Question

For the following decimal expansions, decide whether these are rational or not. If these are rational, then write the note on prime factors of its denominator.

(i) 0.120120012000120000…

(ii) 43.123456789

(iii) 27.14\(\over{28}\)57

Answer 1

(i) 0.120120012000120000…

The decimal expansion of this number is non-terminating and non-repeating 

So it cannot be written in the form of p/q

So, this number is not rational number

(ii) 43.123456789 = (3123456789)/(1000000000)

This number is of the form p/q

This is a rational number

q = 1000000000 = (10)⁹ 

= (2 x 5)⁹ = 2⁹ x 5⁹

So, prime factor of q is 2 and 5.

(iii) 27.14\(\over{28}\)57

= 27.142857 142857 142857…

The decimal expansion of this number is non-terminating and recurring.

So, it can be written in the form p/q

This is a rational number.

So, besides 2 and 5, one other prime factor of q other prime positive integers is possible.