Properties of Integers

Integers

Definition

• Integers are whole numbers, either positive, negative, or zero, without a fractional part.
• Examples: ..., -3, -2, -1, 0, 1, 2, 3, ...

Properties

• Closure: The result of adding, subtracting, or multiplying 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: Integers can be distributed over addition and subtraction.

Operations

• Addition:
  • The sum of two integers with the same sign is an integer with that sign.
  • The sum of two integers with different signs is an integer with the sign of the integer with the larger absolute value.
• Subtraction:
  • Subtracting an integer is equivalent to adding its additive inverse.
• Multiplication:
  • The product of two integers with the same sign is a positive integer.
  • The product of two integers with different signs is a negative integer.
• Division:
  • The quotient of two integers is an integer if and only if the dividend is divisible by the divisor.

Types of Integers

• Even Integers: Integers that are exactly divisible by 2.
• Odd Integers: Integers that are not exactly divisible by 2.
• Prime Integers: Integers greater than 1 that have exactly two distinct factors: 1 and themselves.
• Composite Integers: Integers greater than 1 that have more than two distinct factors.

What are Integers?

• Whole numbers, either positive, negative, or zero, without a fractional part
• Examples: ..., -3, -2, -1, 0, 1, 2, 3, ...

Properties of Integers

Closure

• The result of adding, subtracting, or multiplying 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

• Integers can be distributed over addition and subtraction

Operations on Integers

Addition

• The sum of two integers with the same sign is an integer with that sign
• The sum of two integers with different signs is an integer with the sign of the integer with the larger absolute value

Subtraction

• Subtracting an integer is equivalent to adding its additive inverse

Multiplication

• The product of two integers with the same sign is a positive integer
• The product of two integers with different signs is a negative integer

Division

• The quotient of two integers is an integer if and only if the dividend is divisible by the divisor

Types of Integers

Even Integers

• Integers that are exactly divisible by 2

Odd Integers

• Integers that are not exactly divisible by 2

Prime Integers

• Integers greater than 1 that have exactly two distinct factors: 1 and themselves

Composite Integers

• Integers greater than 1 that have more than two distinct factors

Learn about the properties of integers, including closure, commutative property, and associative property. Understand the definition and examples of integers.