Integers

• Closure under Addition


  Sum of two integers is always an integer
• Integers are closed under addition
• Closure under Subtraction


  Difference of two integers is always an integer
• Integers are closed under subtraction
• Commutative Property of Addition


  • Addition of integers is commutative
• Subtraction of integers is not commutative
• Associative Property of Addition

  • Addition of integers is associative
• Additive Identity


  Zero is the additive identity for integers
• Multiplication of a Positive and a Negative Integer


  Multiply the absolute values and then add the negative sign
• Integers

  • 3 × (–5) = –15
  • (–4) × 5 = –20
  • (–3) × 5 = –15
  • (–7) × 3 = –21
  • (–5) × 6 = –30
  • (–2) × 9 = –18
• Multiplication of a negative integer and a positive integer is the same as multiplication of the positive integer and the negative integer
• Multiplying integers without using number line

  First find the product of the positive integers, then put the negative sign before the product
• Multiplying a positive integer and a negative integer


  Multiply them as whole numbers and put a minus sign before the product to get a negative integer
• Multiplying two negative integers

  The product is a positive integer. Multiply the two negative integers as whole numbers and put the positive sign before the product.
• a × 1


  Equals a
• Multiplying any integer with -1

  1. (–3) × (–1) = 3

  2. 3 × (–1) = –3

  3. (– 6) × (–1) = 6

  4. (–1) × 13 = –13

  5. (–1) × (–25) = 25

  6. 18 × (–1) = –18
• • 1 is not a multiplicative identity of integers
• Associativity for Multiplication


  • The grouping of integers does not affect the product of integers
  • For any three integers a, b and c: (a × b) × c = a × (b × c)
• Distributive Property


  1. a × (b + c) = a × b + a × c

  2. a × (b - c) = a × b - a × c
• Distributivity of multiplication over addition is true for integers
• 4 × (3 - 8) = 4 × 3 - 4 × 8
• (-5) × [(-4) - (-6)] = [(-5) × (-4)] - [(-5) × (-6)]
• In general, for any three integers a, b and c: a × (b - c) = a × b - a × c
• For any integer a, (–1) × a = -a
• The integer whose product with (–1) is -22 is 22
• The integer whose product with (–1) is 37 is -37
• The integer whose product with (–1) is 0 is 0
• Division of Integers


  Division is the inverse operation of multiplication
• When dividing a negative integer by a positive integer, divide as whole numbers and put a minus sign before the quotient
• When dividing a positive integer by a negative integer, first divide as whole numbers and then put a minus sign before the quotient
• When dividing a negative integer by a negative integer, first divide as whole numbers and then put a positive sign
• Integers are not closed under division
• Division is not commutative for integers
• Any integer divided by 0 is not defined, but 0 divided by any integer other than 0 is 0
• Any integer divided by 1 gives the same integer
• Any integer divided by -1 gives the opposite integer