### Theory:

The decimal expansion of a rational number is will 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 $$x = 0.77777...$$

Note that the number $$x$$.  Only the one-digit $$7$$ repeats here.

Here we have to make multiplies of $$x$$ in such a way that the repeated decimals will be the same.

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 $$x = 0.2363636...$$

Note the number $$x$$ - two of digits$$36$$ repeats here.

Here we have to make multiplies of $$x$$ in such a way that the repeated decimals will be the same.

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

$$10x = 2.363636...$$

Multiply $$x$$ by $$1000$$.

$$1000x = 236.363636...$$

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 $$234/990$$.