Properties of multiplication and division of integers

Properties of Multiplication of Integers

  • Closure under Multiplication
    • Integer \times Integer = Integer
  • Commutativity of Multiplication
    • For any two integers a and b, a\; \times \;b{\rm{ }} = {\rm{ }}b{\rm{ }} \times {\rm{ }}a.
  • Associativity of Multiplication
    • For any three integers a, b and c, \left( {a{\rm{ }} \times {\rm{ }}b} \right){\rm{ }} \times {\rm{ }}c{\rm{ }} = {\rm{ }}a{\rm{ }} \times {\rm{ }}\left( {b{\rm{ }} \times {\rm{ }}c} \right).
  • Distributive Property of Integers
    • Under addition and multiplication, integers show the distributive property.
    • For any integers a, b and c, a{\rm{ }} \times {\rm{ }}\left( {b{\rm{ }} + {\rm{ }}c} \right){\rm{ }} = {\rm{ }}a\; \times {\rm{ }}b{\rm{ }} + {\rm{ }}a{\rm{ }} \times {\rm{ }}c.
  • Multiplication by Zero
    • For any integer a, a{\rm{ }} \times {\rm{ }}0{\rm{ }} = {\rm{ }}0{\rm{ }} \times {\rm{ }}a{\rm{ }} = {\rm{ }}0.
  • Multiplicative Identity
    • 1 is the multiplicative identity for integers.
    • a{\rm{ }} \times {\rm{ }}1{\rm{ }} = {\rm{ }}1{\rm{ }} \times {\rm{ }}a{\rm{ }} = {\rm{ }}a

Properties of Division of Integers

For any integer a,

  • \dfrac{a}{0} is not defined
  • \dfrac{a}{1} = a

Integers are not closed under division.

Example: \left( {-9} \right){\rm{ }} \div {\rm{ }}\left( {-3} \right){\rm{ }} = 3 result is an integer but

 \left( {-3} \right) \div \left( { -9} \right) = \dfrac{3}{9} = 0.33 which is not an integer.

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