Fraction – Negative Denominator or Numerator

 

 

 Motivation:         We have introduced the concept for Natural Numbers, zero,  negative numbers and the associated operations for addition, subtraction, and multiplication. And from the courses in elementary schools, we know the meaning of 

 

 

,  ,  , …

 

 

              But what is the meaning of the following representation :

 

 

,  - , , , …

                or

                                                       

 

 

              It would be difficult to use the concept in elementary school to explain the meaning of those strange symbols. Thus, it is necessary to use a systematic approach to explain it in broader sense.  We consider the following problem

 

                        7  @ = 1

 

             What is the actual value of  “@”  ?  We know that

 

                        7  0 = 0

                        7  1 = 7

 

            So, “@” is between 0 and 1.  However,  between 0 and 1, there is no natural number or integer;  “@” is not in the realm of natural numbers of integers.  The process to find the value of  “@” is known as “Division”.   To represent the solution of the problem, we introduce the following symbol

 

                                  

 

            And we use this symbol to represent the solution of the question above.  So,

                             7   = 1

 

 

              It is natural to consider the following problems

 

7   # = -1

(-7)  $ =1

 

             We can represent the answer “#” as

 

 

              and the answer “$”  as

 

             We might want to ask what is the difference between “#”  and “$” ?  They are the solutions to different problems.  But consider the following scenario:

 

1 =  7  

-1 =  7  #

 

                          So,

7  (  + # ) = 1 + (-1) = 0

 

                            It implies

 + #  = 0          

 

                          On the other hand,

1 = 7  

(-1)  1 = (-1)  7   = (-7)  

{ Aside: we can get  the fact that  is   here }.

                           Thus,

                                                          -1 = (-7)  

 

                        Let’s associate it with the problem :   1 =  (-7)  $

 

 

 

 

 

 

                   We have

                                    (-7)  (  + $ ) = (-1) + 1 = 0

                   That also implies

                                                + $ = 0

                   Previously, we have

                                                + #  = 0 

 

                   So, we can conclude that  “$” and “#” are the same.  In other words,

 

 and  are the same thing.   We just use  -  to represent it.

 

 

                   Let’s consider  the following question

0  $ = 1

 

                   Recall that  one of our basic principles is that anything multiplied by 0 is equal to 0.   So, there is no answer to the question.

 

 

                   How about the question

0  $ = 0

 

                   We know that

0  1 = 0

0  5 = 0

0  6 = 0

…

                   There exist infinitely many possible numbers for “$”.    We write down the definition for the fraction:

 

Definition :  if p and q are integers such that  p  0, then

                              is fraction.

 

 

            

 

 

Multiplication of Fraction ( Review )

 

                  Let’s start with the following problem with our previous approach

 

3   = ?

                 It is known that  is the solution to the following problem

5  @ = @  5 = 2

 

                 It is natural that  we will have the following

                                                  3  @  5 = 3  2 = 6

                                                 5  ( 3  @ ) = 6

 

                 So,    ( 3  @ ) is the solution to the problem

                                                 5  $ = 6

                                                 $ =

 

                  In other words,

                                     3  @ = 3   =

 

                 How about the problem

 

                                                       = ?

 

                We still start from the following

                                                  3  @ = 2

                                                  5  $ = 4

 

                 Then

                                             3  @  5  $ = 2  4

                                             3  5  @  $ = 8

                                             15  (@  $) = 8

 

                  So,

                                     @  $ =

                  In other words,

                                                    =

 

 

                    In general,  we will have the following property

 

Property   Let p, q, r, s be integers such that   p  0 and  r  0 .  Then

                               =

 

 

 Example 1:

                       

 

Example 2: