Multiplication for Negative Integers

 

                We defined the operations “+” and “-“ for negative numbers previously.  Furthermore, we would like to explore more on the other operations we used to have for “Natural Numbers” and extend the concept to non-positive numbers.  For example, what does it mean by  8 multiplied by (-1) ?  We still start from very basic concept.

 

 

Multiplication ( Review )

 

              For natural numbers,  we define the operation “  ” in the following way:

 

               3  1 = 3

               3  2 = 3 + 3 = 6

               3  3 = 3 + 3 + 3 = 9

               3  4 = 3 + 3 + 3 + 3 = 12

               …

 

              And we also have the distributive law:

 

               3  2 = 3  ( 1 + 1 ) =  ( 3  1 ) + ( 3  1 ) = 6

and          3  ( 8 – 2 ) = 3  8 – 3  2 = 24 – 6 = 18

            ( Recall that multiplication needs to be handled first if there is no bracket ).

                3   (8-2) = 3  6

 

           But how about the cases that are not natural numbers?  For example,

 

                    3  0 = ?

                    3  (-1) = ?

 

           What is the “reasonable” result for that instead of just memorizing the result?  For anything multiplied by 0, we can accept the outcome is 0 because

 

                   3  1 = 3   --- it means 3 only appears 1 times.

                   3  0 = 0   -- it means that 3 does not show up at all.

 

            With the result that anything multiplied by 0 is 0,  we can consider the situation for negative number.  We require that distributive law also holds for negative numbers. Thus, we have

 

                3  0 = 3  (1-1)                  

                          = 3  ( 1 + (-1) )

                          = 3  1 + 3  (-1)

 

 

 

 

           We know that    3  0 = 0 and 3  1 = 3.   So,

                   

                          0 = 3 + [ 3  (-1)]

 

           What is the outcome of   3  (-1) ?   Our clue tells us that  3 plus the result is equal to 0 .   So, the result is -3 .  In other words,

 

                           3  (-1) = -3

 

           Remember that the result is coming from two things:

 

1.      everything multiplied by 0 is 0.

2.      distributive law

 

 

           From that, we can extend to other cases.  For example,

 

                    3  (-4) = 3  ( (-1)+(-1)+(-1)+(-1)) = -3-3-3-3 = -12

 

          Sometimes, we will find out that the operation “  ”  is represented by using other symbols like “  ” or “*” .  That is because we try to avoid the confusion between it from x when we handle some equations. Furthermore, the operation symbol is omitted when those operands are alphabetical symbols.   For example,  when you find

 

c=ab

it usually means

c=a  b

 

          Another example is

c=3a

 

         It means c is equal to 3 times a.

 

          Those are just the symbols being used for the operation. Let’s back to the topic for the multiplication. With the previous result established, we consider the following problem:

(-3)  (-4) = ?

 

          Remember that the two basic principles we have

 

1.      everything multiplied by 0 is 0.

2.      distributive law

 

 

 

           0 = 3 + (-3)     :  we have this when we introduce addition for negative numbers.

           0 = 0  (-4)    :  we have it from our basic principles above.

 

           Then

                     0 = ( 3 + (-3))  (-4)     : replacing the 0 with (3+(-3)) from the result above

                     0 = (3  (-4)) + ( (-3)  (-4))      : distributive law

                     0 = (-12) + ((-3)  (-4))              : the previous result for  3  (-4)

 

           Thus, what is the result for  (-3)  (-4) ?  We have the clue that the result plus (-12) is equal to 0.   So, the answer is 12.   Now, we are confident to say

 

(-3)  (-4) = 12

 

           Those are just the logical steps the deduce the result.  When we work on some other problems,  we can just use the generalized principles from the result without going over all the steps over and over again.  In general, we have

 

           (positive number)  ( positive number) = (positive number)

           (positive number)  ( negative number) = (negative number)

           (negative number)  ( positive number ) = (negative number)

           (negative number)  ( negative number ) = (positive number)