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Properties of Integers

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Closure property

integer_properties1Closure property under addition

Integers are closed under addition, i.e. for any two integers, a and b, a+b is an integer.
 
Example: 3+4=7, 3 and 4 are integers and when we add them the answer we get is 7 which is also an integer, hence the property.

Closure property under subtraction

Integers are closed under subtraction, i.e. for any two integers, a and b,
a-b is an integer.
Example: -21 – (-9) = -12, -21 and -9 are integers and when we subtract them the answer we get is -12 which is also an integer, hence the property.

Closure property under multiplication

Integers are closed under multiplication, i.e. for any two integers,
a and b, ab is an integer.
Example: 5x 6 =30, 5 and 6 are integers and when we multiplied them the answer we get is 30 which is also an integer, hence the property.

Closure property under division

Integers are NOT closed under division, i.e. for any two integers, a and b, a/b may not be integer.

Commutative property

Commutative property under addition

Addition is commutative for integers.  For any two integers, a and b,
a + b=b + a
Example: 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

Example: 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

Example: 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

Example: 3/6=1/2
6/3 = 2
∴ 3/6 ≠ 6/3

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

Example: 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

Example: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)

Example:[(-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
Example: -2 (4 + 3) =  -2 (7)  = -14
-2(4+3)=(-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
Example: -2 (4- 3) =  -2 (1)  = -2
-2(4-3)=(-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 an integer a, a+0=0+a=a
Example: 4+0=0+4=4

Identity under multiplication

The integer 1 is the identity under multiplication.  That is, for an integer a, 1×a=a×1=a
Example: (-4)×1=1×(-4)=-4

integer_properties2
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