1. Under what conditions the decimal expansion of a rational number terminates?

We know that a rational number is a number of the form a =  where p, q are integers and
q 0. In the above two examples we have seen that the decimal expansion of a rational number is either terminating or non-terminating and repeating.
Then decimal expansion of a =  will terminate iff the prime factors of q are of the form 2ⁿ5^m where n and m are integers and if q is not of the form 2ⁿ5^m then the decimal expansion will be non-terminating and repeating.

2. Without performing the long division, state whether the following rational numbers will have a terminating or non-terminating repeating decimal expansion:
(i)  
(ii)  
(iii)
(iv)

(i)  
Here denominator q = 64
Prime factors of 64 = 2⁶
which is of the form 2ⁿ5^m with n = 6 and m = 0
Therefore, decimal expansion will terminate.

(ii)  
Here denominator = 80
Prime factors of 80 = 2⁴ ? 5
which is given of the form 2ⁿ5^m with n = 4 and m = 0.
Therefore, decimal expansion will terminate.

(iii)  =  =
Denominator of the above rational number is not of the form 2ⁿ5^m hence the number is repeating.

(iv)
Since, the prime factorisation of denominator is of form 2ⁿ5^m with n = 1, m = 2.
So, the decimal expansion will terminate.

3. Write the decimal expansion using prime factorisation:
(i)  
(ii)   
(iii)

(i)  
              
                = 2.1875

(ii)  
               
                 = 2.125

(iii)  =
                
                
                 = .654