Integers

• Introduction to Integers

  • Natural Numbers: Counting numbers ($1, 2, 3, \dots$)
  • Whole Numbers: Natural numbers plus zero ($0, 1, 2, 3, \dots$)
  • Negative Numbers: Numbers less than zero ($-1, -2, -3, \dots$)
  • Definition: Collection of whole numbers and negative numbers ($\dots, -3, -2, -1, 0, 1, 2, 3, \dots$)

• Representation on a Number Line

  • Center Point: Zero ($0$) as origin
  • Right Side: Positive integers (greater than $0$, marked with $+$)
  • Left Side: Negative integers (less than $0$, marked with $-$)
  • Direction: Value increases to the right and decreases to the left

• Ordering of Integers

  • Comparison Rules:
    • Every positive integer is greater than every negative integer
    • Zero is less than every positive integer
    • Zero is greater than every negative integer
    • For two negative integers, the one with smaller numerical value is greater (e.g., $-2 > -5$)

• Addition of Integers

  • Rule 1 (Same Sign): Add their values and give the common sign ($(+3) + (+4) = +7$ or $(-3) + (-4) = -7$)
  • Rule 2 (Opposite Signs): Subtract the smaller value from the larger and give the sign of the integer with the larger value ($(+5) + (-3) = +2$)
  • Additive Inverse: An integer and its opposite add up to zero ($a + (-a) = 0$)

• Subtraction of Integers

  • Rule: To subtract an integer from another, add the additive inverse of the integer being subtracted to the first integer
  • Formula: $a - b = a + (-b)$
  • Example: $(+6) - (+2) = (+6) + (-2) = +4$