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Properties of Integers

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Operations on Integers

Multiplication and Division of Fraction

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Properties of Integers - Lesson Summary

Closure property under addition:
Integers are closed under addition, i.e. for any two integers,a and b, a+b is an integer.
Eg: 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.
Eg: -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.
Eg: 5×6=30; -9×-3=27
Closure property under division:
Integers are NOT closed under division, i.e. for any two integers,

Eg:
Commutative property under addition:
Addition is commutative for integers. For any two integers, a and b, a+b=b+a
Eg: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-b≠b-a
Eg: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
Eg: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,
Eg:3/6=1/2;
Associative property under addition:

Addition is associative for integers. For any three integers, a, b and c, a+(b+c)=(a+b)+c
Eg: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-c)≠(a-b)-c
Eg: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)
Eg: [(-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
Eg: -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
Eg: -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.
Eg: 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
Eg: (-4)×1=1×(-4)=-4

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