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Operations on Integers
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Operations on Integers

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Questions and Answers

What is the result of adding two integers with the same sign?

  • The result always has the same sign. (correct)
  • The result always has the opposite sign.
  • The result is always zero.
  • The result is always undefined.
  • What is the distributive property of multiplication over addition?

  • a × (b ÷ c) = a × b ÷ a × c
  • a × (b - c) = a × b - a × c
  • a × (b + c) = a × b + a × c (correct)
  • a × (b × c) = a × b × a × c
  • What is the additive inverse of 5?

  • 10
  • 0
  • -5 (correct)
  • 5
  • What is the result of multiplying two integers with different signs?

    <p>The result is always negative.</p> Signup and view all the answers

    What is the associative property of addition?

    <p>(a + b) + c = a + (b + c)</p> Signup and view all the answers

    What is the result of subtracting two integers with the same sign?

    <p>The result always has the opposite sign.</p> Signup and view all the answers

    What is the commutative property of multiplication?

    <p>a × b = b × a</p> Signup and view all the answers

    What is the result of dividing two integers with the same sign?

    <p>The result is always positive.</p> Signup and view all the answers

    What is the subtraction of integers defined as?

    <p>Addition of the additive inverse</p> Signup and view all the answers

    Study Notes

    Operations on Integers

    Addition

    • Commutative property: a + b = b + a
    • Associative property: (a + b) + c = a + (b + c)
    • Distributive property: a + (b + c) = a + b + a + c
    • Additive inverse: a + (-a) = 0
    • Examples:
      • 2 + 3 = 5
      • (-2) + 3 = 1

    Subtraction

    • Defined as addition of the additive inverse: a - b = a + (-b)
    • Not commutative: a - b ≠ b - a
    • Not associative: (a - b) - c ≠ a - (b - c)
    • Examples:
      • 5 - 2 = 3
      • 3 - (-2) = 5

    Multiplication

    • Commutative property: a × b = b × a
    • Associative property: (a × b) × c = a × (b × c)
    • Distributive property: a × (b + c) = a × b + a × c
    • Multiplicative identity: a × 1 = a
    • Examples:
      • 2 × 3 = 6
      • (-2) × 3 = -6

    Division

    • Defined as the inverse operation of multiplication: a ÷ b = a × (1/b)
    • Not commutative: a ÷ b ≠ b ÷ a
    • Not associative: (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)
    • Examples:
      • 6 ÷ 2 = 3
      • (-6) ÷ 2 = -3

    Rules for Operations on Integers

    • When multiplying or dividing two integers with the same sign, the result is always positive.
    • When multiplying or dividing two integers with different signs, the result is always negative.
    • When adding or subtracting two integers with the same sign, the result has the same sign.
    • When adding or subtracting two integers with different signs, the result has the sign of the integer with the larger absolute value.

    Operations on Integers

    Addition

    • Commutative property holds: a + b = b + a
    • Associative property holds: (a + b) + c = a + (b + c)
    • Distributive property holds: a + (b + c) = a + b + a + c
    • Additive inverse exists: a + (-a) = 0
    • Examples of addition: 2 + 3 = 5, (-2) + 3 = 1

    Subtraction

    • Defined as addition of the additive inverse: a - b = a + (-b)
    • Not commutative: a - b ≠ b - a
    • Not associative: (a - b) - c ≠ a - (b - c)
    • Examples of subtraction: 5 - 2 = 3, 3 - (-2) = 5

    Multiplication

    • Commutative property holds: a × b = b × a
    • Associative property holds: (a × b) × c = a × (b × c)
    • Distributive property holds: a × (b + c) = a × b + a × c
    • Multiplicative identity exists: a × 1 = a
    • Examples of multiplication: 2 × 3 = 6, (-2) × 3 = -6

    Division

    • Defined as the inverse operation of multiplication: a ÷ b = a × (1/b)
    • Not commutative: a ÷ b ≠ b ÷ a
    • Not associative: (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)
    • Examples of division: 6 ÷ 2 = 3, (-6) ÷ 2 = -3

    Rules for Operations on Integers

    • Same sign → positive result in multiplication and division
    • Different signs → negative result in multiplication and division
    • Same sign → same sign result in addition and subtraction
    • Different signs → sign of the larger absolute value result in addition and subtraction

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    Description

    Understand the properties and rules of addition and subtraction on integers, including commutative, associative, and distributive properties.

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