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Properties of Integers

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Properties of Integers - Lesson Summary

Closure property
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.

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.

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.

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) = 1 2

Commutative property
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.

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.

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.

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
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.

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.

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].

Associative property under division:
Division is not associative for integers.

Distributive property
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.

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.

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.

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.

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