.

Commutative Property

• The order of integers in an operation does not change the result.

Associative Property

• The order in which integers are grouped in an operation does not change the result.

Distributive Property

• The multiplication of an integer over the sum of two integers is equal to the sum of their individual products.

Operations with Integers

• The sum of two integers is always an integer.
• The difference of two integers is always an integer.
• The product of two integers is always an integer.
• The quotient of two integers is not always an integer, but it can be expressed as an integer or a decimal.

Types of Integers

Even Integers

• Even integers are divisible by 2.
• Examples of even integers include ..., -4, -2, 0, 2, 4, ...

Odd Integers

• Odd integers are not divisible by 2.
• Examples of odd integers include ..., -3, -1, 1, 3, 5, ...

Prime Integers

• Prime integers are only divisible by 1 and themselves.
• Examples of prime integers include 2, 3, 5, 7, ...