3. Multiplication of Integers

Properties of Addition on Integers
 (i)    Closure property of Addition : Sum of two integers is always an integer. (In general, for any two integers a and b, a + b is an integer).
        Ex.    2 + 3 = 5         (2, 3 and 5 all are integers)
            –2 + 3 = 1         (–2, 3 and 1, all are integers)
            2 + (–3) = –1        (2, -3 and –1, all are integers)
            –2 + (–3) = –5        (–2, –3 and –5, all are integers)    

(ii)    Commutative law of Addition: According to this law, if ‘a’ and ‘b’ are two integers then a + b = b + a.
         Ex.    2 + (–3) = (–3) + 2 = –1

(iii)    Associative law of Addition: Accroding to this law, if ‘a’, ‘b’ and ‘c’ are integers, then (a + b) + c = a + (b + c)
         Ex.    (2 + 5) + (–7) = 2 + (5 + –7) = 0 

(iv)    Existance of Additive Identity: For any integer ‘a’, we have a + 0 = 0 + a = a.  Zero (0) is called the additive identity for integers.
         Ex.    –2 + 0 = 0 + (–2) = –2

Properties of Subtraction on Integers :
(i)    Closure property for Subtraction : If ‘a’, and ‘b’ are any two integers then (a – b) will always be an integer.
        Ex.    12 – 9 = 3 (Where 12, 9 and 3 are all integers)

(ii)    Commutative Property for subtraction : Subtraction of integers is not commutative i.e. if ‘a’ and ‘b’ are two integers then a – b≠ b – a. 
        Ex.    2 – 3 ¹ 3 – 2

(iii)    Associative property for Subtraction of Integers : Subtraction of integers is not associative i.e. (a – b) – c ≠ a – (b – c). For exmaple if 2, –3 and 5 are three integers then 
        [(2 – (–3)] –5 = 0 and 2 – [(–3) –5] = 10
        hence [2 – (–3)] –5 ¹ 2 –[(–3) –5]