Integers

• Introduction to Integers
  • Definition: Collection of whole numbers and negative numbers
  • Representation on a Number Line
    • Positive integers to the right of zero
    • Negative integers to the left of zero
  • Order of Integers
    • Greater value to the right ($\text{e.g.}, 3 > -2$)
    • Smaller value to the left ($\text{e.g.}, -5 < -1$)
• Addition of Integers
  • Rules for Addition
    • Same signs: Add magnitudes and keep the common sign ($\text{e.g.}, (+3) + (+4) = +7$, $(-2) + (-3) = -5$)
    • Different signs: Subtract smaller magnitude from larger and use the sign of the larger number ($\text{e.g.}, (-5) + (+3) = -2$)
  • Properties of Addition
    • Closure Property: $a + b$ is an integer
    • Commutative Property: $a + b = b + a$
    • Associative Property: $(a + b) + c = a + (b + c)$
    • Additive Identity: $a + 0 = a$
    • Additive Inverse: $a + (-a) = 0$
• Subtraction of Integers
  • Rule: Add the additive inverse of the integer being subtracted to the first integer
    • Formula: $a - b = a + (-b)$
    • Example: $5 - (-3) = 5 + 3 = 8$
• Multiplication of Integers
  • Rules of Signs
    • Positive $\times$ Positive $=$ Positive ($+$ $\times$ $+$ $=$ $+$)
    • Negative $\times$ Negative $=$ Positive ($-$ $\times$ $-$ $=$ $+$)
    • Positive $\times$ Negative $=$ Negative ($+$ $\times$ $-$ $=$ $-$)
    • Negative $\times$ Positive $=$ Negative ($-$ $\times$ $+$ $=$ $-$)
  • Properties of Multiplication
    • Closure Property: $a \times b$ is an integer
    • Commutative Property: $a \times b = b \times a$
    • Associative Property: $(a \times b) \times c = a \times (b \times c)$
    • Multiplicative Identity: $a \times 1 = a$
    • Property of Zero: $a \times 0 = 0$
    • Distributive Property: $a \times (b + c) = (a \times b) + (a \times c)$
• Division of Integers
  • Rules of Signs (Same as multiplication)
    • Positive $\div$ Positive $=$ Positive
    • Negative $\div$ Negative $=$ Positive
    • Positive $\div$ Negative $=$ Negative
    • Negative $\div$ Positive $=$ Negative
  • Properties of Division
    • Division by zero is not defined ($a \div 0$ is meaningless)
    • $0 \div a = 0$ (for $a \neq 0$)
    • $a \div 1 = a$