algebra division

algebra division

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algebra division

Division
There are two ways to think of division: as implying a 
related multiplication, or as multiplying by the reciprocal.


Division as Related Multiplication


The statement "12 ?3 = 4" is true only because 3 x 4 = 12. 
A division problem is really asking the question "What number 
can I multiply the divisor by to get the dividend?"; and so 
every division equation implies an equivalent multiplication 
equation. In general:

a ?b = c if and only if a = b x c

This also shows why you cannot divide by zero. If we asked 
"What is six divided by zero?" we would mean "What number 
times zero is equal to six?", but any number times zero 
gives zero, so there is no answer to this question.


Multiplicative Inverse (The Reciprocal)


For every real number a (except zero) there exists a real 
number denoted by 1/a such that

a(1/a) = 1 

* The number 1/a is called the reciprocal or multiplicative 
inverse of a. 
* Note that the reciprocal of 1/a is a. The reciprocal of the 
reciprocal gives you back what you started with. 

This allows us to define division as multiplication by 
the reciprocal:

a ?b = a x (1/b)

This is usually the most convenient way to think of 
division when you are doing algebra.


Notation for Division


Instead of using the symbol " ?" to represent division, 
we prefer to write it using the fraction notation:

             a
    a ?b = ---
             b

Sign Rules for Division


Because division can always be written as a multiplication 
by the reciprocal, it obeys the same sign rules as 
multiplication.

If a positive is divided by a negative, or a negative divided 
by a positive, the result is negative:

       a     -a      a
    - --- = ---- = ----
       b      b     -b

but if both numbers are the same sign, the result is positive:

      -a     a  
     ---- = ----
      -b     b