Integer Operations and Properties

Integer Operations

• Addition:
  • Commutative property: a + b = b + a
  • Associative property: (a + b) + c = a + (b + c)
  • Distributive property: a + (-a) = 0
• Subtraction:
  • Defined as adding the additive inverse: a - b = a + (-b)
• Multiplication:
  • Commutative property: a × b = b × a
  • Associative property: (a × b) × c = a × (b × c)
  • Distributive property: a × (b + c) = a × b + a × c
• Division:
  • Defined as multiplying by the multiplicative inverse: a ÷ b = a × (1/b)

Number Lines

• A visual representation of integers on a straight line, with:
  • Positive integers to the right of 0
  • Negative integers to the left of 0
  • 0 in the middle
• Used to visualize and compare integers, and to perform operations like addition and subtraction

Prime Numbers

• A prime number is a positive integer that is divisible only by itself and 1
• Examples: 2, 3, 5, 7, 11, ...
• Characteristics:
  • Greater than 1
  • Not divisible by any other number except 1 and itself

Odd and Even Numbers

• Even numbers:
  • Integers that are divisible by 2 (e.g., ..., -4, -2, 0, 2, 4, ...)
  • Can be written in the form 2k, where k is an integer
• Odd numbers:
  • Integers that are not divisible by 2 (e.g., ..., -3, -1, 1, 3, ...)
  • Can be written in the form 2k + 1, where k is an integer

Place Value

• The value of a digit in a number depends on its position or place
• Places in a number, from right to left:
  • Units (ones) place
  • Tens place
  • Hundreds place
  • Thousands place
  • ...
• Each place has a value 10 times the value of the place to its right