Rational Numbers

 

 

Definition ( Rational Number )

             If a number can be represented  as ,   where p, q  Z, and p  0,

             then we call it a rational number.

 

             Usually, we use Q to stand for the set of all rational numbers.

 

 

Example :   is not a rational number.

          Proof:

                           Assume   is a rational number.

                           Thus,  we can have

                                         =    ,  p, q  Z

                                        and  p, q are relatively prime ( gcd ( p, q) = 1 )

 

                             p  = q

 

                           We square both sides :

                                         2p²  = q²

 

                            p and q are integers      2 | q .

                            So, we can write  q = 2k   where k is an integer .

 

                                      2p² = (2k)² 

                                            2p² = 4k²

                                              p² = 2k²

                              and p, k are integers    2 | p

 

                          2| q, 2 | p  but   gcd( p, q ) =1  

                          

                               We get contradiction here .

                               Thus, the original statement does not hold.

 

                              So, we conclude that   is not a rational number.

 

 

                          

 

 

 

 

Theorem :  For any rational numbers  p, q ( p> q) ,  we can find

                   a rational number t such that 

 

                                                       q < t < p

 

           Proof:

 

                           We only need to set  t =  ( p + q ) .

 

                           Claim 1:   t=  ( p + q ) is a rational number .

                                              p , q are rational numbers

                                      So,  t is also a rational number by multiplying

                                               the denominators of  p and q by 2.

 

                           Claim 2:  t < p

                                         p – t = p -  ( p + q )

                                                 =  ( p + p ) -  ( p+ q)

                                                =  ( p –q ) > 0

                                      So, p > t.

 

                            Claim 3:  t > q

                                        t – q =  ( p + q ) – q

                                                =  ( p + q ) -  ( q + q )

                                               =  ( p – q ) > 0

                                     So, t > q

 

                            Thus, we prove that

                                            q < t < p ,   where t is also a rational number.

 

                         

                             This theorem states that you always can find a rational number between two rational numbers.  But it is still not “dense” enough.  Consider the following diagram:

 

                                           

 

                Is it possible that the rational numbers can cover all the space between 1 and 2 on that line segment ?

 

                The answer is no – because you know that    is between 1 and 2, but it is not a rational number.  You know the length between 1 and 2 is 1 and you might want to ask the length occupied by all the rational numbers between 1 and 2.  That kind of measure problem will be explored in more advanced math.