Write all the property of integer …

The set of integers includes zero, negative and positive numbers without any decimal or fractional parts. They are numbers that represent whole things. For examples 1,0,-3 etc. 

Properties of Integers: Commutative Property for Addition, Associative Property for Addition, Distributive Property, Identity Property for Addition, Identity Property for Multiplication, Inverse Property for Addition and Zero Property for Multiplication.

An integer is a number that can be written without a fractional component.

• Closure property
  (i) Closure property under addition:
  Integers are closed under addition, i.e. for any two integers a and b, a + b is an integer. 
  Ex: 3 + 4 = 7; (– 9) + 7 = – 2.
  
  (ii) Closure property under subtraction:
  Integers are closed under subtraction, i.e. for any two integers a and b, a – b is an integer.
  Ex: (– 21) – (– 9) = (– 12); 8 – 3 = 5.
  
  (iii) Closure property under multiplication:
  Integers are closed under multiplication, i.e. for any two integers a and b, ab is an integer.
  Ex: 5 × 6 = 30; (– 9) × (– 3) = 27.
  
  (iv) Closure property under division:
  Integers are not closed under division, i.e. for any two integers a and b,  a b may not be an integer.
  Ex:(– 2) ÷ (– 4) = {tex}1\over 2{/tex} 
   
• Commutative property
  (i) Commutative property under addition:
  Addition is commutative for integers.  For any two integers a and b, a + b = b + a.
  Ex: 5 + (– 6) = 5 – 6 = – 1;
  (– 6) + 5 = – 6 + 5 = –1
  ∴ 5 + (– 6) = (– 6) + 5.
  
  (ii) Commutative property under subtraction:
  Subtraction is not commutative for integers.  For any two integers a and b, a – b ≠ b – a.
  Ex: 8 – (– 6) = 8 + 6 = 14;
  (– 6) – 8 = – 6 – 8 = – 14
  ∴ 8 – (– 6) ≠ – 6 – 8.
  
  (iii) Commutative property under multiplication:
  Multiplication is commutative for integers.  For any two integers a and b, ab = ba.
  Ex: 9 × (– 6) = – (9 × 6) = – 54;
  (– 6) × 9 = – (6 × 9) = – 54
  ∴ 9 × (– 6) = (– 6) × 9.
  
  (iv) Commutative property under division:
  Division is not commutative for integers.  For any two integers a and b, a ÷ b ≠ b ÷ a.
  Ex: (– 14) ÷ 2 = – 7
  2 ÷ (–14) = –  1 7 
  (– 14) ÷ 2 ≠ 2 ÷ (–14).
   
• Associative property
  (i) Associative property under addition:
  Addition is associative for integers. For any three integers a, b and c, a + (b + c) = (a + b) + c
  Ex: 5 + (– 6 + 4) = 5 + (– 2) = 3;
  (5 – 6) + 4 = (– 1) + 4 = 3
  ∴ 5 + (– 6 + 4) = (5 – 6) + 4.
  
  (ii) Associative property under subtraction:
  Subtraction is associative for integers.  For any three integers a, b and c, a – (b – c) ≠ (a – b) – c
  Ex: 5 – (6 – 4) = 5 – 2 = 3;
  (5 – 6) – 4 = – 1 – 4 = – 5
  ∴ 5 – (6 – 4) ≠ (5 – 6) – 4.
  
  (iii) Associative property under multiplication:
  Multiplication is associative for integers. For any three integers a, b and c, (a × b) × c = a × (b × c)
  Ex: [(– 3) × (– 2)] × 4 = (6 × 4) = 24
  (– 3) × [(– 2) × 4] = (– 3) × (– 8) = 24
  ∴ [(– 3) × (– 2)] × 4 = [(– 3) × (– 2) × 4].
  
  (iv) Associative property under division:
  Division is not associative for integers.
   
• Distributive property
  (i) Distributive property of multiplication over addition:
  For any three integers a, b and c, a × (b + c) = (a × b) + (a × c).
  Ex: – 2 (4 + 3) =  –2 (7)  = –14
                        = (– 2 × 4) + (– 2 × 3)
                        = (– 8) + (– 6)
                        = – 14.
  
  (ii) Distributive property of multiplication over subtraction:
  For any three integers, a, b and c, a × (b - c) = (a × b) – (a × c).
  Ex: – 2 (4 – 3) = – 2 (1) = – 2
                       = (–2 × 4) – (– 2 × 3)
                       = (– 8) – (– 6)
                       = – 2.
  The distributive property of multiplication over the operations of addition and subtraction is true in the case of integers.
   
• (i) Identity under addition:
  Integer 0 is the identity under addition.  That is, for any integer a, a + 0 = 0 + a = a.
  Ex: 4 + 0 = 0 + 4 = 4.
  
  (ii) Identity under multiplication:
  The integer 1 is the identity under multiplication. That is, for any integer a, 1 × a = a × 1 = a.
  Ex: (– 4) × 1 = 1 × (– 4) = – 4.
  When an integer is multiplied by –1, the result is the integer with sign changed i.e. the additive identity of the integer.
  For any integer a, a × –1 = –1 × a = –a.