CLASS-7
PROPERTIES OF ADDITION OF INTEGERS

PROPERTIES OF ADDITION OF INTEGERS -

The properties of addition for integers are similar to the properties of addition for real numbers. Here are the main properties:

1. Closure Property:-

  • Definition: The sum of any two integers is always an integer.
  • Example: 3 + (−4) = −1 (which is an integer).

If ‘a’ and ‘b’ are any two integers,then a+b is also an integer.

Integers

2. Commutative Property:-

  • Definition: The order of addition does not change the sum.
  • Example.1)  5 + (−2) = −2 + 5 = 3.
  • Example.2) (-3) + 7 = 4 and 7 + (-3) = 4, → (-3) + 7 = 7 + (-3)
  • Example.3)  8 + (-11) = -3 and (-11) + 8 = -3, → 8 + (-11) = (-11) + 8
  • In general, we have:-

         If ‘a’ and ‘b’ are two integers, then a + b = b + a.

3. Associative Property:-

  • Definition: The way in which integers are grouped in an addition problem does not affect the sum.
  • Example.1) (2 + 3) + 4 = 2 + (3 + 4) = 9.
  • Example.2) (7 + (-13)) + 24 = (-6) + 24 = 18 and 7 + ((-13) + 24) = 7 + 11 = 18 → (7 + (-13)) + 24 = 7 + ((-13) + 24)
  • Example.3) ((-3) + 5) + (-11) = 2 + (-11) = -9 and (-3) + (5 + (-11)) = (-3) + (-6) = -9 → ((-3) + 5) + (-11) = (-3) + (5 + (-11))
  • In general we have:-

      If a, b and c are any three integers, then (a + b) + c = a + (b + c)


Ø  In view of the associative law of addition,to add any three integers we can add any two integers and then add the sum to the third integer.Hence,we can drop brackets and and write the sum of three integers a, b and c as a + b + c.

                 Thus, (a + b) + c = a + (b + c) = a + b + c.

Ø  In view of the commutative property and associative property of addition, we note that while adding any three or more integers we can group them or change their order in such a way that the calculations become easier.


4. Identity Property:-

  • Definition: The sum of any integer and 0 is the integer itself.
  • Example: 7 + 0 = 7.
  • Observe the following:-                              

                          (-5) + 0 = -5 = 0 + (-5)

                        (-73) + 0 = -73 = 0 + (-73)

  • In general we have:-

                             For every integer ‘a’,  a + 0 = a = 0 + a.


5. Additive Inverse:-

  • Definition: Every integer has an opposite (additive inverse) such that when added together, their sum is 0.
  • Example.1) The additive inverse of 6 is −6, because 6 + (−6) = 0.
  • Example.2) 7 + (-7) = 0 = (-7) + 7.
  • Example.3) (-23) + 23 = 0 = 23 + (-23).
  • In general, we have:-

   For every integer ‘a’,there exists integer -a,

                               such that a + (-a) = 0 = (-a) + a

 Thus,-a is the additive inverse of ‘a’ and ‘a’ is the additive inverse of-a.

                               Hence, -(-a) = a.