Decimal Representation Of Rational Numbers    

Theorem:

Let be a rational number such that and prime factorization of q is of the form where m, n are non-negative integers then x has a decimal representation which terminates.

    For example :     

Theorem:

Let be a rational number such that and prime factorization of q is not of the form , where m, n are non-negative integers, then x has a decimal expansion which is non-terminating repeating.

    For example :     

Solved Examples Based on Decimal Representation Of Rational Numbers    

Question:

Without actually calculating, state whether the following rational numbers have a terminating or non-terminating repeating decimal expansion.

        (i)                 (ii)            (iii)        

Hint: If the denominator is of the form for some non negative integer m and n, then rational number has terminating decimal otherwise non terminating.

Solution:     

(i)    

            Since which is not of the form .

            \    It has non terminating decimal representation.

        (ii)    

            Since q = which is of the form .

            \    It has a terminating decimal representation.

        (iii)    

             Since is not of the form . It has a non-terminating decimal representation.

Question:

What can you say about the prime factorization of the denominators of the following rationales:

        (i)     36.12345            (ii)    

Solution:     

(i)    Since 36.12345 has terminating decimal expansion. So, its denominator is of the form where m, n are non-negative integers.

        (ii)    Since has non terminating repeating decimal expansion. So, its denominator has factors other than 2 or 5.