Properties and Operations of Integers

Properties of Integers

• Integers are whole numbers, either positive, negative, or zero, without a fractional part.
• Integers can be represented on a number line, with positive integers on the right side of zero and negative integers on the left side of zero.
• Integers have the following properties:
  • Closure: The result of an operation on two integers is always an integer.
  • Commutativity: The order of integers in an operation does not change the result.
  • Associativity: The order in which integers are grouped in an operation does not change the result.
  • Distributivity: The operation of addition and multiplication can be distributed over integers.

Addition and Subtraction of Integers

• Rules for adding and subtracting integers:
  • Same signs (both positive or both negative): Add the numbers and keep the same sign.
  • Different signs (one positive and one negative): Subtract the numbers and take the sign of the larger number.
• Examples:
  • 2 + 3 = 5 (both positive)
  • (-2) + (-3) = -5 (both negative)
  • 2 + (-3) = -1 (different signs)
  • (-2) + 3 = 1 (different signs)
• Subtraction of integers can be thought of as adding the additive inverse (i.e., changing the sign of the second number).

Multiplication and Division of Integers

• Rules for multiplying integers:
  • Same signs (both positive or both negative): Multiply the numbers and keep the same sign.
  • Different signs (one positive and one negative): Multiply the numbers and take the negative sign.
• Examples:
  • 2 × 3 = 6 (both positive)
  • (-2) × (-3) = 6 (both negative)
  • 2 × (-3) = -6 (different signs)
  • (-2) × 3 = -6 (different signs)
• Division of integers is not commutative, meaning the order of the numbers matters.
• Rules for dividing integers:
  • Divide the absolute values (without signs) and then apply the sign rules for multiplication.

Integer Operations

• Order of Operations (PEMDAS):
  1. Parentheses (evaluate expressions inside parentheses first)
  2. Exponents (evaluate any exponential expressions next)
  3. Multiplication and Division (evaluate from left to right)
  4. Addition and Subtraction (evaluate from left to right)
• When performing multiple operations with integers, follow the order of operations to ensure accurate results.
• Examples of integer operations:
  • 2 × (-3) + 5 =? (follow order of operations)
  • (-2) + 3 × (-4) =? (follow order of operations)