Properties of Integers with examples

Properties of  Integers

Addition Properties of Integers:

1. Closure Property:  The sum of two Integers is always an Integers.

Example: Let two Integers are -4 and -6 then 

                          (-4) + (-6) = (-10) is an Integer.

2. Commutative Property: Let a and b are two Integers, then

                                a + b = b + a 

Example: Let two integers are -5 and 6 then 

                 (-5) + 6 = 1 

                 6 + (-5) = 1  

then        (-5) + 6 = 6 + (-5) 

Hence commutative property is verified

3. Associative Property:  Let a, b and c are three integers, then

                                            a + ( b + c) = (a + b ) + c
Example: Let three integers are 3, 5 and (-6) then
                  3 + { 5 + (-6) } = 3 + {5 - 6 } = 3 + (-1) = 2
                  {3 + 5 } + (-6)  =  8 - 6  = 2
 then          3 + { 5 + (-6) } =  { 3 + 5 }+ (-6) 
Hence Associative property is verified

4. Existence of Additive Identity:  For any integers a ,  we have             

                               a + 0 = 0 +a = a 

0 is called the additive identity.
Example: 4 is any integer, 
                 4 + 0 = 0 + 4 = 4

 Multiplication Properties of Integers:

1. Closure Property:  The product of two integers is always an Integers.

Example: Let two Integers are -3 and 5, then 

                           -3 x 5 = -15 is an integers.

2. Commutative Property: Let a and b are two integers,

                                               a x b = b x a 

Example: Let two integers are -3 and -7 then 

                 (-3) x (-7) = 21 

                 (-7) x (-3) = 21  

then        (-3) x (-7) = (-7) x (-3) 

Hence commutative property is verified

3. Associative Property:  Let a, b and c are three integers, then

                                            a x ( b x c) = (a x b ) x c

Example: Let three integers are -3, 5 and 6, then

                  -3 x ( 5 x 6 ) = -3 x 30 = -90

                  (-3 x 5 ) x 6  = -15 x 6  = -90

 then          -3 x ( 5 x 6 ) =  (-3 x 5 ) x 6

Hence Associative property is verified

4. Existence of Multiplicative Identity: For any integer a, we have         

                            a x 1 = 1 x a = a 

0 is called the additive identity.

Example: 5 is any whole number, 

                 5 x 1 = 1 x 5 = 5

5. Distributive Property of Multiplication over Addition: Let a, b and c are three integers,

                 a x ( b + c ) = a x b + a x c

Example: Let 2, -3 and 4 are three integers,

                2 x ( -3 + 4 ) = 2 x 1 = 2

               2 x -3 + 2 x 4 = -6 + 8 = 2 

Hence Distributive property is verified.

6. Distributive Property of Multiplication over Subtraction: Let a, b and c are three integers,

                     a x ( b - c ) = a x b - a x c

Example: Let 2, -3 and 4 are three integers,

                2 x ( -3 - 4 ) = 2 x -7 = -14

               2 x -3 + 2 x -4 = -6 + (-8) = -14 

Hence Distributive property is verified.

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