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 \(0.77777... = 0.\)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 \(x = 0.77777...\)
 
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 \(0.\)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 \(x = 0.2363636...\)
 
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 \(0.2\)36¯ is \(234/990\).