### 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 \(2\).

\(10x\ =\ 2.363636...\)

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**\(234/990\).