Integers: Definition and Properties

Integers

Definition

• A set of whole numbers, including:
  • Positive integers (1, 2, 3, ...)
  • Negative integers (-1, -2, -3, ...)
  • Zero (0)

Properties

• Closure: The result of adding, subtracting, multiplying, or dividing integers is always an integer.
• Commutative Property: The order of integers does not change the result of addition or multiplication.
• Associative Property: The order in which integers are added or multiplied does not change the result.
• Distributive Property: Multiplication distributes over addition.

Types of Integers

• Even integers: Divisible by 2 (..., -4, -2, 0, 2, 4, ...)
• Odd integers: Not divisible by 2 (..., -3, -1, 1, 3, ...)
• Prime integers: Only divisible by 1 and themselves (2, 3, 5, 7, 11, ...)

Operations on Integers

• Addition: Combine two or more integers to get a sum.
• Subtraction: Find the difference between two integers.
• Multiplication: Combine two or more integers to get a product.
• Division: Divide one integer by another, resulting in a quotient and possibly a remainder.

Real-World Applications

• Counting and measuring quantities
• Representing temperatures (e.g., -5°C, 0°C, 5°C)
• Financial transactions (e.g., +$10, -$5, $0)

Integers

Definition

• Integers are a set of whole numbers including positive integers, negative integers, and zero.

Properties

• Integers are closed under addition, subtraction, multiplication, and division.
• The commutative property applies to integers in addition and multiplication.
• The associative property applies to integers in addition and multiplication.
• The distributive property applies to integers, where multiplication distributes over addition.

Types of Integers

Even Integers

• Even integers are divisible by 2, such as ..., -4, -2, 0, 2, 4, ...

Odd Integers

• Odd integers are not divisible by 2, such as ..., -3, -1, 1, 3, ...

Prime Integers

• Prime integers are only divisible by 1 and themselves, such as 2, 3, 5, 7, 11, ...

Operations on Integers

Addition

• Addition combines two or more integers to get a sum.

Subtraction

• Subtraction finds the difference between two integers.

Multiplication

• Multiplication combines two or more integers to get a product.

Division

• Division divides one integer by another, resulting in a quotient and possibly a remainder.

Real-World Applications

• Integers are used for counting and measuring quantities.
• Integers represent temperatures, such as -5°C, 0°C, and 5°C.
• Integers are used in financial transactions, such as +$10, -$5, and $0.