Properties of Multiplication of Integers

Properties of Multiplication of Integers :
    (i)    Closure property for Multiplication : The product of two integers is always an integer. a x b is an integer, for all Integers a and b.
           Ex.    7 × 9 = 63    (7, 9 and 63 are all integers)
                  –7 × 9 = –63    (–7, 9 and –63 are all integers)
                   7 × –9 = –63    (7, –9 and –63 are all integers)
                –7 ×–9 = 63    (–7, –9 and 63 are all integers) 

(ii)    Commutative law for multiplication of Integers : For any two integers a and b, the commutative law holds true i.e., a × b = b ×a.
         Ex.    –2 × 3 = 3 × –2 = –6

(iii)    Associative law for multiplication of Integers : For any three integers a, b and c, the associative law holds true i.e. (a x b) x c = a × (b × c). If a = –2, b = –3 and c = 4, then (–2 × –3) × 4 = –2 × (–3 × 4) = 24.

(iv)    Distributive law of Multiplication over addition : For any three integers a, b and c, the distributive law holds true i.e., a × ( b + c) = a × b + a × c. 
        If a = 5, b = –3 & c = –7 then 5 × (–3 + –7) = 5 × –3 + 5 × –7 = –50

(v)    Existance of Multiplicative Identity : For every integer a, we have a × 1 = 1 x a = a. 
        Where ‘1’ is the multiplicative identity. 
        Ex.    –5 × 1 = 1 × –5 = –5

(vi)  Existance of multiplicative Inverse : For every non-zero integer x, there exists a multiplicative 
       

(vii)    Property of Zero: In general, for every integer a, we have a × 0 = 0 × a = 0