Theorem 1.5 :  Let  x  be  a  rational  number  whose  decimal  expansion  terminates. 
Then  x  can  be  expressed  in  the form $p/q$
where  p  and  q  are  coprime,  and  the 
prime factorisation of q is of the form $2^n5^m$, where n, m are non-negative integers 



example 
(i) $3/8$ = $3/2^3$ 
  => ${3× 5^3}/{2^3× 5^3}$
  => ${375}/{10^3}$
  => $0.375$

 ii) $14588/625$ 
    => ${2^2  × 7 × 521 }/{5^4}$
	=>${2^6  × 7 × 521 }/{2^4×5^4}$
	=> ${233408}/{10^4}$
	=> $23.3408$

Theorem 1.6 : 
Let $x = p/q$  be a rational number, such that the prime factorisation of  $q$  is  of  the  form  $2^n5^m$, 
where  n,  m  are  non-negative  integers.
 Then  $x$  has  a decimal  expansion  which  terminates. 

Theorem 1.7 :  
Let $x = p/q$ be a rational number, such that the prime factorisation  of  $q$  is NOT of  the  form  $2^n5^m$, 
where  n,  m  are  non-negative  integers.
 Then, $x$ has a decimal  expansion  which  is  non-terminating  repeating  (recurring).