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 \(0.77777... = 0.\)7¯ is a rational number. That is, show that \(0.\)7¯ can be expressed in \(p/q\), where '\(p\)' and '\(q\)' are integers with \(q\)\(0\).
 
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 \(0.\)7¯ is \(7/9\).
 
 
2. Prove that \(0 .2363636...\ =\ 0.2\)36¯ is a rational number. That is, show that \(0.2\)36¯ can be expressed in \(p/q\), where '\(p\)' and '\(q\)' are integers with \(q\)\(0\).
 
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 \(0.2\)36¯ is \(234/990\).