Integers And Absolute Values

Integers And Absolute Values

Multiplication of Integers
Let a and b are any two integers then 
i)\( ( + a) \times ( + b) = + ab,\,\,\,ex:( + 7) \times ( + 6) = + 42 \) 
ii)\( ( + a) \times ( - b) = - ab,\,\,\,ex:( + 8) \times ( - 2) = - 16 \) 
iii)\( ( - a) \times ( - b) = + ab,\,\,\,ex:( - 11) \times ( - 3) = 33 \)
Note :1)\( + \times + = + \)     2)\( - \times + = - \)     3)\( - \times - = + \)     4)\( + \times - = - \)    

Properties of Multiplication of Integers
1. Closure property :- Observe the following 
1)\( 25 \times 75 = 1875 \)         2)\( 10 \times ( - 20) = - 200 \)         3)\( ( - 25) + ( - 25) = + 625 \) 
We can see that product of integers is always an integer.  So, the integers are closed under mulatiplication
Conclusion:- If a and b are any two integers then ‘ab’ is also an integer.
\( \therefore \)Multiplication is closed over integer set.
This is called closure property

2. Commutative property:-
Observe the following
1) i)\( 25 \times 75 = 1875 \)        ii)\( 75 \times 25 = 1875 \) 
2) i)\( - 10 \times - 20 = 200 \)     ii)\( - 20 \times - 10 = 200 \) 
    For any two integers a and b we can see that 
    This property is called commutative property for integer over multiplication for integer.

3. Distributive property :-
Observe the following
1)\( 2 \times (3 + 4) = (2 \times 3) + (2 \times 4) \) 
    \( 2 \times 7 = 6 + 8 \)    
        14 = 14
2)i)\( 2 \times [( - 3) + 4] = 2 \times ( + 1) = + 2 \)
   ii) \( [2 \times ( - 1)] + (2 \times 4) = - 6 + 8 = + 2 \)
    \( \therefore \,\,\,\,2 \times [( - 3) + 4] = [2 \times ( - 3)] + (2 \times 4) \)
  1^st  Integer (2^nd integer +3^rd integer) = (1^st integer \( \times \) 2^nd integer)+ (1^st integer \( \times \) 3^rd integer)
    \( \therefore \, \) We can say that distributive property hold good multiplication over addition