Closure property
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
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