### Theory:

The decimal expansion of a rational number is terminating or non-terminating and recurring. Conversely, the decimal expansion of a number is terminating, or non-terminating recurring is a rational number.
Example:
1. Prove that $$0.77777... = 0.$$$\overline{7}$ is a rational number. That is, show that $$0.$$$\overline{7}$ can be expressed in $$p/q$$, where '$$p$$' and '$$q$$' are integers with $$q$$$\ne$$$0$$.

Solution:

Let us take the provided number as '$$x$$'.

That is $$x\ =\ 0.77777...$$

Now observe the number '$$x$$'. The only repeated digit is $$7$$.

Here we have to make multiples of '$$x$$' in such a way that its decimal part will be the same as the given number.

Let us multiply '$$x$$' by $$10$$.

$$10x\ =\ 7.77777...$$

Now subtract '$$x$$' from $$10x$$,

$$10x - x = 7.77777... - 0.777777...$$

$$9x = 7$$

$$x = 7/9$$

Therefore, the fractional form of the rational number $$0.$$$\overline{7}$ is $$7/9$$.

2. Prove that $$0 .2363636...\ =\ 0.2$$$\overline{36}$ is a rational number. That is, show that $$0.2$$$\overline{36}$ can be expressed in $$p/q$$, where '$$p$$' and '$$q$$' are integers with $$q$$$\ne$$$0$$.

Solution:

Let us take the provided number as '$$x$$'.

That is $$x = 0.2363636...$$

Now observe the number '$$x$$'. There are two repeated digits. That is $$36$$.

Here we have to make multiples of '$$x$$' in such a way that its decimal part will be the same as the given number.

Also, we have one non-repeating number $$2$$.

Let us multiply '$$x$$' by $$10$$ to get the repeated decimals separately.

$$10x\ =\ 2.363636...$$

Now multiply '$$x$$' by $$1000$$ to get the same decimal pars of $$10x$$,

$$1000x\ =\ 236.363636...$$

Subtract $$10x$$ from $$1000x$$.

$$1000x - 10x = 236.363636... - 2.363636...$$

$$990x\ =\ 234$$

$$x\ =\ 234/990$$

Therefore, the fractional form of the rational number $$0.2$$$\overline{36}$ is $$234/990$$.