Integers Definition and Properties

Definition and Properties

• An integer is a whole number, either positive, negative, or zero, without a fractional part.
• Integers are denoted by the symbol "Z" and include the set {..., -3, -2, -1, 0, 1, 2, 3, ...}.
• Integers can be represented on the number line, with positive integers to the right of zero and negative integers to the left of zero.

Types of Integers

• Positive integers: integers greater than zero, denoted by Z+ (e.g., 1, 2, 3, ...).
• Negative integers: integers less than zero, denoted by Z- (e.g., -1, -2, -3, ...).
• Non-positive integers: integers less than or equal to zero, denoted by Z0- (e.g., 0, -1, -2, ...).
• Non-negative integers: integers greater than or equal to zero, denoted by Z0+ (e.g., 0, 1, 2, ...).

Operations on Integers

• 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 may be a fraction or a decimal.

Important Concepts

• Even integers: integers that are divisible by 2, denoted by 2Z (e.g., ..., -4, -2, 0, 2, 4, ...).
• Odd integers: integers that are not divisible by 2, denoted by 2Z + 1 (e.g., ..., -3, -1, 1, 3, ...).
• Prime integers: integers greater than 1 that have only two factors: 1 and themselves (e.g., 2, 3, 5, 7, ...).
• Composite integers: integers greater than 1 that have more than two factors (e.g., 4, 6, 8, 9, ...).

Definition and Properties of Integers

• An integer is a whole number with no fractional part, either positive, negative, or zero.
• Integers are denoted by the symbol "Z" and include {..., -3, -2, -1, 0, 1, 2, 3,...}.
• Integers can be represented on the number line, with positive integers to the right of zero and negative integers to the left.

Types of Integers

• Positive integers are integers greater than zero, denoted by Z+ (e.g., 1, 2, 3,...).
• Negative integers are integers less than zero, denoted by Z- (e.g., -1, -2, -3,...).
• Non-positive integers are integers less than or equal to zero, denoted by Z0- (e.g., 0, -1, -2,...).
• Non-negative integers are integers greater than or equal to zero, denoted by Z0+ (e.g., 0, 1, 2,...).

Operations on Integers

• The sum of two integers is always an integer (addition).
• The difference of two integers is always an integer (subtraction).
• The product of two integers is always an integer (multiplication).
• The quotient of two integers is not always an integer, but may be a fraction or a decimal (division).

Important Concepts

• Even integers are integers divisible by 2, denoted by 2Z (e.g., ..., -4, -2, 0, 2, 4,...).
• Odd integers are integers not divisible by 2, denoted by 2Z + 1 (e.g., ..., -3, -1, 1, 3,...).
• Prime integers are integers greater than 1 with only two factors: 1 and themselves (e.g., 2, 3, 5, 7,...).
• Composite integers are integers greater than 1 with more than two factors (e.g., 4, 6, 8, 9,...).