- Since this number has a terminating decimal expansion of the form and q is of the form . Hence, prime factors of q will either be 2 or 5 or both.
 The decimal expansion is neither terminating nor recurring. Hence, the given number is an irrational number.
Since the decimal expansion is non-terminating yet recurring.Hence the given number is of the form and q will not be of the form . Therefore, the prime factors of q will also have a factor other then 2 or 5.

Let is a rational number. Therefore. we can find two integers a, b (b  0)
such that   =
Let a and b have a common factor other than 1. Then we can divide
them by the common factor. and assume that a and b are co-prime.
a =
=
Therefore. is divisible by 5 and it can be said that a is divisible by 5. Let a
= 5k. where k is an integer
=
= This mans that is divisible by 5 and hence, b is divisible
by 5.
This implies that a and b have 5 as a common factor. And this is a
contradiction to the fact that a and b are co-prime.
Hence, cannot be expressed as  and it can be said that  is irrational.

- Since this number has a terminating decimal expansion of the form and q is of the form . Hence, prime factors of q will either be 2 or 5 or both.
 The decimal expansion is neither terminating nor recurring. Hence, the given number is an irrational number.
Since the decimal expansion is non-terminating yet recurring.Hence the given number is of the form and q will not be of the form . Therefore, the prime factors of q will also have a factor other then 2 or 5.