algebra multiplication

algebra multiplication

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algebra multiplication

Multiplication as Repeated Addition

We think of a multiplication statement like "2 x 3" as 
meaning "Add two threes together", or

3 + 3

and"4 x 9" as "add 4 nines together", or

9 + 9 + 9 + 9.

In general, a x b means to add b’s together such that the 
number of b’s is equal to a:

a x b = b + b + b + . . . + b (a times)


Multiplication with Signed Numbers

We can apply this same rule to make sense out of what we 
mean by a positive number times a negative number. 
For example,

3 x (-4) 

just means to take 3 of the number "negative four" and add 
them together:

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

Unfortunately, this scheme breaks down when we try to multiply 
a negative number times a number. It doesn’t make sense to try 
to write down a number a negative number of times. There are 
two ways to look at this problem.

One way is to use the fact that multiplication obeys the 
commutative law, which means that the order of multiplication 
does not matter: 

a x b = b x a. 

This lets us write a negative times a positive as a positive 
times a negative and proceed as before:

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

However, we are still in trouble when it comes to multiplying 
a negative times a negative. A better way to look at this 
problem is to demand that multiplication obey a consistent 
pattern. If we look at a multiplication table for positive 
numbers and then extend it to include negative numbers, the 
results in the table should continue to change in the same 
pattern.

For example, consider the following multiplication table:

    a  b    a x b 
    -------------
    3  2      6 
    2  2      4 
    1  2      2 
    0  2      0 
 

The numbers in the last column are decreasing by 2 each time, 
so if we let the values for a continue into the negative 
numbers we should keep decreasing the product by 2:

    a  b    a x b 
    ------------- 
    3  2      6 
    2  2      4 
    1  2      2 
    0  2      0 
   -1  2     -2 
   -2  2     -4 
   -3  2     -6 
 

We can make a bigger multiplication table that shows many 
different possibilities. By keeping the step sizes the same 
in each row and column, even as we extend into the negative 
numbers, we see that the following sign rules hold for 
multiplication:


Sign Rules for Multiplication


    (+)(+) = (+)
    (-)(-) = (+)
    (-)(+) = (-)
    (+)(-) = (-)